fixed burgers initialization

diffphys-code-burgers.ipynb

@@ -28,13 +28,13 @@
    "metadata": {},
    "outputs": [],
    "source": [
-    "!pip install --upgrade --quiet phiflow\n",
-    "#!pip install --upgrade --quiet git+https://github.com/tum-pbs/PhiFlow@develop \n",
+    "#!pip install --upgrade --quiet phiflow\n",
     "from phi.tf.flow import *\n",
     "\n",
     "N = 128\n",
+    "DX = 2/N\n",
     "STEPS = 32\n",
-    "DT = 1./STEPS\n",
+    "DT = 1/STEPS\n",
     "NU = 0.01/(N*np.pi)\n",
     "\n",
     "# allocate velocity grid\n",
@@ -92,6 +92,7 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
+      "WARNING:tensorflow:From /Users/thuerey/miniconda3/envs/tf/lib/python3.8/site-packages/tensorflow/python/ops/math_grad.py:297: setdiff1d (from tensorflow.python.ops.array_ops) is deprecated and will be removed after 2018-11-30.\n",
       "Instructions for updating:\n",
       "This op will be removed after the deprecation date. Please switch to tf.sets.difference().\n",
       "Loss: 0.382915\n"
@@ -136,7 +137,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": "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 truncated
+      "image/png": "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 truncated
       "text/plain": [
        "<Figure size 432x288 with 1 Axes>"
       ]
@@ -193,10 +194,10 @@
      "output_type": "stream",
      "text": [
       "Optimization step 0, loss: 0.382915\n",
-      "Optimization step 1, loss: 0.329677\n",
-      "Optimization step 2, loss: 0.284684\n",
-      "Optimization step 3, loss: 0.246604\n",
-      "Optimization step 4, loss: 0.214305\n"
+      "Optimization step 1, loss: 0.326882\n",
+      "Optimization step 2, loss: 0.281032\n",
+      "Optimization step 3, loss: 0.242804\n",
+      "Optimization step 4, loss: 0.210666\n"
      ]
     }
    ],
@@ -237,7 +238,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
        "<Figure size 432x288 with 1 Axes>"
       ]
@@ -289,16 +290,16 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "Optimization step 0, loss: 0.186823\n",
-      "Optimization step 5, loss: 0.098101\n",
-      "Optimization step 10, loss: 0.055802\n",
-      "Optimization step 15, loss: 0.033452\n",
-      "Optimization step 20, loss: 0.020761\n",
-      "Optimization step 25, loss: 0.013248\n",
-      "Optimization step 30, loss: 0.008643\n",
-      "Optimization step 35, loss: 0.005733\n",
-      "Optimization step 40, loss: 0.003848\n",
-      "Runtime 204.01s\n"
+      "Optimization step 0, loss: 0.183476\n",
+      "Optimization step 5, loss: 0.096224\n",
+      "Optimization step 10, loss: 0.054792\n",
+      "Optimization step 15, loss: 0.032819\n",
+      "Optimization step 20, loss: 0.020334\n",
+      "Optimization step 25, loss: 0.012852\n",
+      "Optimization step 30, loss: 0.008185\n",
+      "Optimization step 35, loss: 0.005186\n",
+      "Optimization step 40, loss: 0.003263\n",
+      "Runtime 56.25s\n"
      ]
     }
    ],
@@ -330,7 +331,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Thinking back to the PINN version from {doc}`diffphys-code-burgers`, we have a much lower error here after only 50 steps (by ca. an order of magnitude), and the runtime is also lower (roughly by a factor of 2).\n",
+    "Thinking back to the PINN version from {doc}`diffphys-code-burgers`, we have a much lower error here after only 50 steps (by ca. an order of magnitude), and the runtime is also lower (roughly by a factor of 1.5 to 2). This behavior stems fro\n",
     "\n",
     "Let's plot again how well our solution at $t=0.5$ (blue) matches the constraints (green) now:"
    ]
@@ -342,7 +343,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
        "<Figure size 432x288 with 1 Axes>"
       ]
@@ -376,7 +377,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
        "<Figure size 432x288 with 1 Axes>"
       ]
@@ -392,7 +393,7 @@
     "pltx = np.linspace(-1,1,N)\n",
     "\n",
     "# ground truth state at time=0 , move down\n",
-    "INITIAL_GT = np.asarray( [-np.sin(np.pi * x) * 1. for x in np.linspace(-1,1,N)] ) # 1D array\n",
+    "INITIAL_GT = np.asarray( [-np.sin(np.pi * x) for x in np.linspace(-1+DX/2,1-DX/2,N)] ) # 1D numpy array\n",
     "fig.plot(pltx, INITIAL_GT.flatten()      , lw=2, color='forestgreen', label=\"Ground truth initial state\")  # ground truth initial state of sim\n",
     "fig.plot(pltx, velocity.values.numpy('x'), lw=2, color='mediumblue',  label=\"Optimized initial state\") # manual\n",
     "plt.xlabel('x'); plt.ylabel('u'); plt.legend(); plt.title(\"Initial State After 50 Optimization Steps\");\n"
@@ -414,7 +415,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
        "<Figure size 1152x360 with 2 Axes>"
       ]
@@ -475,7 +476,7 @@
     },
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
        "<Figure size 1152x360 with 2 Axes>"
       ]
@@ -519,15 +520,15 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "MAE PINN: 0.22991 \n",
-      "MAE DP:   0.05811\n",
+      "MAE PINN: 0.23281 \n",
+      "MAE DP:   0.06382\n",
       "\n",
       "Error GT to PINN (top) , DiffPhys (bottom):\n"
      ]
     },
     {
      "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>"
       ]

overview-burgers-forw.ipynb

@@ -2,6 +2,7 @@
  "cells": [
   {
    "cell_type": "markdown",
+   "metadata": {},
    "source": [
     "# Simple Forward Simulation of Burgers Equation with phiflow\n",
     "\n",
@@ -22,45 +23,43 @@
     "    \\nu \\nabla\\cdot \\nabla u\n",
     "$$ \n",
     "\n"
-   ],
-   "metadata": {}
+   ]
   },
   {
    "cell_type": "markdown",
+   "metadata": {},
    "source": [
     "## Importing and loading phiflow\n",
     "\n",
     "Let's get some preliminaries out of the way: first we'll import the phiflow library, more specifically the `numpy` operators for fluid flow simulations: `phi.flow` (differentiable versions for a DL framework _X_ are loaded via `phi.X.flow` instead).\n",
     "\n",
     "**Note:** Below, the first command with a \"!\" prefix will install the [phiflow python package from GitHub](https://github.com/tum-pbs/PhiFlow) via `pip` in your python environment once you uncomment it. We've assumed that phiflow isn't installed, but if you have already done so, just comment out the first line (the same will hold for all following notebooks)."
-   ],
-   "metadata": {}
+   ]
   },
   {
    "cell_type": "code",
    "execution_count": 1,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Using phiflow version: 2.0.3\n"
+     ]
+    }
+   ],
    "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 *\n",
     "\n",
     "from phi import __version__\n",
     "print(\"Using phiflow version: {}\".format(phi.__version__))"
-   ],
-   "outputs": [
-    {
-     "output_type": "stream",
-     "name": "stdout",
-     "text": [
-      "Using phiflow version: 2.0.0rc2\n"
-     ]
-    }
-   ],
-   "metadata": {}
+   ]
   },
   {
    "cell_type": "markdown",
+   "metadata": {},
    "source": [
     "Next we can define and initialize the necessary constants (denoted by upper-case names):\n",
     "our simulation domain will have `N=128` cells as discretization points for the 1D velocity $u$ in a periodic domain $\\Omega$ for the interval $[-1,1]$. We'll use 32 time `STEPS` for a time interval of 1, giving us `DT=1/32`. Additionally, we'll use a viscosity `NU` of $\\nu=0.01/\\pi$.\n",
@@ -70,73 +69,89 @@
     "Phiflow is object-oriented and centered around field data in the form of grids (internally represented by a tensor object). I.e. you assemble your simulation by constructing a number of grids, and updating them over the course of time steps. \n",
     "\n",
     "Phiflow internally works with tensors that have named dimensions. This will be especially handy later on for 2D simulations with additional batch and channel dimensions, but for now we'll simply convert the 1D array into a phiflow tensor that has a single spatial dimension `'x'`.\n"
-   ],
-   "metadata": {}
+   ]
   },
   {
    "cell_type": "code",
-   "execution_count": 2,
+   "execution_count": 3,
+   "metadata": {},
+   "outputs": [],
    "source": [
     "N = 128\n",
+    "DX = 2./N\n",
     "STEPS = 32\n",
     "DT = 1./STEPS\n",
     "NU = 0.01/(N*np.pi)\n",
     "\n",
-    "# initialization of velocities\n",
-    "INITIAL_NUMPY = np.asarray( [-np.sin(np.pi * x) * 1. for x in np.linspace(-1,1,N)] ) # 1D numpy array\n",
+    "# initialization of velocities, cell centers of a CenteredGrid have DX/2 offsets for linspace()\n",
+    "INITIAL_NUMPY = np.asarray( [-np.sin(np.pi * x) for x in np.linspace(-1+DX/2,1-DX/2,N)] ) # 1D numpy array\n",
     "\n",
     "INITIAL = math.tensor(INITIAL_NUMPY, spatial('x') ) # convert to phiflow tensor\n"
-   ],
-   "outputs": [],
-   "metadata": {}
+   ]
   },
   {
    "cell_type": "markdown",
+   "metadata": {},
    "source": [
     "\n",
     "Next, we initialize a 1D `velocity` grid from the `INITIAL` numpy array that was converted into a tensor.\n",
     "The extent of our domain $\\Omega$ is specifiied via the `bounds` parameter $[-1,1]$, and the grid uses periodic boundary conditions (`extrapolation.PERIODIC`). These two properties are the main difference between a tensor and a grid: the latter has boundary conditions and a physical extent.\n",
     "\n",
-    "Just to illustrate, we'll also print some info about the velocity object: it's a `phi.math` tensor with a size of 128. Note that the actual grid content is contained in the `values` of the grid. Below we're printing five entries by using the `numpy()` function to convert the content of the phiflow tensor into a numpy array. For tensors with more dimensions, we'd need to specify the order here, e.g., `'y,x,vector'` for a 2D velocity field. (If we'd call `numpy('x,vector')` below, this would convert the 1D array into one with an additional dimension for each 1D velocity component.)"
-   ],
-   "metadata": {}
-  },
-  {
-   "cell_type": "code",
-   "execution_count": 3,
-   "source": [
-    "velocity = CenteredGrid(INITIAL, extrapolation.PERIODIC, x=N, bounds=Box[-1:1])\n",
-    "#velocity = CenteredGrid(Noise(), extrapolation.PERIODIC, x=N, bounds=Box[-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",
-    "print(\"Velocity tensor entries 10 to 14: \" + format( velocity.values.numpy()[10:15] ))"
-   ],
-   "outputs": [
-    {
-     "output_type": "stream",
-     "name": "stdout",
-     "text": [
-      "Velocity tensor shape: (xˢ=128)\n",
-      "Velocity tensor type: <class 'phi.math._tensors.CollapsedTensor'>\n",
-      "Velocity tensor entries 10 to 14: [0.47480196 0.51774486 0.55942075 0.59972764 0.6385669 ]\n"
-     ]
-    }
-   ],
-   "metadata": {}
-  },
-  {
-   "cell_type": "markdown",
-   "source": [
-    "## Running the simulation\n",
-    "\n",
-    "Now we're ready to run the simulation itself. To ccompute the diffusion and advection components of our model equation we can simply call the existing `diffusion` and `semi_lagrangian` operators in phiflow: `diffuse.explicit(u,...)` computes an explicit diffusion step via central differences for the term $\\nu \\nabla\\cdot \\nabla u$ of our model. Next, `advect.semi_lagrangian(f,u)` is used for a stable first-order approximation of the transport of an arbitrary field `f` by a velocity `u`. In our model we have $\\partial u / \\partial{t} + u \\nabla f$, hence we use the `semi_lagrangian` function to transport the velocity with itself in the implementation:"
-   ],
-   "metadata": {}
+    "Just to illustrate, we'll also print some info about the velocity object: it's a `phi.math` tensor with a size of 128. Note that the actual grid content is contained in the `values` of the grid. Below we're printing five entries by using the `numpy()` function to convert the content of the phiflow tensor into a numpy array. For tensors with more dimensions, we'd need to specify the additional dimenions here, e.g., `'y,x,vector'` for a 2D velocity field. (For tensors with a single dimensions we could leave it out.)"
+   ]
   },
   {
    "cell_type": "code",
    "execution_count": 4,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Velocity tensor shape: (xˢ=128)\n",
+      "Velocity tensor type: <class 'phi.math._tensors.CollapsedTensor'>\n",
+      "Velocity tensor entries 10 to 14: [0.49289819 0.53499762 0.57580819 0.61523159 0.65317284]\n"
+     ]
+    }
+   ],
+   "source": [
+    "velocity = CenteredGrid(INITIAL, extrapolation.PERIODIC, x=N, bounds=Box[-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",
+    "\n",
+    "print(\"Velocity tensor shape: \"   + format( velocity.shape )) # == velocity.values.shape\n",
+    "print(\"Velocity tensor type: \"    + format( type(velocity.values) ))\n",
+    "print(\"Velocity tensor entries 10 to 14: \" + format( velocity.values.numpy('x')[10:15] ))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Running the simulation\n",
+    "\n",
+    "Now we're ready to run the simulation itself. To ccompute the diffusion and advection components of our model equation we can simply call the existing `diffusion` and `semi_lagrangian` operators in phiflow: `diffuse.explicit(u,...)` computes an explicit diffusion step via central differences for the term $\\nu \\nabla\\cdot \\nabla u$ of our model. Next, `advect.semi_lagrangian(f,u)` is used for a stable first-order approximation of the transport of an arbitrary field `f` by a velocity `u`. In our model we have $\\partial u / \\partial{t} + u \\nabla f$, hence we use the `semi_lagrangian` function to transport the velocity with itself in the implementation:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "New velocity content at t=1.0: [[0.0057228 ]\n",
+      " [0.01716715]\n",
+      " [0.02861034]\n",
+      " [0.040052  ]\n",
+      " [0.05149214]]\n"
+     ]
+    }
+   ],
    "source": [
     "velocities = [velocity]\n",
     "age = 0.\n",
@@ -146,21 +161,12 @@
     "    age += DT\n",
     "    velocities.append(v2)\n",
     "\n",
-    "print(\"New velocity content at t={}: {}\".format( age, velocities[-1].values.numpy()[0:5] ))"
-   ],
-   "outputs": [
-    {
-     "output_type": "stream",
-     "name": "stdout",
-     "text": [
-      "New velocity content at t=1.0: [0.00274862 0.01272991 0.02360343 0.03478042 0.0460869 ]\n"
-     ]
-    }
-   ],
-   "metadata": {}
+    "print(\"New velocity content at t={}: {}\".format( age, velocities[-1].values.numpy('x,vector')[0:5] ))"
+   ]
   },
   {
    "cell_type": "markdown",
+   "metadata": {},
    "source": [
     "Here we're actually collecting all time steps in the list `velocities`. This is not necessary in general (and could consume lots of memory for long-running sims), but useful here to plot the evolution of the velocity states later on.\n",
     "\n",
@@ -169,12 +175,36 @@
     "## Visualization\n",
     "\n",
     "We can visualize this 1D case easily in a graph: the following code shows the initial state in blue, and then times $10/32, 20/32, 1$ in green, cyan and purple. "
-   ],
-   "metadata": {}
+   ]
   },
   {
    "cell_type": "code",
-   "execution_count": 5,
+   "execution_count": 6,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "<matplotlib.legend.Legend at 0x7f9fd19fd940>"
+      ]
+     },
+     "execution_count": 6,
+     "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": [
     "# get \"velocity.values\" from each phiflow state with a channel dimensions, i.e. \"vector\"\n",
     "vels = [v.values.numpy('x,vector') for v in velocities] # gives a list of 2D arrays \n",
@@ -187,47 +217,37 @@
     "fig.plot(np.linspace(-1,1,len(vels[20].flatten())), vels[20].flatten(), lw=2, color='cyan',  label=\"t=0.625\")\n",
     "fig.plot(np.linspace(-1,1,len(vels[32].flatten())), vels[32].flatten(), lw=2, color='purple',label=\"t=1\")\n",
     "pylab.xlabel('x'); pylab.ylabel('u'); pylab.legend()\n"
-   ],
-   "outputs": [
-    {
-     "output_type": "execute_result",
-     "data": {
-      "text/plain": [
-       "<matplotlib.legend.Legend at 0x7f9ca2d4ef70>"
-      ]
-     },
-     "metadata": {},
-     "execution_count": 5
-    },
-    {
-     "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": {}
+   ]
   },
   {
    "cell_type": "markdown",
+   "metadata": {},
    "source": [
     "This nicely shows the shock developing in the center of our domain, which forms from the collision of the two initial velocity \"bumps\", the positive one on left (moving right) and the negative one right of the center (moving left).\n",
     "\n",
     "As these lines can overlap quite a bit we'll also use a different visualization in the following chapters that shows the evolution over the course of all time steps in a 2D image. Our 1D domain will be shown along the Y-axis, and each point along X will represent one time step.\n",
     "\n",
     "The code below converts our collection of velocity states into a 2D array, repeating individual time steps 8 times to make the image a bit wider. This is purely optional, of course, but makes it easier to see what's happening in our Burgers simulation."
-   ],
-   "metadata": {}
+   ]
   },
   {
    "cell_type": "code",
-   "execution_count": 6,
+   "execution_count": 7,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "image/png": "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 truncated
+      "text/plain": [
+       "<Figure size 1152x360 with 2 Axes>"
+      ]
+     },
+     "metadata": {
+      "needs_background": "light"
+     },
+     "output_type": "display_data"
+    }
+   ],
    "source": [
     "def show_state(a, title):\n",
     "    # we only have 33 time steps, blow up by a factor of 2^4 to make it easier to see\n",
@@ -250,32 +270,18 @@
     "np.savez_compressed(\"./temp/burgers-groundtruth-solution.npz\", np.reshape(vels_img,[N,STEPS+1])) # remove batch & channel dimension\n",
     "\n",
     "show_state(vels_img, \"Velocity\")"
-   ],
-   "outputs": [
-    {
-     "output_type": "display_data",
-     "data": {
-      "text/plain": [
-       "<Figure size 1152x360 with 2 Axes>"
-      ],
-      "image/png": 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truncated
-     },
-     "metadata": {
-      "needs_background": "light"
-     }
-    }
-   ],
-   "metadata": {}
+   ]
   },
   {
    "cell_type": "markdown",
+   "metadata": {},
    "source": [
     "This concludes a first simulation in phiflow. It's not overly complex, but because of that it's a good starting point for evaluating and comparing different physics-based deep learning approaches in the next chapter. But before that, we'll target a more complex simulation type in the next section."
-   ],
-   "metadata": {}
+   ]
   },
   {
    "cell_type": "markdown",
+   "metadata": {},
    "source": [
     "## Next steps\n",
     "\n",
@@ -284,8 +290,7 @@
     "- Feel free to experiment - the setup above is very simple, you can change the simulation parameters, or the initialization. E.g., you can use a noise field via `Noise()` to get more chaotic results (cf. the comment in the `velocity` cell above).\n",
     "\n",
     "- A bit more complicated: extend the simulation to 2D (or higher). This will require changes throughout, but all operators above support higher dimensions. Before trying this, you probably will want to check out the next example, which covers a 2D Navier-Stokes case."
-   ],
-   "metadata": {}
+   ]
   }
  ],
  "metadata": {

physicalloss-code.ipynb

@@ -60,7 +60,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": null,
+   "execution_count": 1,
    "metadata": {
     "colab": {
      "base_uri": "https://localhost:8080/"
@@ -68,7 +68,32 @@
     "id": "L30jgXowoXat",
     "outputId": "bcceefc0-fb33-44c7-959c-0b742e5a1606"
    },
-   "outputs": [],
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "/bin/bash: pip: command not found\n",
+      "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",
+      "WARNING:tensorflow:From /Users/thuerey/Dropbox/mbaDevelSelected/phiflow-v1.5/phi/tf/util.py:119: The name tf.AUTO_REUSE is deprecated. Please use tf.compat.v1.AUTO_REUSE instead.\n",
+      "\n",
+      "WARNING:tensorflow:From /Users/thuerey/Dropbox/mbaDevelSelected/phiflow-v1.5/phi/tf/profiling.py:12: The name tf.RunOptions is deprecated. Please use tf.compat.v1.RunOptions instead.\n",
+      "\n",
+      "WARNING:tensorflow:From /Users/thuerey/Dropbox/mbaDevelSelected/phiflow-v1.5/phi/tf/profiling.py:13: The name tf.RunMetadata is deprecated. Please use tf.compat.v1.RunMetadata instead.\n",
+      "\n"
+     ]
+    },
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "/Users/thuerey/Dropbox/mbaDevelSelected/phiflow-v1.5/phi/tf/flow.py:15: UserWarning: TensorFlow-CUDA solver is not available. To compile it, download phiflow sources and run\n",
+      "$ python setup.py tf_cuda\n",
+      "before reinstalling phiflow.\n",
+      "  warnings.warn(\"TensorFlow-CUDA solver is not available. To compile it, download phiflow sources and run\\n$ python setup.py tf_cuda\\nbefore reinstalling phiflow.\")\n"
+     ]
+    }
+   ],
    "source": [
     "!pip install --quiet phiflow==1.5.1\n",
     "\n",
@@ -186,8 +211,10 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
+      "WARNING:tensorflow:From <ipython-input-2-f2ebc84042fb>:7: dense (from tensorflow.python.layers.core) is deprecated and will be removed in a future version.\n",
       "Instructions for updating:\n",
       "Use keras.layers.Dense instead.\n",
+      "WARNING:tensorflow:From /usr/local/lib/python3.7/site-packages/tensorflow_core/python/layers/core.py:187: Layer.apply (from tensorflow.python.keras.engine.base_layer) is deprecated and will be removed in a future version.\n",
       "Instructions for updating:\n",
       "Please use `layer.__call__` method instead.\n"
      ]
@@ -232,12 +259,20 @@
      "output_type": "stream",
      "text": [
       "Size of grid_u: (1, 128, 33, 1)\n",
+      "WARNING:tensorflow:From /Users/thuerey/Dropbox/mbaDevelSelected/phiflow-v1.5/phi/tf/session.py:17: The name tf.Session is deprecated. Please use tf.compat.v1.Session instead.\n",
+      "\n",
+      "WARNING:tensorflow:From /Users/thuerey/Dropbox/mbaDevelSelected/phiflow-v1.5/phi/tf/session.py:18: The name tf.get_default_graph is deprecated. Please use tf.compat.v1.get_default_graph instead.\n",
+      "\n",
+      "WARNING:tensorflow:From /Users/thuerey/Dropbox/mbaDevelSelected/phiflow-v1.5/phi/tf/session.py:28: The name tf.global_variables_initializer is deprecated. Please use tf.compat.v1.global_variables_initializer instead.\n",
+      "\n",
+      "WARNING:tensorflow:From /Users/thuerey/Dropbox/mbaDevelSelected/phiflow-v1.5/phi/tf/session.py:29: The name tf.train.Saver is deprecated. Please use tf.compat.v1.train.Saver instead.\n",
+      "\n",
       "Randomly initialized network state:\n"
      ]
     },
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
        "<Figure size 1152x360 with 2 Axes>"
       ]
@@ -302,7 +337,17 @@
     "id": "bfZvwnFooXa1",
     "outputId": "f0188a14-5922-4faa-b1aa-fb6482bd81f3"
    },
-   "outputs": [],
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "WARNING:tensorflow:From /usr/local/lib/python3.7/site-packages/tensorflow_core/python/ops/math_grad.py:1375: where (from tensorflow.python.ops.array_ops) is deprecated and will be removed in a future version.\n",
+      "Instructions for updating:\n",
+      "Use tf.where in 2.0, which has the same broadcast rule as np.where\n"
+     ]
+    }
+   ],
    "source": [
     "# Boundary loss\n",
     "N_SAMPLE_POINTS_BND = 100\n",
@@ -350,20 +395,20 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "Step 0, loss: 0.108430\n",
-      "Step 1, loss: 0.103907\n",
-      "Step 2, loss: 0.100539\n",
-      "Step 1000, loss: 0.057177\n",
-      "Step 2000, loss: 0.053230\n",
-      "Step 3000, loss: 0.049326\n",
-      "Step 4000, loss: 0.046316\n",
-      "Step 5000, loss: 0.044153\n",
-      "Step 6000, loss: 0.041827\n",
-      "Step 7000, loss: 0.039517\n",
-      "Step 8000, loss: 0.037237\n",
-      "Step 9000, loss: 0.035256\n",
-      "Step 10000, loss: 0.033931\n",
-      "Runtime 110.15s\n"
+      "Step 0, loss: 0.095709\n",
+      "Step 1, loss: 0.093502\n",
+      "Step 2, loss: 0.091750\n",
+      "Step 1000, loss: 0.050177\n",
+      "Step 2000, loss: 0.044144\n",
+      "Step 3000, loss: 0.039773\n",
+      "Step 4000, loss: 0.034952\n",
+      "Step 5000, loss: 0.032054\n",
+      "Step 6000, loss: 0.030908\n",
+      "Step 7000, loss: 0.029630\n",
+      "Step 8000, loss: 0.028747\n",
+      "Step 9000, loss: 0.028692\n",
+      "Step 10000, loss: 0.020781\n",
+      "Runtime 88.29s\n"
      ]
     }
    ],
@@ -411,7 +456,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": "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 truncated
+      "image/png": "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 truncated
       "text/plain": [
        "<Figure size 1152x360 with 2 Axes>"
       ]
@@ -441,7 +486,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 10,
+   "execution_count": 9,
    "metadata": {
     "colab": {
      "base_uri": "https://localhost:8080/",
@@ -454,16 +499,16 @@
     {
      "data": {
       "text/plain": [
-       "<matplotlib.legend.Legend at 0x7f84721c3fd0>"
+       "<matplotlib.legend.Legend at 0x11f9b5b90>"
       ]
      },
-     "execution_count": 10,
+     "execution_count": 9,
      "metadata": {},
      "output_type": "execute_result"
     },
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
        "<Figure size 432x288 with 1 Axes>"
       ]
@@ -503,7 +548,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 11,
+   "execution_count": 10,
    "metadata": {
     "colab": {
      "base_uri": "https://localhost:8080/",
@@ -516,16 +561,16 @@
     {
      "data": {
       "text/plain": [
-       "<matplotlib.legend.Legend at 0x7f847223da60>"
+       "<matplotlib.legend.Legend at 0x11fa4db90>"
       ]
      },
-     "execution_count": 11,
+     "execution_count": 10,
      "metadata": {},
      "output_type": "execute_result"
     },
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
        "<Figure size 432x288 with 1 Axes>"
       ]
@@ -566,7 +611,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 12,
+   "execution_count": 11,
    "metadata": {
     "colab": {
      "base_uri": "https://localhost:8080/",
@@ -586,16 +631,16 @@
     {
      "data": {
       "text/plain": [
-       "<matplotlib.legend.Legend at 0x7f84722de760>"
+       "<matplotlib.legend.Legend at 0x11faef650>"
       ]
      },
-     "execution_count": 12,
+     "execution_count": 11,
      "metadata": {},
      "output_type": "execute_result"
     },
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
        "<Figure size 432x288 with 1 Axes>"
       ]
@@ -645,7 +690,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 13,
+   "execution_count": 12,
    "metadata": {
     "colab": {
      "base_uri": "https://localhost:8080/",
@@ -665,16 +710,16 @@
     {
      "data": {
       "text/plain": [
-       "<matplotlib.legend.Legend at 0x7f8470cfb1f0>"
+       "<matplotlib.legend.Legend at 0x11fb76c50>"
       ]
      },
-     "execution_count": 13,
+     "execution_count": 12,
      "metadata": {},
      "output_type": "execute_result"
     },
     {
      "data": {
-      "image/png": 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truncated
+      "image/png": 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truncated
       "text/plain": [
        "<Figure size 432x288 with 1 Axes>"
       ]
@@ -709,7 +754,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 14,
+   "execution_count": 13,
    "metadata": {
     "colab": {
      "base_uri": "https://localhost:8080/",
@@ -723,22 +768,22 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "Mean absolute error for re-simulation across 16 steps: 0.01067\n"
+      "Mean absolute error for re-simulation across 16 steps: 0.02124\n"
      ]
     },
     {
      "data": {
       "text/plain": [
-       "<matplotlib.legend.Legend at 0x7f8472550e50>"
+       "<matplotlib.legend.Legend at 0x11fc06910>"
       ]
      },
-     "execution_count": 14,
+     "execution_count": 13,
      "metadata": {},
      "output_type": "execute_result"
     },
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
        "<Figure size 432x288 with 1 Axes>"
       ]
@@ -775,7 +820,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 15,
+   "execution_count": 14,
    "metadata": {
     "colab": {
      "base_uri": "https://localhost:8080/",
@@ -788,7 +833,7 @@
    "outputs": [
     {
      "data": {
-      "image/png": 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truncated
       "text/plain": [
        "<Figure size 1152x360 with 2 Axes>"
       ]
@@ -819,7 +864,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 16,
+   "execution_count": 15,
    "metadata": {
     "colab": {
      "base_uri": "https://localhost:8080/"
@@ -886,7 +931,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.8.5"
+   "version": "3.7.6"
   }
  },
  "nbformat": 4,