phiflow version updates, preparing for 2.2
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@@ -28,7 +28,7 @@
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"metadata": {},
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"outputs": [],
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"source": [
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"!pip install --upgrade --quiet phiflow\n",
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"!pip install --upgrade --quiet phiflow==2.2\n",
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"from phi.tf.flow import *\n",
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"\n",
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"N = 128\n",
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@@ -38,11 +38,11 @@
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"NU = 0.01/(N*np.pi)\n",
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"\n",
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"# allocate velocity grid\n",
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"velocity = CenteredGrid(0, extrapolation.PERIODIC, x=N, bounds=Box[-1:1])\n",
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"velocity = CenteredGrid(0, extrapolation.PERIODIC, x=N, bounds=Box(x=(-1,1)))\n",
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"\n",
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"# and a grid with the reference solution \n",
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"REFERENCE_DATA = math.tensor([0.008612174447657694, 0.02584669669548606, 0.043136357266407785, 0.060491074685516746, 0.07793926183951633, 0.0954779141740818, 0.11311894389663882, 0.1308497114054023, 0.14867023658641343, 0.1665634396808965, 0.18452263429574314, 0.20253084411376132, 0.22057828799835133, 0.23865132431365316, 0.25673879161339097, 0.27483167307082423, 0.2929182325574904, 0.3109944766354339, 0.3290477753208284, 0.34707880794585116, 0.36507311960102307, 0.38303584302507954, 0.40094962955534186, 0.4188235294008765, 0.4366357052408043, 0.45439856841363885, 0.4720845505219581, 0.4897081943759776, 0.5072391070000235, 0.5247011051514834, 0.542067187709797, 0.5593576751669057, 0.5765465453632126, 0.5936507311857876, 0.6106452944663003, 0.6275435911624945, 0.6443221318186165, 0.6609900633731869, 0.67752574922899, 0.6939334022562877, 0.7101938106059631, 0.7263049537163667, 0.7422506131457406, 0.7580207366534812, 0.7736033721649875, 0.7889776974379873, 0.8041371279965555, 0.8190465276590387, 0.8337064887158392, 0.8480617965162781, 0.8621229412131242, 0.8758057344502199, 0.8891341984763013, 0.9019806505391214, 0.9143881632159129, 0.9261597966464793, 0.9373647624856912, 0.9476871303793314, 0.9572273019669029, 0.9654367940878237, 0.9724097482283165, 0.9767381835635638, 0.9669484658390122, 0.659083299684951, -0.659083180712816, -0.9669485121167052, -0.9767382069792288, -0.9724097635533602, -0.9654367970450167, -0.9572273263645859, -0.9476871280825523, -0.9373647681120841, -0.9261598056102645, -0.9143881718456056, -0.9019807055316369, -0.8891341634240081, -0.8758057205293912, -0.8621229450911845, -0.8480618138204272, -0.833706571569058, -0.8190466131476127, -0.8041372124868691, -0.7889777195422356, -0.7736033858767385, -0.758020740007683, -0.7422507481169578, -0.7263049162371344, -0.7101938950789042, -0.6939334061553678, -0.677525822052029, -0.6609901538934517, -0.6443222327338847, -0.6275436932970322, -0.6106454472814152, -0.5936507836778451, -0.5765466491708988, -0.5593578078967361, -0.5420672759411125, -0.5247011730988912, -0.5072391580614087, -0.4897082914472909, -0.47208460952428394, -0.4543985995006753, -0.4366355580500639, -0.41882350871539187, -0.40094955631843376, -0.38303594105786365, -0.36507302109186685, -0.3470786936847069, -0.3290476440540586, -0.31099441589505206, -0.2929180880304103, -0.27483158663081614, -0.2567388003912687, -0.2386513127155433, -0.22057831776499126, -0.20253089403524566, -0.18452269630486776, -0.1665634500729787, -0.14867027528284874, -0.13084990929476334, -0.1131191325854089, -0.09547794429803691, -0.07793928430794522, -0.06049114408297565, -0.0431364527809777, -0.025846763281087953, -0.00861212501518312] , math.spatial('x'))\n",
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"SOLUTION_T16 = CenteredGrid( REFERENCE_DATA, extrapolation.PERIODIC, x=N, bounds=Box[-1:1])"
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"SOLUTION_T16 = CenteredGrid( REFERENCE_DATA, extrapolation.PERIODIC, x=N, bounds=Box(x=(-1,1)))"
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]
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},
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{
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@@ -136,7 +136,7 @@
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"outputs": [
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{
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"data": {
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"image/png": 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truncated
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"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
@@ -237,7 +237,7 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAZAAAAEWCAYAAABIVsEJAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjMuMywgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy/Il7ecAAAACXBIWXMAAAsTAAALEwEAmpwYAABLJUlEQVR4nO3dd3gVVfrA8e+bntBLpEhHuhCQIqAURRQBy9rQdRVXWdSVJoqiWFFZFKWpq6Iirr3sbxURBUVQpCggTXqR3kInpCfv74+ZxEsKJCHJ3Ju8n+e5D/fOnJl5M/dy33vOmTlHVBVjjDEmv4K8DsAYY0xgsgRijDGmQCyBGGOMKRBLIMYYYwrEEogxxpgCsQRijDGmQCyBGETkIhHZJCJxInKt1/EUFRFZIyLdC7jtNyLSv3AjOruYjPGaJZBSRETmicgREQnPsmo08IqqllXVL0REReS8Qj72NhFJcJNUnIjMPkP5ziLyg4icEJFjIvKViDTPx/GmicizvstUtYWqzitI/Kp6paq+W5BtiyqmMxyrhYjMFpHDInJURJaJSG93XXcR2VXYxzxb7udzwGnW13M/myEF3P9fRWS7iJwUkS9EpPJpyqpbLuPz+lZBjlnSWQIpJUSkHtAFUODqLKvrAmsK6Tin+899lZukyqrq5afZRydgNvAlUBOoD6wEFohIg8KIsxT4CvgOqA6cAwwBjnsakYdEpAXwBnAbUA2IB/59hs1ifD6vuSa2Uk1V7VEKHsATwAJgPDDDZ/kWIB1IAOKARThJ5qT7up9bri+wAjgKLARa+exjG/AwsApIAkJyOP424LI8xjof+HcOy78B/uM+7w7sAh4FDrr7v9VdNxBIAZLdv+GrrDEATwGfAe8DJ4DVQGPgEeAAsBO43OfY84AB7vOV7n4zHgp0d9d9BuwDjgE/AS3yEVM4MBHY4z4mAuFZ/t4H3Pj2An/P5fxVdWOqmMO6Mu57ne4Tf02cH5Mj3c/DIeBToLK7TT13fwPduPYCD/rsswOwFCdB7QfG5xJXJWAGEAsccZ/Xctc9B6QBiW5Mr+Sw/Q43joy4O+Xj8z8G+NDndUP3vSiXS3kFzvP6/62/PzwPwB7F9EbDZuCfQFv3i6yaz7rMLzH39Sn/eYA27pfWhUAw0N/dJtxn+xVAbSAyl+Nvc79cYnFqFzG5lItyv0guyWHd34G97vPuQCpOQgwHuuEkvSbu+mnAsznE4JtAEoErgBDgP8AfwCggFPgH8IfPtvNwE0iWfQ4E1gPl3dd3AuX4Mxms8Cl7pphGA4txagzROIn6mSx/72g3vt44v6Ir5RCTAJtwvqCv9X2vffa1K8uyoe6xa7mxvwF85K6r534mPsJJQC3d9zEj7kXAbe7zskDHXN7bKsD17ntcDifZfnGmc+yzPiOOEJ9lF+P8qMntcbFb7kvg4Sz7iwPa5nIsxUmW+4D/A+p5/X/YHx/WhFUKiMjFOM1Un6rqMpxfmX/Nxy4GAm+o6i+qmqZOX0AS0NGnzGRV3amqCbns41acL4C6wFxglohUzKFcZZxfw3tzWLcX59e1r8dVNUlVfwS+Bm7K498EMF9VZ6lqKs6XWTQwVlVTgI+BernECGSe12eBq1X1OICqTlXVE6qahJOkYkSkQh7juRUYraoHVDUWeBqnySVDirs+RVVn4nwBNsm6E3W+AS/BSU4vAXtF5CcRaXSaY98DjFLVXT6x35ClSfJpVT2pqquBd4BbfOI6T0Sqqmqcqi7O6QCqekhV/6uq8ap6AqfW0e30p+T0VPVnVa14msfPbtGyOLVCX8dwEllOuuF8XpviJJIZBe17KcksgZQO/YHZqnrQff2huyyv6gIPuJ2xR0XkKE5to6ZPmZ2n24GqLlDVBPfL4184vw675FD0CE7zSo0c1tXAaa7KLKuqJ31eb88S05ns93meABxU1TSf1+B88WQjIrVxmnn6q+pGd1mwiIwVkS0ichznCxyyJ73c1MT5GzJk/XsOuckuQ3xu8bmJYJCqNsR5/07i1LJyUxf4n8/7uw6nJljNp4zve+wb2104zX/rRWSJiPTN6QAiEiUib7gd2cdxmvgqikjwaeIqLHFA+SzLyuM0X2ajqj+parKqHsWpndUHmhVphAHIEkgJJyKROL/Ku4nIPhHZB9yP88s4Jo+72Qk8l+WXXZSqfuRTJr/DOitOU8upC52EsAi4MYdtbgLm+LyuJCJlfF7Xwfm1WJB48sw9p18AE1X1G59VfwWuAS4DKuD8goU//84zxbQH54s8g+/fU2CquhN4FTj/NHHsBK7M8h5HqOpunzK1c4pNVTep6i04TW/PA59neV8yPIBTY7pQVcsDXd3leT0/2daLSBefK6VyemT8SFkDxPhs1wCnqW7jGY7pe+xsn9fSzhJIyXctzi/J5kBr99EMp6P69ly22Q/4Xu30JnCPiFwojjIi0kdEcqv+n0JE6rj3moSJSISIjMD5Vb4gl01GAv1FZIiIlBORSu7lr51wmnV8Pe3utwtOR/9nufwNhWkqsF5VX8iyvBxO094hnHb+MVnWnymmj4DHRCRaRKriXPjwfn6Dc8/X0yJynogEufu6E6ePIyOOKlma1l4HnhORuu4+okXkmiy7ftytRbTA6Y/6xC37NxGJVtV0nJolOLXIrMrh1OyOupfQPpll/ZnOT6y738wyqjpf/7xSKqfHfLfoB8BVbsIpg9OX9H9uU9op3EugW7s1yrI4zYC7cWplxoclkJKvP/COqu5Q1X0ZD+AV4NZc2nWfAt51mzNuUtWlOJ3Kr+A0MW0G7shHDOWA19xtdwO9cH7tHsqpsNtufQVwHU6/x3acjvyLVXWTT9F97j734HxB3KOq6911bwPN3b/hi3zEmhc3A3/J4Zfuf9xYdwNr+fMLO8OZYnoW52qmVThXhf3mLsuvZJzaz/c4V0b9jpPY7gBwz9FHwFY3lprAJGA6MFtETrixX5hlvz/ivPdzgBdVNeNenl7AGhGJc/dzcy59YROBSJxmyMXAt1nWT8LpdzkiIpOzbqyq8Tj9JgvcuDtmLZMbVV2D08/zAc4FIeVwLioBMm8UfdR9WQ0nOR4HtuKcy75u35jxIU5/mzGBRZy7t99X1Voeh1LiufcQ/QGEZumDMaWc1UCMMcYUiCUQY4wxBWJNWMYYYwrEaiDGGGMKpFTdWVm1alWtV6+e12EYY0xAWbZs2UFVjc66vFQlkHr16rF06VKvwzDGmIAiIttzWm5NWMYYYwrEEogxxpgCsQRijDGmQDztAxGRqTjjFx1Q1fNzWC84wxtkzH1wh6r+5q7rDzzmFn1Wz3K6UWMMpKSksGvXLhITE70OxXggIiKCWrVqERoamqfyXneiT8MZXym3YaavBBq5jwtxxlO60GcgtnY4o2QuE5HpqnqkyCM2pgTbtWsX5cqVo169eji/30xpoaocOnSIXbt2Ub9+/Txt42kTlqr+BBw+TZFrcKYwVXeSmooiUgNnoL3vVPWwmzS+wxnQzRhzFhITE6lSpYolj1JIRKhSpUq+ap/+3gdyLqdOYrPLXZbb8mxEZKCILBWRpbGxsUUWqDElhSWP0iu/773XTVhFTlWnAFMA2rVrZ+O2mGKVnJbMst3LWL1vNetj15OUmsTVza+mR8MehASV+P9+poTz90/wbk6dBa2Wu2w30D3L8nnFFpUxp3Ey+SRztszh243f8vO2nzmZcvKU9d9u+pbqZaszrvc4Otfp7FGU/is4OJiWLVuSmppK/fr1ee+996hYsWKu5WNjY+nbty/JyclMnjyZLl1yminZFAV/TyDTgUEi8jFOJ/oxVd0rIrOAMSJSyS13OfCIV0Eao6os2bWEj1d9zOxNs0lI/XM+pcZVG3NhrQtpek5T4pPj+WDlB2w7so1PVn1iCSQHkZGRrFixAoD+/fvz6quvMmrUqFzLz5kzh5YtW/LWW2/l+RhpaWkEBxfHVOwlm9eX8X6EU5OoKiK7cK6sCgVQ1deBmTiX8G7GuYz37+66wyLyDLDE3dVoVT1dZ7wxRSIpNYn/rf0f05ZNY9OhPydLbFOzDX2b9qVnw56cW+HU7rnzqpzH3//7d44k2EWDZ9KpUydWrVoFwJYtW7jvvvuIjY0lKiqKN998k8TERB566CESEhJYunQpixYtYv78+Tz55JMkJSXRsGFD3nnnHcqWLUu9evXo168f3333HQ899BCVK1fOtVz//v356quvSElJ4bPPPqNp06bExcUxePBgli5diojw5JNPcv311zN79uwc91MaeJpAVPWWM6xX4L5c1k3FmZvamGJ3OP4wH6z8gPeWv8eheGdm3ugy0dzY8kZuOv8malesneu2FSMrAvh9Amn4YsMi2e+WB7fkqVxaWhpz5szhrrvuAmDgwIG8/vrrNGrUiF9++YV//vOf/PDDD4wePZqlS5fyyiuvcPDgQZ599lm+//57ypQpw/PPP8/48eN54oknAKhSpQq//fYbBw8e5Lrrrsu1XNWqVfntt9/497//zYsvvshbb73FM888Q4UKFVi9ejUAR44cOePxSjp/b8Iyxq/sO7GP1399nc9Wf0ZiqnO5Y4tqLRjQbgBXNr6S0OAz34BVObIy4P8JxCsJCQm0bt2a3bt306xZM3r27ElcXBwLFy7kxhtvzCyXlJSUbdvFixezdu1aLrroIgCSk5Pp1KlT5vp+/frlqdx1110HQNu2bfm///s/AL7//ns+/vjjzDKVKlVixowZp91PSWcJxJg8iD0Zy+u/vM6HKz8kOS0ZgG71u/GP9v+gY+2O+br8sVKk03Xn7wkkrzWFwpbRBxIfH88VV1zBq6++yh133EHFihUz+0Zyo6r07NmTjz76KMf1ZcqUyVO58PBwwOnQT03NfRr4M+2npPP3+0CM8dSh+EP8a96/6P5md6b9No3ktGR6N+7NzP4zmXr9VDrV6ZTva+ejQqMICw4jMTWRhJSEM29QSkVFRTF58mReeukloqKLine truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
@@ -342,7 +342,7 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
@@ -376,7 +376,7 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": "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 truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
@@ -414,7 +414,7 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 1152x360 with 2 Axes>"
|
||||
]
|
||||
@@ -475,7 +475,7 @@
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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|
||||
"text/plain": [
|
||||
"<Figure size 1152x360 with 2 Axes>"
|
||||
]
|
||||
@@ -527,7 +527,7 @@
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAp0AAAEyCAYAAAC4d2uYAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjMuMywgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy/Il7ecAAAACXBIWXMAAAsTAAALEwEAmpwYAABLDElEQVR4nO29fbBlZ3Xe+ay97+3WB1hCEshCEhapyONgJ4aEEp7CM0VMSDBDWePEUYQnMTiK5alCZXvKU2NBqmKPKaqUTILDDC7iDqiMUjaCsk3QOKpgwCZOagyWIGT4chIFQ5BGSEhqfaLue8/ea/7Y50oX0a3uPs86d639nudX1dX3nnv2Pu/ZH+/77Odda73m7hBCCCGEEGKddNkNEEIIIYQQ7SPRKYQQQggh1o5EpxBCCCGEWDsSnUIIIYQQYu1IdAohhBBCiLUj0SmEEEIIIdbO2kSnmb3WzP6jmd1lZjeu63OEEEIIIUR9bB11Os2sB/CfALwGwN0A7gDwBnf/YviHCSGEEEKI8qzL6bwKwF3u/mV33wFwK4Cr1/RZQgghhBCiOFtr2u+lAL627/e7AbziZG++4PC2X3rO4TU1ZUZocSg4LLsJ+eg6AKBrQexD94TuhyC+9NijD7j787Pbcbr8tdf+BX/wgcfPeLtPf/pPP+Lur11DkyjWJTpPiZldD+B6AHjh2Yfx2//dX8xqighiHGt0iu412rEqc28/UOM7jAXawFLhOLK08B0icFfebgVe9rHf+2p2G86EBx94HJ+6821nvN2W/e2L1tAcmnWJznsAXL7v98uWrz2Fux8BcAQAvve87/Dd3dWbYpb/GFyhDS2gAaqdY9CC6GNp4Vy28B2EmCsOxzgO2c0IY12i8w4AV5rZizGJzWsB/PhJ3+3AYuipD8wWfdmfDwBdgTYIHom1GCSWJnQcYpBTKXJwuC+yGxHGWkSnuy/M7AYAHwHQA7jZ3b9w8vcbdne3V/48VvCZObqO3cdIbR+BS3SWEP8sEgntoHNZAwlGMVsccJfTeUrc/XYAt5/Oe0cYdobVm9KTgs/M0Y2s6OQ6tQixxArnCLLFtwZ5IdpDolFsKg7HKKczGDdqen2wjp5a7jtOLHUBYov9DuuouXomTMKZC5OoIJyzyRbuoh0k1oSYO+uZXjezmwG8HsD97v59J/i7AXgngNcB+CaAN7n7Z9jPLSE6I5xO1ilk4+j6jtu+M6dFI3sMeNFrdBuGgTuOLUyvd52EghBCCGCNMZ2/DuBdAG45yd9/GMCVy3+vAPBuPEvpy9OlhOicEolWH2i948VONhHxmKxbG+GUVhC+DBWuo1FGJwC53kIIAXf4GC863f0PzeyKZ3nL1QBu8UkYfNLMzjezS9z9XuZzS4jO0Tmnc8uH9MztCLGT/R0iBCN7HLLdXriln4cWiBDvVeq+zhkJdyEaICem80SL/FwKYP6i0wHsjnmJOJ05wCaHcaGM6NzTs89ZoeDmAcKV2jxk5ZJso7HC5Dp9LRRI6KrgWmfTgnCXcBabzcrT6xeZ2Z37fj+yrI+eShHRadgdyQQUYoAxAz/Sk6K17xzGKiZSLbHT80NATCcrvCMcY9q1Js9jhOjNPgYRSPgKIEY4S7iK+eLAuLvKhg+4+8uJDz7lIj+rUEJ0woEF6XQyg2wXYY+xonUcMSWLZbaB29zMs5uADpP4XZWQ0lVya0NCFFqIz80WvhWOQQUkXMVccU8rDn8bgBvM7FZMCUSPsPGcQBHROcLo6XWmVucA3qHrWLnVAUYn8vBtoDYPCBFg3VZabAXEdNLCOUKnBIyPzD0RsaqS3FqeiGOQ/R2qwApXidYYNq+knANrSCQys/cDeBWmafi7AfwigG0AcPd/hqnO+usA3IWpZNJPRnxuCdHpAHaJzrFzw0gopg7ALnkd00Jl7Oki93zaM1fvdIChJyXXwD58kKIViHFbMz8f4IWrgU8KY6kQZsAit7bGdxBitvh6RKe7v+EUf3cAb47+3BKiE86JjV0APSWW+I5xd+ypaXozB8j6jBHF5bmpaYCVXBGilXetObJFa0gbAvQmG9saIXYy17GvEGIA5Iu+Fr5DBHJKxcpoRaJYYpzO1WHjAAGgGw0DVt9H3zntVEZM8TOYO78PMq61c26QM/N00Rji8CW3oTOHE/cDUEO0MoxFym9lH4cIWvgOQqyCwWFrcDqzKCI6jRJ9u2NHOZ1uTseU0ovIjCPvdLIxoaRL2Bkw0IqJFc5kA5wPEcgWrREOX7ZoBWJCBBgiVtiiz4PEkhCbzZqm17MoITqn6XXC6SRHyIhSP6xTCnTpTqf1I0B8D3dPXxVpGLm4VAAYkkMEIkRrtrMTIdjoNgQ4rdkOWwsJWdnXAdDGd9D0vGiBEqLTwYm2iHuJTWBZhHRKpGgkm2AjuYMOAHkcWYutx0jHpbIuHyt2WEbkO5URq1NFtIHBYbOf4gfy3dYWYjorPESxjKNJeM4SOZ3hTKJz9e2nvoDr2Finko4Jpbae6Mn+ZPCOSoYaAtSO0YXZ2fPgGFmHLCAulSV7il9xqRMtiFYxoXMhcnCYEoniWRA3dO/AgujcIxKJWKe068aAZCjWKR0pwdUhIBmKDORzUvCNzi8WQDt87BoBRURr9iDLulMxU9vc9opLnZDgi0FT9DPEAYzsOt11KCE6HVwt68E5h8vBD9TsrTi4Ucu3D5iy+BnMjRJcIwxdsuPMJkNNjjknudjxjRVsIyxEeDK0MsWfLVZamOKvIForkH0eWsFJc2V+KHt9LTAd0wLAFjMtDH5qmnFqo+jJNrhzZZ+mwZF1W7kTYQYqGaqD024pmww1iXcO1i3tA4TO3Kf4q8SlbvoUf0TpqezvIMTquJzOaNy5mM7OONHXGff5QH5Mp0W0gdy+B+DJAxwt+MzoZCgjbcaIuNLsZCgrINiYMmpAjTqbIYIrWbRGkO2WVkgkqtAGTc8noJJJ68CoAWp0rlMZnXeG2AE24jmGFo0hbm1uBj5GzpXo4JPwJGBja4GRitfobP7JUBGDa0TWdLZbG1JztYBYYsgWWntU+B4V2iAOHpPTWQt38M4QPb3O3sz8AMmKRr7WKN+p0fXtjYtnHNz4WqPkuWwlGYppQwnBGODWsk9RNRKBqM2VDCWeQk7pCrim19cCM0AYeNHIlqjkpxIdpNagRSOTyARgiqVMXtmJzsA358s20TqFFwns9RyRDFUhC5/NXmfFRoTYmbu71UoyVAtxpRXawLCptUbldBbD3TDScU8cbEyoGahRujPQx4At+4SO9Zt5d2ookIHPxpWyTukI3p1qJRkqO65UyVA1kqEiyHZL5y4YxarI6QzHwTkznXE3ZETcFDtADg7KXhqdd8i2AjpF5tboEJBIBO5cdO5YVMjAJ+icFwoViuyziUARg3T29DqLkqGe2gFFC4ItIkyhgvBlp+jnhrnL6awGWxPQ3eis64gVJOlYwICyT6wzw9Ya5ae2nVs/HkavnU4nlY1kYhwZ1wq0E1fK0HmBGpXU1ss2kNsrGaqduNJs0VghA3+WSHS2B9sx80nXRqlGC4jjY51KgE+H4ksmsULFaZcvIqmNuh75cqnNxJU2UWR/5oN0DcFGbQ4LCNXIPwY1rqPs4zg75HTGY+A7BXYJSRa2QzA4Na/audPODr1edkDGMZvMxLq1k+PMMdAWW8AKWYorBciwGyC/yD6Q75ZWiCvNFhpKhprI/g4RbZglEp0iGjauFWZ0MlM3ckJhcgk5ItawZ25PWniDr3FpDtoyzs7ArxBXSjulAWKJvyN4sqfXK4hWJUNNZLulml5fBYeNESNTDcqIzuzVVPnBJSDbl0wkisgWZoTvVkC36uzTPPn5XcDyjRHXEr8iUW4GfoW40pAMfPp6nH+R/WyUDPXUDmgqXA8V3NJZ4ZDTGY9TF9JG2u3PwMEPcHRhdvAWHV3gfuzIJBzemwpx+JhqDgGrKtExnQXaUCEDP+JBcPZF9ukW5LehhWQoIKKeNEeFZCiRSwnRaeA6BbY+ZQS8w8aLRtZnpCtRuIF9HKfXj2cHBzd05IpEEfVOufshIgknIKaTjYFrYGWniLJR2W4pi5KhJiokQ2XTgnA/eFSnMx7jYsjo1Yi4zcNgnEbjzDH684Gl2CHbwJ6LiE6NFb7s4NSxix2QBfIB0C5lSExnclxrRAZ+dq1RQMlQQI240uxp4ZD44uRwjwqhFhmYV1EpPDVEJ8i6hNklagrgAckn/HHkg+1Z4TuA7NicnZSNmcJiB9mBLnBP3hGmDHwgJq6UoZXpzOzp9YjxoUQbsoVvgHBnH2Bmh9Zej8cA9NR1FJAw0IBbGlK2idoBX1aEru1ok1O48uebpw8OBu5csvUpgYj4Xt72VgZ+zPR+9nKkLSRDVci6riDYsqf4K5yHFJS9HsskOql6QZRY6c0xzDxuKgK6wH1ABn3EeWCeCSOeoS09ptMp4Q3wTmmPURn4iIkrZVAy1LIN5PZVyjYxVBBsEWWnKjjvB4q7ROc6YESnO1lfEgHLYCZPK4fsI+BeTk+GCkigyS775AExnSz04OQdvZxodvZ7hQx8etGMAslQHfLdUpYWptebEM4F6t5moBWJgjEDtriEY+pm6JEfJ2IWIzwZIpJwItagZxhAPs07Py3LhzmAEo5TIWuOiOVE4blxpS1k4PODfH4yFJC/MlSEaE13CUmnMmKMyxbOQH5c6sEjpzMcdno9Ims6QnPS09MF3FKGiMzCCKeUmVoepkZQdAGFCbmFAvhptAV5Q2yBL2XGVhEAuJqtFTLwI2LN554MBQTU+qxQRYDcvokM+gKlp2aHQ6IzHudEZ8cljwC86ORL/fBPcOwgTS91bfnJTB3ImM4CZZ9Yp3NqBL+cKPXx4IUC7daCK381mmGLPJsRcaUMLSRDLRtBwYcI5McSVhCtFdqQLZwPHjmd4fCJRJzQAPjpG3bdcxiXDOXgXRVWdHrA1HS2WxxR9mkIqO9CpdUVyMAHuZxoRAYLine truncated
|
||||
"text/plain": [
|
||||
"<Figure size 1152x360 with 2 Axes>"
|
||||
]
|
||||
|
||||
+284
-287
@@ -2,6 +2,9 @@
|
||||
"cells": [
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "o4JZ84moBKMr"
|
||||
},
|
||||
"source": [
|
||||
"# Differentiable Fluid Simulations\n",
|
||||
"\n",
|
||||
@@ -28,13 +31,13 @@
|
||||
" \\frac{\\partial s}{\\partial{t}} + \\mathbf{u} \\cdot \\nabla s &= 0 \n",
|
||||
"\\end{aligned}$$\n",
|
||||
"\n"
|
||||
],
|
||||
"metadata": {
|
||||
"id": "o4JZ84moBKMr"
|
||||
}
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "pwmMK-LW_trZ"
|
||||
},
|
||||
"source": [
|
||||
"## Formulation\n",
|
||||
"\n",
|
||||
@@ -57,124 +60,141 @@
|
||||
"Hence, the optimization can only change $\\mathbf{u}_0$ to align $y_{t_e,i}$ with the references $y^*_{t_e,i}$ as closely as possible.\n",
|
||||
"\n",
|
||||
"\n"
|
||||
],
|
||||
"metadata": {
|
||||
"id": "pwmMK-LW_trZ"
|
||||
}
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "omVCYK5f_tra"
|
||||
},
|
||||
"source": [
|
||||
"## Starting the Implementation\n",
|
||||
"\n",
|
||||
"First, let's get the loading of python modules out of the way. By importing `phi.torch.flow`, we get fluid simulation functions that work within pytorch graphs and can provide gradients (`phi.tf.flow` would be the alternative for tensorflow).\n",
|
||||
"\n"
|
||||
],
|
||||
"metadata": {
|
||||
"id": "omVCYK5f_tra"
|
||||
}
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 1,
|
||||
"source": [
|
||||
"!pip install --upgrade --quiet phiflow\n",
|
||||
"from phi.torch.flow import * \n",
|
||||
"import pylab # for visualizations later on\n"
|
||||
],
|
||||
"outputs": [
|
||||
{
|
||||
"output_type": "stream",
|
||||
"name": "stdout",
|
||||
"text": []
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"colab": {
|
||||
"base_uri": "https://localhost:8080/"
|
||||
},
|
||||
"id": "da1uZcDXdVcF",
|
||||
"outputId": "66973d88-ec2c-4137-ed5b-1696848be36f"
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": []
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"!pip install --upgrade --quiet phiflow==2.2\n",
|
||||
"from phi.torch.flow import * \n",
|
||||
"import pylab # for visualizations later on"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "rdSTbMoaS0Uz"
|
||||
},
|
||||
"source": [
|
||||
"## Batched simulations\n",
|
||||
"\n",
|
||||
"Now we can set up the simulation, which will work in line with the previous \"regular\" simulation example from the {doc}`overview-ns-forw`. However, now we'll directly include an additional dimension, similar to a mini-batch used for NN training. For this, we'll introduce a named dimension called `inflow_loc`. This dimension will exist \"above\" the previous spatial dimensions `y`, `x` and the channel dimensions `vector`. As indicated by the name `inflow_loc`, the main differences for this dimension will lie in different locations of the inflow, in order to obtain different flow simulations. The named dimensions in phiflow make it very convenient to broadcast information across matching dimensions in different tensors.\n",
|
||||
"\n",
|
||||
"The `Domain` object is allocated just like before, but the `INFLOW_LOCATION` tensor now receives a string\n",
|
||||
"`'inflow_loc,vector'` that indicates the names of the two dimensions. This leads to the creation of an `inflow_loc` dimensions in addition to the two spatial dimensions (the `vector` part).\n"
|
||||
],
|
||||
"metadata": {
|
||||
"id": "rdSTbMoaS0Uz"
|
||||
}
|
||||
"Now we can set up the simulation, which will work in line with the previous \"regular\" simulation example from the {doc}`overview-ns-forw`. However, now we'll directly include an additional dimension, similar to a mini-batch used for NN training. For this, we'll introduce a named dimension called `inflow_loc`. This dimension will exist \"above\" the previous spatial dimensions `y`, `x` which are declared as dimensions for the `vector` channel. As indicated by the name `inflow_loc`, the main differences for this dimension will lie in different locations of the inflow, in order to obtain different flow simulations. The named dimensions in phiflow make it very convenient to broadcast information across matching dimensions in different tensors.\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 2,
|
||||
"source": [
|
||||
"# closed domain\n",
|
||||
"INFLOW_LOCATION = tensor([(12, 4), (13, 6), (14, 5), (16, 5)], batch('inflow_loc'), channel('vector'))\n",
|
||||
"INFLOW = (1./3.) * CenteredGrid(Sphere(center=INFLOW_LOCATION, radius=3), extrapolation.BOUNDARY, x=32, y=40, bounds=Box[0:32, 0:40])\n",
|
||||
"BND = extrapolation.ZERO # closed, boundary conditions for velocity grid below\n",
|
||||
"\n",
|
||||
"# uncomment this for a slightly different open domain case\n",
|
||||
"#INFLOW_LOCATION = tensor([(11, 6), (12, 4), (14, 5), (16, 5)], batch('inflow_loc'), channel('vector'))\n",
|
||||
"#INFLOW = (1./4.) * CenteredGrid(Sphere(center=INFLOW_LOCATION, radius=3), extrapolation.BOUNDARY, x=32, y=40, bounds=Box[0:32, 0:40])\n",
|
||||
"#BND = extrapolation.BOUNDARY # open boundaries \n",
|
||||
"\n",
|
||||
"INFLOW.shape"
|
||||
],
|
||||
"outputs": [
|
||||
{
|
||||
"output_type": "execute_result",
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"(inflow_locᵇ=4, xˢ=32, yˢ=40)"
|
||||
]
|
||||
},
|
||||
"metadata": {
|
||||
"tags": []
|
||||
},
|
||||
"execution_count": 2
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"colab": {
|
||||
"base_uri": "https://localhost:8080/"
|
||||
},
|
||||
"id": "mphMP0sYIOz-",
|
||||
"outputId": "0c3cea02-a1b2-4d88-e756-afbaef0544b0"
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"(inflow_locᵇ=4, xˢ=32, yˢ=40)"
|
||||
]
|
||||
},
|
||||
"execution_count": 2,
|
||||
"metadata": {
|
||||
"tags": []
|
||||
},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"# closed domain\n",
|
||||
"INFLOW_LOCATION = tensor([(12, 4), (13, 6), (14, 5), (16, 5)], batch('inflow_loc'), channel(vector=\"x,y\"))\n",
|
||||
"INFLOW = (1./3.) * CenteredGrid(Sphere(center=INFLOW_LOCATION, radius=3), extrapolation.BOUNDARY, x=32, y=40, bounds=Box(x=(0,32), y=(0,40)))\n",
|
||||
"BND = extrapolation.ZERO # closed, boundary conditions for velocity grid below\n",
|
||||
"\n",
|
||||
"# uncomment this for a slightly different open domain case\n",
|
||||
"#INFLOW_LOCATION = tensor([(11, 6), (12, 4), (14, 5), (16, 5)], batch('inflow_loc'), channel(vector=\"x,y\"))\n",
|
||||
"#INFLOW = (1./4.) * CenteredGrid(Sphere(center=INFLOW_LOCATION, radius=3), extrapolation.BOUNDARY, x=32, y=40, bounds=Box(x=(0,32), y=(0,40)))\n",
|
||||
"#BND = extrapolation.BOUNDARY # open boundaries \n",
|
||||
"\n",
|
||||
"INFLOW.shape"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "VSisaefX_trh"
|
||||
},
|
||||
"source": [
|
||||
"The last statement verifies that our `INFLOW` grid likewise has an `inflow_loc` dimension in addition to the spatial dimensions `x` and `y`. You can test for the existence of a tensor dimension in phiflow with the `.exists` boolean, which can be evaluated for any dimension name. E.g., above `INFLOW.inflow_loc.exists` will give `True`, while `INFLOW.some_unknown_dim.exists` will give `False`. The $^b$ superscript indicates that `inflow_loc` is a batch dimension.\n",
|
||||
"\n",
|
||||
"Phiflow tensors are automatically broadcast to new dimensions via their names, and hence typically no-reshaping operations are required. E.g., you can easily add or multiply tensors with differing dimensions. Below we'll multiply a staggered grid with a tensor of ones along the `inflow_loc` dimension to get a staggered velocity that has `x,y,inflow_loc` as dimensions via `StaggeredGrid(...) * math.ones(batch(inflow_loc=4))`."
|
||||
],
|
||||
"metadata": {
|
||||
"id": "VSisaefX_trh"
|
||||
}
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"source": [
|
||||
"We can easily simulate a few steps now starting with these different initial conditions. Thanks to the broadcasting, the exact same code we used for the single forward simulation in the overview chapter will produce four simulations with different smoke inflow positions."
|
||||
],
|
||||
"metadata": {
|
||||
"id": "SDmFUYlI_tri"
|
||||
}
|
||||
},
|
||||
"source": [
|
||||
"We can easily simulate a few steps now starting with these different initial conditions. Thanks to the broadcasting, the exact same code we used for the single forward simulation in the overview chapter will produce four simulations with different smoke inflow positions."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 3,
|
||||
"metadata": {
|
||||
"colab": {
|
||||
"base_uri": "https://localhost:8080/",
|
||||
"height": 263
|
||||
},
|
||||
"id": "2PccI2r5_trj",
|
||||
"outputId": "8dc97fa4-6c27-4406-e0c1-cdfb9c95b5e5"
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": "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 truncated
|
||||
"text/plain": [
|
||||
"<Figure size 720x432 with 4 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {
|
||||
"needs_background": "light",
|
||||
"tags": []
|
||||
},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"smoke = CenteredGrid(0, extrapolation.BOUNDARY, x=32, y=40, bounds=Box[0:32, 0:40]) # sampled at cell centers\n",
|
||||
"velocity = StaggeredGrid(0, BND, x=32, y=40, bounds=Box[0:32, 0:40]) # sampled in staggered form at face centers \n",
|
||||
"smoke = CenteredGrid(0, extrapolation.BOUNDARY, x=32, y=40, bounds=Box(x=(0,32), y=(0,40))) # sampled at cell centers\n",
|
||||
"velocity = StaggeredGrid(0, BND, x=32, y=40, bounds=Box(x=(0,32), y=(0,40))) # sampled in staggered form at face centers \n",
|
||||
"\n",
|
||||
"def step(smoke, velocity):\n",
|
||||
" smoke = advect.mac_cormack(smoke, velocity, dt=1) + INFLOW\n",
|
||||
@@ -193,44 +213,24 @@
|
||||
" axes[i].imshow(smoke_final.values.numpy('inflow_loc,y,x')[i,...], origin='lower', cmap='magma')\n",
|
||||
" axes[i].set_title(f\"Inflow {INFLOW_LOCATION.numpy('inflow_loc,vector')[i]}\" + (\", Reference\" if i==3 else \"\"))\n",
|
||||
"pylab.tight_layout()\n"
|
||||
],
|
||||
"outputs": [
|
||||
{
|
||||
"output_type": "display_data",
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"<Figure size 720x432 with 4 Axes>"
|
||||
],
|
||||
"image/png": 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truncated
|
||||
},
|
||||
"metadata": {
|
||||
"needs_background": "light",
|
||||
"tags": []
|
||||
}
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"colab": {
|
||||
"base_uri": "https://localhost:8080/",
|
||||
"height": 263
|
||||
},
|
||||
"id": "2PccI2r5_trj",
|
||||
"outputId": "8dc97fa4-6c27-4406-e0c1-cdfb9c95b5e5"
|
||||
}
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "V1tfVLqJ_trj"
|
||||
},
|
||||
"source": [
|
||||
"The last image shows the state of the advected smoke fields after 20 simulation steps. The final smoke shape of simulation `[3]` with an inflow at `(16,5)`, with the straight plume on the far right, will be our **reference state** below. The initial velocity of the other three will be modified in the optimization procedure below to match this reference.\n",
|
||||
"\n",
|
||||
"(As a small side note: phiflow tensors will keep track of their chain of operations using the backend they were created for. E.g. a tensor created with NumPy will keep using NumPy/SciPy operations unless a PyTorch or TensorFlow tensor is also passed to the same operation. Thus, it is a good idea to verify that tensors are using the right backend once in a while, e.g., via `GRID.values.default_backend`.)\n"
|
||||
],
|
||||
"metadata": {
|
||||
"id": "V1tfVLqJ_trj"
|
||||
}
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "vjIJq_Uz_trk"
|
||||
},
|
||||
"source": [
|
||||
"## Gradients\n",
|
||||
"\n",
|
||||
@@ -245,17 +245,30 @@
|
||||
"It is important that our initial velocity has the `inflow_loc` dimension before we record the gradients, such that we have the full \"mini-batch\" of four versions of our velocity (three of which will be updated via gradients in our optimization later on). To get the appropriate velocity tensor, we initialize a `StaggeredGrid` with a tensor of zeros along the `inflow_loc` batch dimension. As the staggered grid already has `y,x` and `vector` dimensions, this gives the desired four dimensions, as verified by the print statement below.\n",
|
||||
"\n",
|
||||
"Phiflow provides a unified API for gradients across different platforms by using functions that need to return a loss values, in addition to optional state values. It uses a loss function based interface, for which we define the `simulate` function below. `simulate` computes the $L^2$ error outlined above and returns the evolved `smoke` and `velocity` states after 20 simulation steps.\n"
|
||||
],
|
||||
"metadata": {
|
||||
"id": "vjIJq_Uz_trk"
|
||||
}
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 4,
|
||||
"metadata": {
|
||||
"colab": {
|
||||
"base_uri": "https://localhost:8080/"
|
||||
},
|
||||
"id": "_j7fUGvQ_trl",
|
||||
"outputId": "f06dea67-cc48-4af1-a963-46500f90d138"
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"Velocity dimensions: (inflow_locᵇ=4, xˢ=32, yˢ=40, vectorᵛ=2)\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"initial_smoke = CenteredGrid(0, extrapolation.BOUNDARY, x=32, y=40, bounds=Box[0:32, 0:40])\n",
|
||||
"initial_velocity = StaggeredGrid(math.zeros(batch(inflow_loc=4)), BND, x=32, y=40, bounds=Box[0:32, 0:40])\n",
|
||||
"initial_smoke = CenteredGrid(0, extrapolation.BOUNDARY, x=32, y=40, bounds=Box(x=(0,32), y=(0,40)))\n",
|
||||
"initial_velocity = StaggeredGrid(math.zeros(batch(inflow_loc=4)), BND, x=32, y=40, bounds=Box(x=(0,32), y=(0,40)))\n",
|
||||
"print(\"Velocity dimensions: \"+format(initial_velocity.shape))\n",
|
||||
"\n",
|
||||
"def simulate(smoke: CenteredGrid, velocity: StaggeredGrid):\n",
|
||||
@@ -267,97 +280,84 @@
|
||||
" #loss = field.l2_loss(diffuse.explicit(smoke - field.stop_gradient(smoke.inflow_loc[-1]), 1, 1, 10))\n",
|
||||
" \n",
|
||||
" return loss, smoke, velocity\n"
|
||||
],
|
||||
"outputs": [
|
||||
{
|
||||
"output_type": "stream",
|
||||
"name": "stdout",
|
||||
"text": [
|
||||
"Velocity dimensions: (inflow_locᵇ=4, xˢ=32, yˢ=40, vectorᵛ=2)\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"colab": {
|
||||
"base_uri": "https://localhost:8080/"
|
||||
},
|
||||
"id": "_j7fUGvQ_trl",
|
||||
"outputId": "f06dea67-cc48-4af1-a963-46500f90d138"
|
||||
}
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "pjqeVedM_trm"
|
||||
},
|
||||
"source": [
|
||||
"Phiflow's `field.functional_gradient()` function is the central function to compute gradients. Next, we'll use it to obtain the gradient with respect to the initial velocity. Since the velocity is the second argument of the `simulate()` function, we pass `wrt=[1]`. (Phiflow also has a `field.spatial_gradient` function which instead computes derivatives of tensors along spatial dimensions, like `x,y`.)\n",
|
||||
"\n",
|
||||
"`functional_gradient` generates a gradient function. As a demonstration, the next cell evaluates the gradient once with the initial states for smoke and velocity. The last statement prints a summary of a part of the resulting gradient tensor.\n"
|
||||
],
|
||||
"metadata": {
|
||||
"id": "pjqeVedM_trm"
|
||||
}
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 5,
|
||||
"source": [
|
||||
"sim_grad = field.functional_gradient(simulate, wrt=[1], get_output=False)\n",
|
||||
"(velocity_grad,) = sim_grad(initial_smoke, initial_velocity)\n",
|
||||
"\n",
|
||||
"print(\"Some gradient info: \" + format(velocity_grad))\n",
|
||||
"print(format(velocity_grad.values.inflow_loc[0].vector[0])) # one example, location 0, x component, automatically prints size & content range\n"
|
||||
],
|
||||
"outputs": [
|
||||
{
|
||||
"output_type": "stream",
|
||||
"name": "stdout",
|
||||
"text": [
|
||||
"Some gradient info: StaggeredGrid[(inflow_locᵇ=4, xˢ=32, yˢ=40, vectorᵛ=2), size=(32, 40), extrapolation=0]\n",
|
||||
"(xˢ=31, yˢ=40) float32 -17.366662979125977 < ... < 14.014090538024902\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"colab": {
|
||||
"base_uri": "https://localhost:8080/"
|
||||
},
|
||||
"id": "grYi0l-g_trn",
|
||||
"outputId": "93dae39b-1b31-429a-cf81-f615e23bbf7d"
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"Some gradient info: StaggeredGrid[(inflow_locᵇ=4, xˢ=32, yˢ=40, vectorᵛ=2), size=(32, 40), extrapolation=0]\n",
|
||||
"(xˢ=31, yˢ=40) float32 -17.366662979125977 < ... < 14.014090538024902\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"sim_grad = field.functional_gradient(simulate, wrt=[1], get_output=False)\n",
|
||||
"(velocity_grad,) = sim_grad(initial_smoke, initial_velocity)\n",
|
||||
"\n",
|
||||
"print(\"Some gradient info: \" + format(velocity_grad))\n",
|
||||
"print(format(velocity_grad.values.inflow_loc[0].vector[0])) # one example, location 0, x component, automatically prints size & content range\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "-drJBAe2_tro"
|
||||
},
|
||||
"source": [
|
||||
"The last two lines just print some information about the resulting gradient field. Naturally, it has the same shape as the velocity itself: it's a staggered grid with four inflow locations. The last line shows how to access the x-components of one of the gradients.\n",
|
||||
"\n",
|
||||
"We could use this to take a look at the content of the computed gradient with regular plotting functions, e.g., by converting the x component of one of the simulations to a numpy array via `velocity_grad.values.inflow_loc[0].vector[0].numpy('y,x')`. However, below we'll use phiflow's `view()` function instead. It automatically analyzes the grid content and provides UI buttons to choose different viewing modes. You can use them to show arrows, single components of the 2-dimensional velocity vectors, or their magnitudes. (Because of its interactive nature, the corresponding image won't show up outside of Jupyter, though.)\n"
|
||||
],
|
||||
"metadata": {
|
||||
"id": "-drJBAe2_tro"
|
||||
}
|
||||
"We could use this to take a look at the content of the computed gradient with regular plotting functions, e.g., by converting the x component of one of the simulations to a numpy array via `velocity_grad.values.inflow_loc[0].vector[0].numpy('y,x')`. An interactive alternative would be phiflow's `view()` function, which automatically analyzes the grid content and provides UI buttons to choose different viewing modes. You can use them to show arrows, single components of the 2-dimensional velocity vectors, or their magnitudes. Because of its interactive nature, the corresponding image won't show up outside of Jupyter, though, and hence we're showing the vector length below via `plot()` instead.\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"source": [
|
||||
"# neat phiflow helper function:\n",
|
||||
"view(velocity_grad)\n"
|
||||
],
|
||||
"outputs": [],
|
||||
"metadata": {
|
||||
"id": "2LTHHjtZ_tro"
|
||||
}
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"# neat phiflow helper function:\n",
|
||||
"vis.plot(field.vec_length(velocity_grad)) # show magnitude"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"source": [
|
||||
"Not surprisingly, the fourth gradient on the left is zero (it's already matching the reference). The other three gradients have detected variations for the initial round inflow positions shown as positive and negative regions around the circular shape of the inflow. The ones for the larger distances on the left are also noticeably larger."
|
||||
],
|
||||
"metadata": {
|
||||
"id": "J7IgFG4o_trp"
|
||||
}
|
||||
},
|
||||
"source": [
|
||||
"Not surprisingly, the fourth gradient on the left is zero (it's already matching the reference). The other three gradients have detected variations for the initial round inflow positions shown as positive and negative regions around the circular shape of the inflow. The ones for the larger distances on the left are also noticeably larger."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "3mpyowRYUSS4"
|
||||
},
|
||||
"source": [
|
||||
"\n",
|
||||
"\n",
|
||||
@@ -370,27 +370,22 @@
|
||||
"The following cell implements a simple steepest gradient descent optimization: it re-evaluates the gradient function, and iterates several times to optimize $\\mathbf{u}_0$ with a learning rate (step size) `LR`.\n",
|
||||
"\n",
|
||||
"`field.functional_gradient` has a parameter `get_output` that determines whether the original results of the function (`simulate()` in our case) are returned, or only the gradient. As it's interesting to track how the loss evolves over the course of the iterations, let's redefine the gradient function with `get_output=True`. \n"
|
||||
],
|
||||
"metadata": {
|
||||
"id": "3mpyowRYUSS4"
|
||||
}
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 7,
|
||||
"source": [
|
||||
"sim_grad_wloss = field.functional_gradient(simulate, wrt=[1], get_output=True) # if we need outputs...\n",
|
||||
"\n",
|
||||
"LR = 1e-03 \n",
|
||||
"for optim_step in range(80): \n",
|
||||
" (loss, _smoke, _velocity), (velocity_grad,) = sim_grad_wloss(initial_smoke, initial_velocity)\n",
|
||||
" initial_velocity = initial_velocity - LR * velocity_grad\n",
|
||||
" if optim_step<3 or optim_step%10==9: print('Optimization step %d, loss: %f' % (optim_step, np.sum(loss.numpy()) ))\n"
|
||||
],
|
||||
"metadata": {
|
||||
"colab": {
|
||||
"base_uri": "https://localhost:8080/"
|
||||
},
|
||||
"id": "0U8h7_-b_trq",
|
||||
"outputId": "11df5469-5c45-41b4-ba21-81114d2a3431"
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"output_type": "stream",
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"Optimization step 0, loss: 1193.145020\n",
|
||||
"Optimization step 1, loss: 1165.816650\n",
|
||||
@@ -405,51 +400,30 @@
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"colab": {
|
||||
"base_uri": "https://localhost:8080/"
|
||||
},
|
||||
"id": "0U8h7_-b_trq",
|
||||
"outputId": "11df5469-5c45-41b4-ba21-81114d2a3431"
|
||||
}
|
||||
"source": [
|
||||
"sim_grad_wloss = field.functional_gradient(simulate, wrt=[1], get_output=True) # if we need outputs...\n",
|
||||
"\n",
|
||||
"LR = 1e-03 \n",
|
||||
"for optim_step in range(80): \n",
|
||||
" (loss, _smoke, _velocity), (velocity_grad,) = sim_grad_wloss(initial_smoke, initial_velocity)\n",
|
||||
" initial_velocity = initial_velocity - LR * velocity_grad\n",
|
||||
" if optim_step<3 or optim_step%10==9: print('Optimization step %d, loss: %f' % (optim_step, np.sum(loss.numpy()) ))\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "PI3xY-7m_trr"
|
||||
},
|
||||
"source": [
|
||||
"The loss should have gone down significantly, from above 1100 to below 700, and now we can also visualize the initial velocities that were obtained in the optimization. \n",
|
||||
"\n",
|
||||
"The following images show the resulting three initial velocities in terms of their x (first set), and y components (second set of images). We're skipping the fourth set with `inflow_loc[0]` because it only contains zeros.\n"
|
||||
],
|
||||
"metadata": {
|
||||
"id": "PI3xY-7m_trr"
|
||||
}
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 8,
|
||||
"source": [
|
||||
"fig, axes = pylab.subplots(1, 3, figsize=(10, 4))\n",
|
||||
"for i in range(INFLOW.shape.get_size('inflow_loc')-1):\n",
|
||||
" im = axes[i].imshow(initial_velocity.staggered_tensor().numpy('inflow_loc,y,x,vector')[i,...,0], origin='lower', cmap='magma')\n",
|
||||
" axes[i].set_title(f\"Ini. vel. X {INFLOW_LOCATION.numpy('inflow_loc,vector')[i]}\")\n",
|
||||
" pylab.colorbar(im,ax=axes[i])\n",
|
||||
"pylab.tight_layout()\n"
|
||||
],
|
||||
"outputs": [
|
||||
{
|
||||
"output_type": "display_data",
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"<Figure size 720x288 with 6 Axes>"
|
||||
],
|
||||
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAsgAAAEXCAYAAACwK7IRAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADh0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uMy4yLjIsIGh0dHA6Ly9tYXRwbG90bGliLm9yZy+WH4yJAAAgAElEQVR4nO29e7QlVX3v+/3uR+9+0C9oxBY4gtrRYM4I3LRoNBoOzzbRYM41CvF68Rw86B3hxjz0iMmIRBJzUJMQx/GVVhDMURHx1ceBkgbhoDcJ9kY5CCjStCDd0mB3Qz/o5977d/+o2rhq1tx71apVtVbNWd9Pjxq9ZtWcs+Zau741Z836/eaPZgYhhBBCCCFEwsiwGyCEEEIIIUST0ABZCCGEEEKIDjRAFkIIIYQQogMNkIUQQgghhOhAA2QhhBBCCCE6GBt2A4Sog/POO9127tzdU5m77vrxzWa2rqYmCSHmQHoVIizaoFkNkEWU7Ny5G3d+9x97KjM2+h9W1dQcIcQ8SK9ChEUbNKsBsogTAzAzM+xWCCGKIL0KERYt0KwGyCJSLHrxChEP0qsQYRG/ZuWkJ+JlZqa3rQAk15F8gORmkpd5jr+K5PdITpF8vXNsmuTd6bahom8pRBzUoFchRI1ErlnNIIs4MQAVh1EnOQrgowDOAbAVwCaSG8zs/o5sPwXwFgDv9FRxwMxOrbRRQsRADXoVQtRICzSrAbKIlFpe/5wOYLOZbQEAktcDOB/AMwNkM3s4PRbe47IQQyP+17VCxEX8mtUAWcRL7+JdRXKyI73ezNZ3pI8H8GhHeiuAl/ZQ/8K0/ikAV5rZV3ttoBDREnlnK0R0RK5ZDZBFnJTzsN1hZmtraM0szzWzbSSfB+BbJH9gZg/VeD4hwqAFHvFCREULNKsBsoiUWl7/bANwYkf6hHRfsRaZbUv/30LydgCnAdAAWYgWvK4VIi7i16wGyCJO6nm63QRgDcmTkQyMLwDw+0UKklwJYL+ZHSK5CsArAHyw6gYKESQtmI0SIipaoFkNkEWkGGjVitfMpkheCuBmAKMArjGz+0heAWDSzDaQfAmArwBYCeC1JN9nZi8G8MsA/jF13htBYoN8/xynEqJlVK9XIUSdxK9ZDZBFvNTwdGtmNwG4ydn33o7Pm5CYXrjl/gXAv6+8QULEQuSzUUJER+Sa1QBZxIkBmIl7jUYhokF6FSIsWqBZDZBFpMTvQCBEPEivQoRF/JrVAFnESQscCISIBulViLBogWZHht0AIWrDZnrbhBDDowa9klxH8gGSm0le5jn+KpLfIzlF8vXOsWmSd6fbhoq+pRDxEHkfqxlkESnxv/4RIh6q1yvJUQAfBXAOkqiXm0hucFaP+SmAtwB4p6eKA2Z2aqWNEiIa4u9jNUAWcdICBwIhoqEevZ4OYLOZbQEAktcDOB/AMwNkM3s4PRZ3Ty9E1bSgj9UAWURK/E+3QsRDKb2uIjnZkV5vZus70scDeLQjvRXAS3uof2Fa/xSSdcu/2msDhYiX+PtYDZBFvEQuXiGione97jCztXU0JeW5ZraN5PMAfIvkD8xMoeGFmCXyPlYDZBEnBjBy8QoRDfXodRuAEzvSJ6T7ijXJbFv6/xaStwM4DYAGyEIArehjtYqFiBQDrMdNCDEkatHrJgBrSJ5McgGACwAUWo2C5EqSE+nnVQBegQ7bZSFE/H2sZpBFvET+dCtEVFSsVzObInkpgJsBjAK4xszuI3kFgEkz20DyJQC+AmAlgNeSfJ+ZvRjALwP4x9R5bwSJDbIGyEJ0EnkfqwGyiJMWLGIuRDTUpFczuwnATc6+93Z83oTE9MIt9y8A/n3lDRIiFlrQx8rEQgghhBBCiA40gywixaJfo1GIeJBehQiL+DWrGWQRJ7Ovf3rZhBDDQXoVIixq0GyfoeEvIvlgul1UxVds7QCZ5Deq+hHnOccZJLfWWPcMyX0k19Vxjh7b8y2SB0l+Z9hteQZ1uNEgvVbenmtJHqjr+5ZCeo0Kabby9jxE8jDJ/zHstjxDhZrtCA3/agCnALiQ5ClOttnQ8J9zyh4N4HIkgYBOB3A5yZX9fr2oBsgkHyZ5dpG8ZvZqM7uu7jYVheRrSW5P/9Cz+84nuY3k8jmK/czMjjKzb6b5V5PcQPJnJI3kSc45/jZ9utpL8kck/++Sbb0mrf8Fs/vM7EwAby9TXz3EvwRN6EivXfX6QZKPktxD8hGSf9ZjG48l+TmSu0k+SfKzs8fM7C1IOqKGIL2GgDQ7v2Y76j2a5M97mTAi+RaS0+mAfHY7Y/a4mT0fwN8Ura9+KtfsM6HhzewwgNnQ8L84o9nDZnYPAHe0fR6AjWa2y8yeBLARQN8PNVENkEPGzP4ngG8BuAoASK4A8HEA/4+Z7S5YzQyAbwL4P+c4/jSA1wJYDuAiAB8m+fJe2knyNwA8v5cyQ0GvbEWNDEivVwN4kZktA/ByAG8i+R97aOaXAWwH8O8APAvA3/ZQdrBIr6JmBqTZWT4A4Iclmvmv6YB8dru9RB2DoZxmV5Gc7Ngu6ajRFxr++IKt6afsnEQ7QE6fxr6Tzpo+SfInJF/dcfx2km8tUM8bSU46+/6Y5Ib080R6jp+SfJzkJ0guKtnsPwTwapLnIRHx/zKzQgvbA4CZPW5mH0OyQL7v+OVm9iMzmzGzOwF8G8CvF62f5BiA/w7g/y1aZqjMWG+bGBrSq/f4A2b2dMeuGQAv8OV1IXkukihy7zKz3WZ2xMy+X7RtQ0F6DQpp1k866fQrAD5dso3h0Ltmd5jZ2o5t/bC/wnxEO0BOeSmABwCsAvBBAFeTZI91/E8ALyS5pmPf7+MXNjBXAvglAKci6byOB/BelMDMdgB4B4DPAngNEjHXQnqDeQmA+3oo9scA7khfcTQc04xUeEivDiQvI7kPyYzIEji2d/PwMiS/5XUkd5LcRPI3q25fdUivgSLNdsDEjvYjAC5FMsfaK6eR3EHyxyT/Ip2UaiiVa7af0PB9hZWfi9gHyI+Y2SfNbBrAdQBWAziulwrMbD+ArwG4EABSEb8IwIb0RnAJgD9ObV/2IrERuqCPNv8bEhOIfzazn/dRTzc+AeB/I4ky1RWSJwJ4G0remAaOXtmGiPTqYGZXAlgK4P8A8E8Air4KPgHAuQBuA/BsAH8H4GtMwiY3D+k1VKTZLH8I4E4zu6tE2TuQzDw/C4kJx4UA3lVh26qles2WDg2PZBxzLpMQ8SuR3PsKjW3mI/YB8vbZD6kIAeCoEvV8Dql4kTzZfjWt71gAiwHcRfIpkk8hsU86tnyTsR7AZwD8FsnC5g+9QPJDSIT4BrPC3i7/AOCKHmy1ho9e2YaG9OrBEr4P4ACA9xUsdgDAw2Z2dWpecT0SG71X1NHGSpBeQ0SaTSH5HCQD5D8vUz51TvtJagL5AwBXAHh9t3JDpULNmtkUkpn3m5HYb98wGxqe5O8AAMmXMFm15PeQhIK/Ly27C8BfIRlkb0IyVtnV79dr8PR9o9gI4FiSpyIR8R+n+3cg6YhebGZ9T+eTvBjJa4LXAJgE8CmSp6UenZVA8n1IvNd/08z29FD0LAC/QfKDHfv+leQ7zKzoa98BYoBplqmlRKNXhzEUd5C9B4lDbicNHlVKry0nBs2ejmQG/f7UymQRgEUktwM4Pp1l7wUD0Ku5ygCpXrNlQ8Onx64BcE2V7Yl9BrkSzOwIgC8C+BCAo5GIGWY2A+CTAK4i+SwAIHl86gDQE+nT54cA/BczO4TEBGInenwaJbkQwESanEjTs8feg+Tp/Gwz29ljE38JwK8isQM7Nd33WgBf6bGewWDQjFRLiUGvJEdIvi19ZUiSpwP4AwC3Fqz6KwBWMlk8f5TJovonAPj/emnfwJBeW00MmgXwDQAn4Rd95HsBfB/AqUUGxyRfTfK49POLAPwFEtOTZtICzWqADIDkK1NHmPn4HICzAXwxfRUwy7sBbAbwbyT3ALgFwAvnOM83OPdaph8DcL2ZfRtIXqsC+C8A/ojki4t/GxwAMPtdfpSmZ/kbJEs+beYv1lkstLaqmT1hZttnt3T3DjM7MG/BYSKbxihpkV5/F8BDAPYC+B9IVpD570UqTV8v/g6AdyKxW74MwPmpk1IzkV6jpQ2aNbNDTh+5G8CRjv6yG2cBuIfk00hmUb+MRq177CFyzbK4CapoEiRfhcRW5xCAN5pZ3wbpfbZnIxLP+e+a2VnDbAsArP2l423Tx97WU5mRcy6/y8zW1tQk0WIaqNerkdjxPWFmhZaOqxPpVTSNBmr2ASQreNxgZv95mG0B2qFZ2SAHipndgcTGqRGY2TnDbkMWC/KJVcRJA/V6MYCLh92OXyC9imbRQM16Z82HR/ya1QBZxEuANk9CtBbpVYiwiFyzhW2QU0eP75P8epo+meSdJDeT/EK6bp0QzcCQeNj2skWE9CqCouV6BaRZERgt0GwvM8jvQLI23bI0/QEAV5nZ9SQ/geR13cfnq2D5+GJ71oLlpRoqxCyb92/fYWZd1sEM02u2QvrW67KxxXbshPQq+mOL9FoU9bGiEaiPTSg0QCZ5AoDfBvB+AH/CZJG/M5EsGQYkEXT+El3E+6wFy/HhX/lPpRsrBAD89nf/2yOFMkYu3rmoSq/HTizHB3+5QWaqIkhef9f7pdcuqI8VTUJ9bELRGeR/APBfkYQ8BYBjADzVsRTLViTelTlIXoIkVCSOXbDMl0WI6pkNg9lOKtHrKulVDIp26xVQHytCowWa7TpAJvkaJEsB3UXyjF5PYGbrkYR2xJolqzOPGzPWPUhMkZ/fKqrHfRiyAkFs3DJFvtMgn7mKhOEZYfcWjTgV0fkW7nFvHZ59dM7ty1OkfV4if7r1UaVen+/otVD5EpopQlW6KnLuur5DVRTRmqvPInW4u8rcF4qce05aqFeg3X2s91xd9FekjiLfuy7K9lfd+tgiZdTHVksRJ71XAPgdkg8DuB7Line truncated
|
||||
},
|
||||
"metadata": {
|
||||
"needs_background": "light",
|
||||
"tags": []
|
||||
}
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"colab": {
|
||||
"base_uri": "https://localhost:8080/",
|
||||
@@ -457,34 +431,34 @@
|
||||
},
|
||||
"id": "FdL_orrX_trr",
|
||||
"outputId": "b716256c-9b8a-40d9-fc01-d75e261a728c"
|
||||
}
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 9,
|
||||
"source": [
|
||||
"fig, axes = pylab.subplots(1, 3, figsize=(10, 4))\n",
|
||||
"for i in range(INFLOW.shape.get_size('inflow_loc')-1):\n",
|
||||
" im = axes[i].imshow(initial_velocity.staggered_tensor().numpy('inflow_loc,y,x,vector')[i,...,1], origin='lower', cmap='magma')\n",
|
||||
" axes[i].set_title(f\"Ini. vel. Y {INFLOW_LOCATION.numpy('inflow_loc,vector')[i]}\")\n",
|
||||
" pylab.colorbar(im,ax=axes[i])\n",
|
||||
"pylab.tight_layout()\n"
|
||||
],
|
||||
"outputs": [
|
||||
{
|
||||
"output_type": "display_data",
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 720x288 with 6 Axes>"
|
||||
],
|
||||
"image/png": "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 truncated
|
||||
]
|
||||
},
|
||||
"metadata": {
|
||||
"needs_background": "light",
|
||||
"tags": []
|
||||
}
|
||||
},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"fig, axes = pylab.subplots(1, 3, figsize=(10, 4))\n",
|
||||
"for i in range(INFLOW.shape.get_size('inflow_loc')-1):\n",
|
||||
" im = axes[i].imshow(initial_velocity.staggered_tensor().numpy('inflow_loc,y,x,vector')[i,...,0], origin='lower', cmap='magma')\n",
|
||||
" axes[i].set_title(f\"Ini. vel. X {INFLOW_LOCATION.numpy('inflow_loc,vector')[i]}\")\n",
|
||||
" pylab.colorbar(im,ax=axes[i])\n",
|
||||
"pylab.tight_layout()\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 9,
|
||||
"metadata": {
|
||||
"colab": {
|
||||
"base_uri": "https://localhost:8080/",
|
||||
@@ -492,24 +466,70 @@
|
||||
},
|
||||
"id": "mLhwxsZd_trs",
|
||||
"outputId": "5bb43835-6e21-4d97-bb07-17b17215bbc8"
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 720x288 with 6 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {
|
||||
"needs_background": "light",
|
||||
"tags": []
|
||||
},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"fig, axes = pylab.subplots(1, 3, figsize=(10, 4))\n",
|
||||
"for i in range(INFLOW.shape.get_size('inflow_loc')-1):\n",
|
||||
" im = axes[i].imshow(initial_velocity.staggered_tensor().numpy('inflow_loc,y,x,vector')[i,...,1], origin='lower', cmap='magma')\n",
|
||||
" axes[i].set_title(f\"Ini. vel. Y {INFLOW_LOCATION.numpy('inflow_loc,vector')[i]}\")\n",
|
||||
" pylab.colorbar(im,ax=axes[i])\n",
|
||||
"pylab.tight_layout()\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "3mitRMPx_trt"
|
||||
},
|
||||
"source": [
|
||||
"## Re-simulation\n",
|
||||
"\n",
|
||||
"We can also visualize how the full simulation over the course of 20 steps turns out, given the new initial velocity conditions for each of the inflow locations. This is what happened internally at optimization time for every gradient calculation, and what was measured by our loss function. Hence, it's good to get an intuition for which solutions the optimization has found.\n",
|
||||
"\n",
|
||||
"Below, we re-run the forward simulation with the new initial conditions from `initial_velocity`:\n"
|
||||
],
|
||||
"metadata": {
|
||||
"id": "3mitRMPx_trt"
|
||||
}
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 10,
|
||||
"metadata": {
|
||||
"colab": {
|
||||
"base_uri": "https://localhost:8080/",
|
||||
"height": 263
|
||||
},
|
||||
"id": "OINe99pf_trt",
|
||||
"outputId": "a7c9ade3-3685-40e6-a5f1-97a5084deec9"
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": "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 truncated
|
||||
"text/plain": [
|
||||
"<Figure size 720x432 with 4 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {
|
||||
"needs_background": "light",
|
||||
"tags": []
|
||||
},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"smoke = initial_smoke \n",
|
||||
"velocity = initial_velocity\n",
|
||||
@@ -522,45 +542,45 @@
|
||||
" axes[i].imshow(smoke_final.values.numpy('inflow_loc,y,x')[i,...], origin='lower', cmap='magma')\n",
|
||||
" axes[i].set_title(f\"Inflow {INFLOW_LOCATION.numpy('inflow_loc,vector')[i]}\" + (\", Reference\" if i==3 else \"\"))\n",
|
||||
"pylab.tight_layout()\n"
|
||||
],
|
||||
"outputs": [
|
||||
{
|
||||
"output_type": "display_data",
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"<Figure size 720x432 with 4 Axes>"
|
||||
],
|
||||
"image/png": 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truncated
|
||||
},
|
||||
"metadata": {
|
||||
"needs_background": "light",
|
||||
"tags": []
|
||||
}
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"colab": {
|
||||
"base_uri": "https://localhost:8080/",
|
||||
"height": 263
|
||||
},
|
||||
"id": "OINe99pf_trt",
|
||||
"outputId": "a7c9ade3-3685-40e6-a5f1-97a5084deec9"
|
||||
}
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "ootUIrQq_tru"
|
||||
},
|
||||
"source": [
|
||||
"Naturally, the image on the right is the same (this is the reference), and the other three simulations now exhibit a shift towards the right. As the differences are a bit subtle, let's visualize the difference between the target configuration and the different final states.\n",
|
||||
"\n",
|
||||
"The following images contain the difference between the evolved simulated and target density. Hence, dark regions indicate where the target should be, but isn't. The top row shows the original states with the initial velocity being zero, while the bottom row shows the versions after the optimization has tuned the initial velocities. Hence, in each column you can compare before (top) and after (bottom): \n"
|
||||
],
|
||||
"metadata": {
|
||||
"id": "ootUIrQq_tru"
|
||||
}
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 14,
|
||||
"metadata": {
|
||||
"colab": {
|
||||
"base_uri": "https://localhost:8080/",
|
||||
"height": 410
|
||||
},
|
||||
"id": "7lm9Rrel640-",
|
||||
"outputId": "6ef7914a-8f63-49af-9a1a-fde1aed37682"
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 720x432 with 12 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {
|
||||
"needs_background": "light",
|
||||
"tags": []
|
||||
},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"fig, axes = pylab.subplots(2, 3, figsize=(10, 6))\n",
|
||||
"for i in range(INFLOW.shape.get_size('inflow_loc')-1):\n",
|
||||
@@ -571,44 +591,24 @@
|
||||
" axes[1,i].imshow(smoke.values.numpy('inflow_loc,y,x')[i,...] - smoke_final.values.numpy('inflow_loc,y,x')[3,...], origin='lower', cmap='magma')\n",
|
||||
" axes[1,i].set_title(f\"Result {INFLOW_LOCATION.numpy('inflow_loc,vector')[i]}\")\n",
|
||||
" pylab.colorbar(im,ax=axes[1,i])"
|
||||
],
|
||||
"outputs": [
|
||||
{
|
||||
"output_type": "display_data",
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"<Figure size 720x432 with 12 Axes>"
|
||||
],
|
||||
"image/png": "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 truncated
|
||||
},
|
||||
"metadata": {
|
||||
"needs_background": "light",
|
||||
"tags": []
|
||||
}
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"colab": {
|
||||
"base_uri": "https://localhost:8080/",
|
||||
"height": 410
|
||||
},
|
||||
"id": "7lm9Rrel640-",
|
||||
"outputId": "6ef7914a-8f63-49af-9a1a-fde1aed37682"
|
||||
}
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "cuGIjhGx_trw"
|
||||
},
|
||||
"source": [
|
||||
"These difference images clearly show that the optimization managed to align the upper region of the plumes very well. Each original image (at the top) shows a clear misalignment in terms of a black halo, while the states after optimization largely overlap the target smoke configuration of the reference, and exhibit differences closer to zero for the front of each smoke cloud.\n",
|
||||
"\n",
|
||||
"Note that all three simulations need to \"work\" with a fixed inflow, hence they cannot simply \"produce\" marker density out of the blue to match the target. Also each simulation needs to take into account how the non-linear model equations change the state of the system over the course of 20 time steps. So the optimization goal is quite difficult, and it is not possible to exactly satisfy the constraints to match the reference simulation in this scenario. E.g., this is noticeable at the stems of the smoke plumes, which still show a black halo after the optimization. The optimization was not able to shift the inflow position, and hence needs to focus on aligning the upper regions of the plumes.\n"
|
||||
],
|
||||
"metadata": {
|
||||
"id": "cuGIjhGx_trw"
|
||||
}
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "jHdQureE_trx"
|
||||
},
|
||||
"source": [
|
||||
"## Conclusions\n",
|
||||
"\n",
|
||||
@@ -616,13 +616,13 @@
|
||||
"complex PDEs. Above, we've optimized for a mini-batch of 20 steps of a full Navier-Stokes solver.\n",
|
||||
"\n",
|
||||
"This is a powerful basis to bring NNs into the picture. As you might have noticed, our degrees of freedom were still a regular grid, and we've jointly solved a single inverse problem. There were three cases to solve as a mini-batch, of course, but nonetheless the setup still represents a direct optimization. Thus, in line with the PINN example of {doc}`physicalloss-code` we've not really dealt with a _machine learning_ task here. However, DP training allows for a range of flexible compinations with NNs that will be the topic of the next chapters.\n"
|
||||
],
|
||||
"metadata": {
|
||||
"id": "jHdQureE_trx"
|
||||
}
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "ffaDxZ3x_try"
|
||||
},
|
||||
"source": [
|
||||
"## Next steps\n",
|
||||
"\n",
|
||||
@@ -631,10 +631,7 @@
|
||||
"- Modify the setup of the simulation to differ more strongly across the four instances, run longer, or use a finer spatial discretization (i.e. larger grid size). Note that this will make the optimization problem tougher, and hence might not converge directly with this simple setup.\n",
|
||||
"\n",
|
||||
"- As a larger change, add a multi-resolution optimization to handle cases with larger differences. I.e., first solve with a coarse discretization, and then uses this solution as an initial guess for a finer discretization. \n"
|
||||
],
|
||||
"metadata": {
|
||||
"id": "ffaDxZ3x_try"
|
||||
}
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
|
||||
@@ -168,7 +168,7 @@
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"!pip install --upgrade --quiet phiflow\n",
|
||||
"!pip install --upgrade --quiet phiflow==2.1\n",
|
||||
"#!pip install --upgrade --quiet git+https://github.com/tum-pbs/PhiFlow@develop\n",
|
||||
"\n",
|
||||
"from phi.tf.flow import *\n",
|
||||
|
||||
@@ -50,7 +50,7 @@
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"!pip install --upgrade --quiet phiflow\n",
|
||||
"!pip install --upgrade --quiet phiflow==2.2\n",
|
||||
"from phi.flow import *\n",
|
||||
"\n",
|
||||
"from phi import __version__\n",
|
||||
@@ -116,10 +116,10 @@
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"velocity = CenteredGrid(INITIAL, extrapolation.PERIODIC, x=N, bounds=Box[-1:1])\n",
|
||||
"velocity = CenteredGrid(INITIAL, extrapolation.PERIODIC, x=N, bounds=Box(x=(-1,1)))\n",
|
||||
"vt = advect.semi_lagrangian(velocity, velocity, DT)\n",
|
||||
"#velocity = CenteredGrid(lambda x: -math.sin(np.pi * x), extrapolation.PERIODIC, x=N, bounds=Box[-1:1])\n",
|
||||
"#velocity = CenteredGrid(Noise(), extrapolation.PERIODIC, x=N, bounds=Box[-1:1]) # random init\n",
|
||||
"#velocity = CenteredGrid(lambda x: -math.sin(np.pi * x), extrapolation.PERIODIC, x=N, bounds=Box(x=(-1,1)))\n",
|
||||
"#velocity = CenteredGrid(Noise(), extrapolation.PERIODIC, x=N, bounds=Box(x=(-1,1))) # random init\n",
|
||||
"\n",
|
||||
"print(\"Velocity tensor shape: \" + format( velocity.shape )) # == velocity.values.shape\n",
|
||||
"print(\"Velocity tensor type: \" + format( type(velocity.values) ))\n",
|
||||
@@ -194,7 +194,7 @@
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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||||
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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
@@ -237,7 +237,7 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 1152x360 with 2 Axes>"
|
||||
]
|
||||
|
||||
+226
-226
@@ -2,6 +2,9 @@
|
||||
"cells": [
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "o4JZ84moBKMr"
|
||||
},
|
||||
"source": [
|
||||
"# Navier-Stokes Forward Simulation\n",
|
||||
"\n",
|
||||
@@ -19,101 +22,137 @@
|
||||
"We'll solve this PDE on a closed domain with Dirichlet boundary conditions $\\mathbf{u}=0$ for the velocity, and Neumann boundaries $\\frac{\\partial p}{\\partial x}=0$ for pressure, on a domain $\\Omega$ with a physical size of $100 \\times 80$ units. \n",
|
||||
"[[run in colab]](https://colab.research.google.com/github/tum-pbs/pbdl-book/blob/main/overview-ns-forw.ipynb)\n",
|
||||
"\n"
|
||||
],
|
||||
"metadata": {
|
||||
"id": "o4JZ84moBKMr"
|
||||
}
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Implementation\n",
|
||||
"\n",
|
||||
"As in the previous section, the first command with a \"!\" prefix installs the [phiflow python package from GitHub](https://github.com/tum-pbs/PhiFlow) via `pip` in your python environment. (Skip or modify this command if necessary.)"
|
||||
],
|
||||
"metadata": {}
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 1,
|
||||
"source": [
|
||||
"!pip install --upgrade --quiet phiflow \n",
|
||||
"#!pip install --upgrade --quiet git+https://github.com/tum-pbs/PhiFlow@develop\n",
|
||||
"\n",
|
||||
"from phi.flow import * # The Dash GUI is not supported on Google colab, ignore the warning\n",
|
||||
"import pylab"
|
||||
],
|
||||
"outputs": [],
|
||||
"metadata": {
|
||||
"colab": {
|
||||
"base_uri": "https://localhost:8080/"
|
||||
},
|
||||
"id": "da1uZcDXdVcF",
|
||||
"outputId": "1082dc87-796c-4b57-e72e-5790fc1444c9"
|
||||
}
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"!pip install --upgrade --quiet phiflow==2.2\n",
|
||||
"#!pip install --upgrade --quiet git+https://github.com/tum-pbs/PhiFlow@develop\n",
|
||||
"\n",
|
||||
"from phi.flow import * # The Dash GUI is not supported on Google colab, ignore the warning\n",
|
||||
"import pylab"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "BVV1IKVqDfLl"
|
||||
},
|
||||
"source": [
|
||||
"## Setting up the simulation\n",
|
||||
"\n",
|
||||
"The following code sets up a few constants, which are denoted by upper case names. We'll use $40 \\times 32$ cells to discretize our domain, introduce a slight viscosity via $\\nu$, and define the time step to be $\\Delta t=1.5$. \n",
|
||||
"\n",
|
||||
"We're creating a first `CenteredGrid` here, which is initialized by a `Sphere` geometry object. This will represent the inflow region `INFLOW` where hot smoke is generated."
|
||||
],
|
||||
"metadata": {
|
||||
"id": "BVV1IKVqDfLl"
|
||||
}
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 2,
|
||||
"metadata": {
|
||||
"id": "WrA3IXDxv31P"
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"DT = 1.5\n",
|
||||
"NU = 0.01\n",
|
||||
"\n",
|
||||
"INFLOW = CenteredGrid(Sphere(center=(30,15), radius=10), extrapolation.BOUNDARY, x=32, y=40, bounds=Box[0:80, 0:100]) * 0.2\n"
|
||||
],
|
||||
"outputs": [],
|
||||
"metadata": {
|
||||
"id": "WrA3IXDxv31P"
|
||||
}
|
||||
"INFLOW = CenteredGrid(Sphere(center=tensor([30,15], channel(vector='x,y')), radius=10), extrapolation.BOUNDARY, x=32, y=40, bounds=Box(x=(0,80),y=(0,100))) * 0.2\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "ExA0Pi2sFVka"
|
||||
},
|
||||
"source": [
|
||||
"The inflow will be used to inject smoke into a second centered grid `smoke` that represents the marker field $d$ from above. Note that we've defined a `Box` of size $100 \\times 80$ above. This is the physical scale in terms of spatial units in our simulation, i.e., a velocity of magnitude $1$ will move the smoke density by 1 unit per 1 time unit, which may be larger or smaller than a cell in the discretized grid, depending on the settings for `x,y`. You could parametrize your simulation grid to directly resemble real-world units, or keep appropriate conversion factors in mind. \n",
|
||||
"\n",
|
||||
"The inflow sphere above is already using the \"world\" coordinates: it is located at $x=30$ along the first axis, and $y=15$ (within the $100 \\times 80$ domain box).\n",
|
||||
"\n",
|
||||
"Next, we create grids for the quantities we want to simulate. For this example, we require a velocity field and a smoke density field."
|
||||
],
|
||||
"metadata": {
|
||||
"id": "ExA0Pi2sFVka"
|
||||
}
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 3,
|
||||
"source": [
|
||||
"smoke = CenteredGrid(0, extrapolation.BOUNDARY, x=32, y=40, bounds=Box[0:80, 0:100]) # sampled at cell centers\n",
|
||||
"velocity = StaggeredGrid(0, extrapolation.ZERO, x=32, y=40, bounds=Box[0:80, 0:100]) # sampled in staggered form at face centers "
|
||||
],
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"metadata": {}
|
||||
"source": [
|
||||
"smoke = CenteredGrid(0, extrapolation.BOUNDARY, x=32, y=40, bounds=Box(x=(0,80),y=(0,100))) # sampled at cell centers\n",
|
||||
"velocity = StaggeredGrid(0, extrapolation.ZERO, x=32, y=40, bounds=Box(x=(0,80),y=(0,100))) # sampled in staggered form at face centers "
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"We sample the smoke field at the cell centers and the velocity in [staggered form](https://tum-pbs.github.io/PhiFlow/Staggered_Grids.html). The staggered grid internally contains 2 centered grids with different dimensions, and can be converted into centered grids (or simply numpy arrays) via the `unstack` function, as explained in the link above.\n",
|
||||
"\n",
|
||||
"Next we define the update step of the simulation, which calls the necessary functions to advance the state of our fluid system by `dt`. The next cell computes one such step, and plots the marker density after one simulation frame."
|
||||
],
|
||||
"metadata": {}
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 4,
|
||||
"metadata": {
|
||||
"colab": {
|
||||
"base_uri": "https://localhost:8080/",
|
||||
"height": 282
|
||||
},
|
||||
"id": "WmGZdOwswOva",
|
||||
"outputId": "3ae4d68d-b586-4bbe-eca9-a223d7720949"
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"Max. velocity and mean marker density: [0.15530995, 0.008125]\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"<matplotlib.image.AxesImage at 0x7fa2f8d16eb0>"
|
||||
]
|
||||
},
|
||||
"execution_count": 4,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": "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",
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {
|
||||
"needs_background": "light"
|
||||
},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"def step(velocity, smoke, pressure, dt=1.0, buoyancy_factor=1.0):\n",
|
||||
" smoke = advect.semi_lagrangian(smoke, velocity, dt) + INFLOW\n",
|
||||
@@ -128,49 +167,11 @@
|
||||
"print(\"Max. velocity and mean marker density: \" + format( [ math.max(velocity.values) , math.mean(smoke.values) ] ))\n",
|
||||
"\n",
|
||||
"pylab.imshow(np.asarray(smoke.values.numpy('y,x')), origin='lower', cmap='magma')"
|
||||
],
|
||||
"outputs": [
|
||||
{
|
||||
"output_type": "stream",
|
||||
"name": "stdout",
|
||||
"text": [
|
||||
"Max. velocity and mean marker density: [0.15530995, 0.008125]\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"output_type": "execute_result",
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"<matplotlib.image.AxesImage at 0x7fa2f8d16eb0>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"execution_count": 4
|
||||
},
|
||||
{
|
||||
"output_type": "display_data",
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
],
|
||||
"image/png": "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"
|
||||
},
|
||||
"metadata": {
|
||||
"needs_background": "light"
|
||||
}
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"colab": {
|
||||
"base_uri": "https://localhost:8080/",
|
||||
"height": 282
|
||||
},
|
||||
"id": "WmGZdOwswOva",
|
||||
"outputId": "3ae4d68d-b586-4bbe-eca9-a223d7720949"
|
||||
}
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"A lot has happened in this `step()` call: we've advected the smoke field, added an upwards force via a Boussinesq model, advected the velocity field, and finally made it divergence free via a pressure solve.\n",
|
||||
"\n",
|
||||
@@ -185,21 +186,16 @@
|
||||
"The variables we created for the fields of the simulation here are instances of the class `Grid`.\n",
|
||||
"Like tensors, grids also have the `shape` attribute which lists all batch, spatial and channel dimensions.\n",
|
||||
"[Shapes in phiflow](https://tum-pbs.github.io/PhiFlow/Math.html#shapes) store not only the sizes of the dimensions but also their names and types."
|
||||
],
|
||||
"metadata": {}
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 5,
|
||||
"source": [
|
||||
"print(f\"Smoke: {smoke.shape}\")\n",
|
||||
"print(f\"Velocity: {velocity.shape}\")\n",
|
||||
"print(f\"Inflow: {INFLOW.shape}, spatial only: {INFLOW.shape.spatial}\")\n"
|
||||
],
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"output_type": "stream",
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"Smoke: (xˢ=32, yˢ=40)\n",
|
||||
"Velocity: (xˢ=32, yˢ=40, vectorᵛ=2)\n",
|
||||
@@ -207,60 +203,55 @@
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {}
|
||||
"source": [
|
||||
"print(f\"Smoke: {smoke.shape}\")\n",
|
||||
"print(f\"Velocity: {velocity.shape}\")\n",
|
||||
"print(f\"Inflow: {INFLOW.shape}, spatial only: {INFLOW.shape.spatial}\")\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Note that the phiflow output here indicates the type of a dimension, e.g., $^S$ for a spatial, and $^V$ for a vector dimension. Later on for learning, we'll also introduce batch dimensions.\n",
|
||||
"\n",
|
||||
"The actual content of a shape object can be obtained via `.sizes`, or alternatively we can query the size of a specific dimension `dim` via `.get_size('dim')`. Here are two examples:"
|
||||
],
|
||||
"metadata": {}
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 6,
|
||||
"source": [
|
||||
"print(f\"Shape content: {velocity.shape.sizes}\")\n",
|
||||
"print(f\"Vector dimension: {velocity.shape.get_size('vector')}\")"
|
||||
],
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"output_type": "stream",
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"Shape content: (32, 40, 2)\n",
|
||||
"Vector dimension: 2\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {}
|
||||
"source": [
|
||||
"print(f\"Shape content: {velocity.shape.sizes}\")\n",
|
||||
"print(f\"Vector dimension: {velocity.shape.get_size('vector')}\")"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"The grid values can be accessed using the `values` property. This is an important difference to a phiflow tensor object, which does not have `values`, as illustrated in the code example below."
|
||||
],
|
||||
"metadata": {}
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 7,
|
||||
"source": [
|
||||
"print(\"Statistics of the different simulation grids:\")\n",
|
||||
"print(smoke.values)\n",
|
||||
"print(velocity.values)\n",
|
||||
"\n",
|
||||
"# in contrast to a simple tensor:\n",
|
||||
"test_tensor = math.tensor(numpy.zeros([2, 5, 3]), spatial('x,y'), channel('vector'))\n",
|
||||
"print(\"Reordered test tensor shape: \" + format(test_tensor.numpy('vector,y,x').shape) )\n",
|
||||
"#print(test_tensor.values.numpy('y,x')) # error! tensors don't return their content via \".values\"\n"
|
||||
],
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"output_type": "stream",
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"Statistics of the different simulation grids:\n",
|
||||
"(xˢ=32, yˢ=40) float32 0.0 < ... < 0.20000000298023224\n",
|
||||
@@ -269,37 +260,48 @@
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {}
|
||||
"source": [
|
||||
"print(\"Statistics of the different simulation grids:\")\n",
|
||||
"print(smoke.values)\n",
|
||||
"print(velocity.values)\n",
|
||||
"\n",
|
||||
"# in contrast to a simple tensor:\n",
|
||||
"test_tensor = math.tensor(numpy.zeros([3, 5, 2]), spatial('x,y'), channel(vector=\"x,y\"))\n",
|
||||
"print(\"Reordered test tensor shape: \" + format(test_tensor.numpy('vector,y,x').shape) ) # reorder to vector,y,x \n",
|
||||
"#print(test_tensor.values.numpy('y,x')) # error! tensors don't return their content via \".values\""
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Grids have many more properties which are documented [here](https://tum-pbs.github.io/PhiFlow/phi/field/#phi.field.Grid).\n",
|
||||
"Also note that the staggered grid has a [non-uniform shape](https://tum-pbs.github.io/PhiFlow/Math.html#non-uniform-tensors) because the number of faces is not equal to the number of cells (in this example the x component has $31 \\times 40$ cells, while y has $32 \\times 39$). The `INFLOW` grid naturally has the same dimensions as the `smoke` grid.\n"
|
||||
],
|
||||
"metadata": {}
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Time evolution\n",
|
||||
"\n",
|
||||
"With this setup, we can easily advance the simulation forward in time a bit more by repeatedly calling the `step` function."
|
||||
],
|
||||
"metadata": {}
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 8,
|
||||
"source": [
|
||||
"for time_step in range(10):\n",
|
||||
" velocity, smoke, pressure = step(velocity, smoke, pressure, dt=DT)\n",
|
||||
" print('Computed frame {}, max velocity {}'.format(time_step , np.asarray(math.max(velocity.values)) ))\n"
|
||||
],
|
||||
"metadata": {
|
||||
"colab": {
|
||||
"base_uri": "https://localhost:8080/"
|
||||
},
|
||||
"id": "0hZk5HX3w4Or",
|
||||
"outputId": "f7811af7-4b58-4ff6-a8b6-6e7bedefaa6e"
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"output_type": "stream",
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"Computed frame 0, max velocity 0.4609803557395935\n",
|
||||
"Computed frame 1, max velocity 0.8926814794540405\n",
|
||||
@@ -314,53 +316,24 @@
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"colab": {
|
||||
"base_uri": "https://localhost:8080/"
|
||||
},
|
||||
"id": "0hZk5HX3w4Or",
|
||||
"outputId": "f7811af7-4b58-4ff6-a8b6-6e7bedefaa6e"
|
||||
}
|
||||
"source": [
|
||||
"for time_step in range(10):\n",
|
||||
" velocity, smoke, pressure = step(velocity, smoke, pressure, dt=DT)\n",
|
||||
" print('Computed frame {}, max velocity {}'.format(time_step , np.asarray(math.max(velocity.values)) ))\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"source": [
|
||||
"Now the hot plume is starting to rise:"
|
||||
],
|
||||
"metadata": {
|
||||
"id": "GMKKWQBLHIwP"
|
||||
}
|
||||
},
|
||||
"source": [
|
||||
"Now the hot plume is starting to rise:"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 9,
|
||||
"source": [
|
||||
"pylab.imshow(smoke.values.numpy('y,x'), origin='lower', cmap='magma')"
|
||||
],
|
||||
"outputs": [
|
||||
{
|
||||
"output_type": "execute_result",
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"<matplotlib.image.AxesImage at 0x7fa2f902cf40>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"execution_count": 9
|
||||
},
|
||||
{
|
||||
"output_type": "display_data",
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
],
|
||||
"image/png": 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truncated
|
||||
},
|
||||
"metadata": {
|
||||
"needs_background": "light"
|
||||
}
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"colab": {
|
||||
"base_uri": "https://localhost:8080/",
|
||||
@@ -368,20 +341,79 @@
|
||||
},
|
||||
"id": "Mfl80CjZxZcL",
|
||||
"outputId": "92f3a9ba-d403-4799-a543-132ee8ed234c"
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"<matplotlib.image.AxesImage at 0x7fa2f902cf40>"
|
||||
]
|
||||
},
|
||||
"execution_count": 9,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {
|
||||
"needs_background": "light"
|
||||
},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"pylab.imshow(smoke.values.numpy('y,x'), origin='lower', cmap='magma')"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"source": [
|
||||
"Let's compute and show a few more steps of the simulation. Because of the inflow being located off-center to the left (with x position 30), the plume will curve towards the right when it hits the top wall of the domain."
|
||||
],
|
||||
"metadata": {
|
||||
"id": "wnbQJvA-HPSL"
|
||||
}
|
||||
},
|
||||
"source": [
|
||||
"Let's compute and show a few more steps of the simulation. Because of the inflow being located off-center to the left (with x position 30), the plume will curve towards the right when it hits the top wall of the domain."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 10,
|
||||
"metadata": {
|
||||
"colab": {
|
||||
"base_uri": "https://localhost:8080/",
|
||||
"height": 489
|
||||
},
|
||||
"id": "tkhCOzc0ITsj",
|
||||
"outputId": "f6366c12-1eb5-4ff6-e0d7-94b806bfd8e4"
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"Computing time step 0\n",
|
||||
"Computing time step 1\n",
|
||||
"Computing time step 2\n",
|
||||
"Computing time step 10\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": "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 truncated
|
||||
"text/plain": [
|
||||
"<Figure size 1152x360 with 5 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {
|
||||
"needs_background": "light"
|
||||
},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"steps = [[ smoke.values, velocity.values.vector[0], velocity.values.vector[1] ]]\n",
|
||||
"for time_step in range(20):\n",
|
||||
@@ -395,50 +427,45 @@
|
||||
"for i in range(len(steps)):\n",
|
||||
" axes[i].imshow(steps[i][0].numpy('y,x'), origin='lower', cmap='magma')\n",
|
||||
" axes[i].set_title(f\"d at t={i*5}\")"
|
||||
],
|
||||
"outputs": [
|
||||
{
|
||||
"output_type": "stream",
|
||||
"name": "stdout",
|
||||
"text": [
|
||||
"Computing time step 0\n",
|
||||
"Computing time step 1\n",
|
||||
"Computing time step 2\n",
|
||||
"Computing time step 10\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"output_type": "display_data",
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"<Figure size 1152x360 with 5 Axes>"
|
||||
],
|
||||
"image/png": 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truncated
|
||||
},
|
||||
"metadata": {
|
||||
"needs_background": "light"
|
||||
}
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"colab": {
|
||||
"base_uri": "https://localhost:8080/",
|
||||
"height": 489
|
||||
},
|
||||
"id": "tkhCOzc0ITsj",
|
||||
"outputId": "f6366c12-1eb5-4ff6-e0d7-94b806bfd8e4"
|
||||
}
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"We can also take a look at the velocities. The `steps` list above already stores `vector[0]` and `vector[1]` components of the velocities as numpy arrays, which we can show next."
|
||||
],
|
||||
"metadata": {}
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 11,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 1152x360 with 5 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {
|
||||
"needs_background": "light"
|
||||
},
|
||||
"output_type": "display_data"
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": "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 truncated
|
||||
"text/plain": [
|
||||
"<Figure size 1152x360 with 5 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {
|
||||
"needs_background": "light"
|
||||
},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"fig, axes = pylab.subplots(1, len(steps), figsize=(16, 5))\n",
|
||||
"for i in range(len(steps)):\n",
|
||||
@@ -450,54 +477,27 @@
|
||||
" axes[i].imshow(steps[i][2].numpy('y,x'), origin='lower', cmap='magma')\n",
|
||||
" axes[i].set_title(f\"u_y at t={i*5}\")\n",
|
||||
" "
|
||||
],
|
||||
"outputs": [
|
||||
{
|
||||
"output_type": "display_data",
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"<Figure size 1152x360 with 5 Axes>"
|
||||
],
|
||||
"image/png": 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truncated
|
||||
},
|
||||
"metadata": {
|
||||
"needs_background": "light"
|
||||
}
|
||||
},
|
||||
{
|
||||
"output_type": "display_data",
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"<Figure size 1152x360 with 5 Axes>"
|
||||
],
|
||||
"image/png": 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truncated
|
||||
},
|
||||
"metadata": {
|
||||
"needs_background": "light"
|
||||
}
|
||||
}
|
||||
],
|
||||
"metadata": {}
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "ooqVxCPM8PXl"
|
||||
},
|
||||
"source": [
|
||||
"It looks simple here, but this simulation setup is a powerful tool. The simulation could easily be extended to more complex cases or 3D, and it is already fully compatible with backpropagation pipelines of deep learning frameworks. \n",
|
||||
"\n",
|
||||
"In the next chapters we'll show how to use these simulations for training NNs, and how to steer and modify them via trained NNs. This will illustrate how much we can improve the training process by having a solver in the loop, especially when the solver is _differentiable_. Before moving to these more complex training processes, we will cover a simpler supervised approach in the next chapter. This is very fundamental: even when aiming for advanced physics-based learning setups, a working supervised training is always the first step."
|
||||
],
|
||||
"metadata": {
|
||||
"id": "ooqVxCPM8PXl"
|
||||
}
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Next steps\n",
|
||||
"\n",
|
||||
"You could create a variety of nice fluid simulations based on this setup. E.g., try changing the spatial resolution, the buoyancy factors, and the overall length of the simulation run."
|
||||
],
|
||||
"metadata": {}
|
||||
]
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
|
||||
+6
-6
@@ -51,11 +51,11 @@ We'll need a few tools for the derivations below, which are summarized here for
|
||||
|
||||
Not surprisingly, we'll need some Taylor-series expansions. With the notation above it reads:
|
||||
|
||||
$$L(x+\Delta) = L + J^T \Delta + \frac{1}{2} H \Delta^2 + \cdots$$
|
||||
$$L(x+\Delta) = L + J \Delta + \frac{1}{2} H \Delta^2 + \cdots$$
|
||||
|
||||
Then we also need the _Lagrange form_, which yields an exact solution for a $\xi$ from the interval $[x, x+\Delta]$:
|
||||
|
||||
$$L(x+\Delta) = L + J^T \Delta + \frac{1}{2} H(\xi) \Delta^2$$
|
||||
$$L(x+\Delta) = L + J \Delta + \frac{1}{2} H(\xi) \Delta^2$$
|
||||
|
||||
In several instances we'll make use of the fundamental theorem of calculus, repeated here for completeness:
|
||||
|
||||
@@ -134,27 +134,27 @@ with lower loss values.
|
||||
First, we apply the fundamental theorem to L
|
||||
|
||||
$$
|
||||
L(x + \lambda \Delta) = L + \int_0^1 \text{ds}~ J^T(x + s \lambda \Delta) \lambda \Delta \ ,
|
||||
L(x + \lambda \Delta) = L + \int_0^1 \text{ds}~ J(x + s \lambda \Delta) \lambda \Delta \ ,
|
||||
$$
|
||||
|
||||
and likewise express $J$ around this location with it:
|
||||
|
||||
$$\begin{aligned}
|
||||
J^T(x + s \lambda \Delta) &= J^T + \int_0^1 \text{dt}~ H(x + s t \lambda \Delta) s \lambda \Delta
|
||||
J(x + s \lambda \Delta) = J + s \lambda \Delta^T \int_0^1 \text{dt}~ H(x + s t \lambda \Delta)
|
||||
\end{aligned}$$
|
||||
|
||||
Inserting this $J^T$ into $L$ yields:
|
||||
|
||||
$$\begin{aligned}
|
||||
L(x + \lambda \Delta) - L(x)
|
||||
&= \int_0^1 \text{d}s \Big[ J^T + \int_0^1 \text{d}t~ H(x + s t \lambda \Delta) ~ s \lambda \Delta \Big]^T \lambda \Delta \\
|
||||
&= \int_0^1 \text{d}s \Big[ J + s \lambda \Delta^T \int_0^1 \text{d}t~ H(x + s t \lambda \Delta) \Big]^T \lambda \Delta \\
|
||||
&= \int_0^1 \text{d}s \Big[ -H \Delta + \int_0^1 \text{d}t H(x + s t \lambda \Delta) ~ s \lambda \Delta \Big]^T \lambda \Delta \\
|
||||
&= \int_0^1 \text{d}s \int_0^1 \text{d}t \big[ -H \Delta + H(x + s t \lambda \Delta) ~ s \lambda \Delta \big]^T \lambda \Delta \\
|
||||
&= \int_0^1 \text{d}s \int_0^1 \text{d}t \big[ -H \Delta (1+\lambda s) + [ H(x + s t \lambda \Delta) - H ] ~ s \lambda \Delta \big]^T \lambda \Delta \\
|
||||
&= -(H\Delta)^T \big( 1+\frac{\lambda}{2} \big) \lambda \Delta + \lambda^2 \int_0^1 \text{d}s ~ s \int_0^1 \text{d}t \big[ [ H(x + s t \lambda \Delta) - H ] ~ \Delta \big]^T \Delta \\
|
||||
\end{aligned}$$ (newton-step-size-conv1)
|
||||
|
||||
Here we've first used $\Delta = - J^T/H$, and moved it into the integral in the third line.
|
||||
Here we've first used $\Delta = - J^T/H$, and moved it into the integral in the third line, alongside the $s \lambda \Delta$ term.
|
||||
Line four factors out $H \Delta$. This allows us to evaluate the integral for the first term
|
||||
$ -H \Delta (1+\lambda s)$ in the last line of equation {eq}`newton-step-size-conv1` above.
|
||||
|
||||
|
||||
+2
-1
@@ -67,7 +67,8 @@
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"!pip install --upgrade --quiet git+https://github.com/tum-pbs/PhiFlow\n",
|
||||
"!pip install --upgrade --quiet phiflow==2.2\n",
|
||||
"#!pip install --upgrade --quiet git+https://github.com/tum-pbs/PhiFlow\n",
|
||||
"from phi.torch.flow import * # switch to TF with \"phi.tf.flow\""
|
||||
]
|
||||
},
|
||||
|
||||
@@ -785,12 +785,9 @@
|
||||
"metadata": {
|
||||
"colab": {
|
||||
"collapsed_sections": [],
|
||||
"name": "LinearChain_GD_PG_HIG-fromSlack-colab.ipynb",
|
||||
"name": "physgrad-hig-code.ipynb",
|
||||
"provenance": []
|
||||
},
|
||||
"interpreter": {
|
||||
"hash": "60a46218545510a486eb79ffac0761a327b44b02462576ac242e6c15f2abc4b7"
|
||||
},
|
||||
"kernelspec": {
|
||||
"display_name": "Python 3.8.10 64-bit ('tf250': conda)",
|
||||
"name": "python3"
|
||||
|
||||
+1
-21
@@ -74,23 +74,6 @@
|
||||
"name": "stderr",
|
||||
"output_type": "stream",
|
||||
"text": []
|
||||
},
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"Could not load resample cuda libraries: CUDA binaries not found at /Users/thuerey/Dropbox/mbaDevelSelected/phiflow-v1.5/phi/tf/cuda/build/resample.so. Run \"python setup.py cuda\" to compile them\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"name": "stderr",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
" for key in self._mapping:\n",
|
||||
" yield (key, self._mapping[key])\n",
|
||||
"$ python setup.py tf_cuda\n",
|
||||
"before reinstalling phiflow.\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
@@ -210,10 +193,7 @@
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"Instructions for updating:\n",
|
||||
"Use keras.layers.Dense instead.\n",
|
||||
"Instructions for updating:\n",
|
||||
"Please use `layer.__call__` method instead.\n"
|
||||
"\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
|
||||
Reference in new issue
Block a user