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phiflow 2.2 updates

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NT committed 2022-08-27 13:53:10 +02:00
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commit 8502c1e36c
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diffphys-code-burgers.ipynb
+135 -92
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@@ -2,7 +2,9 @@
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"metadata": {
"id": "BiKkPaSYZKnF"
},
"source": [
"# Burgers Optimization with a Differentiable Physics Gradient\n",
"\n",
@@ -25,10 +27,12 @@
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"metadata": {
"id": "W69db7SUZKnI"
},
"outputs": [],
"source": [
"!pip install --upgrade --quiet phiflow==2.2\n",
"#!pip install --upgrade --quiet phiflow==2.2\n",
"from phi.tf.flow import *\n",
"\n",
"N = 128\n",
@@ -47,7 +51,9 @@
},
{
"cell_type": "markdown",
"metadata": {},
"metadata": {
"id": "lewJYuRzZKnL"
},
"source": [
"Below we verify that the fields of our simulation are now backed by TensorFlow."
]
@@ -55,7 +61,10 @@
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"metadata": {
"id": "5u03jWPfZKnM",
"outputId": "6606726e-3498-4419-da44-021f8e30adda"
},
"outputs": [
{
"data": {
@@ -74,7 +83,9 @@
},
{
"cell_type": "markdown",
"metadata": {},
"metadata": {
"id": "FBoE0HsOZKnN"
},
"source": [
"## Gradients\n",
"\n",
@@ -86,38 +97,41 @@
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"metadata": {
"id": "WjRnOIihZKnO",
"outputId": "5851883a-1635-48a9-bb7f-1d254c2983af"
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Instructions for updating:\n",
"This op will be removed after the deprecation date. Please switch to tf.sets.difference().\n",
"Loss: 0.382915\n"
]
}
],
"source": [
"velocities = [velocity]\n",
"\n",
"with math.record_gradients(velocity.values):\n",
"\n",
"def loss_function(velocity):\n",
" velocities = [velocity]\n",
" for time_step in range(STEPS):\n",
" v1 = diffuse.explicit(1.0*velocities[-1], NU, DT, substeps=1)\n",
" v2 = advect.semi_lagrangian(v1, v1, DT)\n",
" velocities.append(v2)\n",
"\n",
" loss = field.l2_loss(velocities[16] - SOLUTION_T16)*2./N # MSE\n",
" return loss, velocities\n",
"\n",
" grad = math.gradients(loss, velocity.values)\n",
"gradient_function = math.gradient(loss_function)\n",
"\n",
"(loss,velocities), grad = gradient_function(velocity)\n",
"\n",
"print('Loss: %f' % (loss))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"metadata": {
"id": "n7PQQAP6ZKnP"
},
"source": [
"Because we're only constraining time step 16, we could actually omit steps 17 to 31 in this setup. They don't have any degrees of freedom and are not constrained in any way. However, for fairness regarding a comparison with the previous case, we include them.\n",
"\n",
@@ -132,11 +146,14 @@
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"metadata": {
"id": "ujFnJy0eZKnQ",
"outputId": "fd89f5c8-c34b-4f0b-a5e4-82443d831f75"
},
"outputs": [
{
"data": {
"image/png": 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truncated
"image/png": "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 truncated
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
@@ -154,19 +171,21 @@
"pltx = np.linspace(-1,1,N)\n",
"\n",
"# first gradient\n",
"fig.plot(pltx, grad.numpy('x') , lw=2, color='green', label=\"Gradient\") \n",
"fig.plot(pltx, grad[0].values.numpy('x') , lw=2, color='green', label=\"Gradient\") \n",
"fig.plot(pltx, velocity.values.numpy('x'), lw=2, color='mediumblue', label=\"u at t=0\")\n",
"plt.xlabel('x'); plt.ylabel('u'); plt.legend();\n",
"\n",
"# some (optional) other fields to plot:\n",
"#fig.plot(pltx, (velocities[16]).values.numpy('x') , lw=2, color='cyan', label=\"u at t=0.5\") \n",
"#fig.plot(pltx, (SOLUTION_T16).values.numpy('x') , lw=2, color='red', label=\"solution at t=0.5\") \n",
"#fig.plot(pltx, (velocities[16] - SOLUTION_T16).values.numpy('x') , lw=2, color='blue', label=\"difference at t=0.5\") \n"
"# fig.plot(pltx, (velocities[16]).values.numpy('x') , lw=2, color='cyan', label=\"u at t=0.5\") \n",
"# fig.plot(pltx, (SOLUTION_T16).values.numpy('x') , lw=2, color='red', label=\"solution at t=0.5\") \n",
"# fig.plot(pltx, (velocities[16] - SOLUTION_T16).values.numpy('x') , lw=2, color='blue', label=\"difference at t=0.5\") \n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"metadata": {
"id": "pEsbDiF4ZKnT"
},
"source": [
"This gives us a \"search direction\" for each velocity variable. Based on a linear approximation, the gradient tells us how to change each of them to increase the loss function (gradients _always_ point \"upwards\"). Thus, we can use the gradient to run an optimization and find an initial state `velocity` that minimizes our loss.\n",
"\n"
@@ -174,7 +193,9 @@
},
{
"cell_type": "markdown",
"metadata": {},
"metadata": {
"id": "m8fc8eMeZKnU"
},
"source": [
"## Optimization \n",
"\n",
@@ -186,7 +207,10 @@
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"metadata": {
"id": "fGVc9zQcZKnV",
"outputId": "b27673a4-1b17-4ba7-a707-523248fb8bf0"
},
"outputs": [
{
"name": "stdout",
@@ -205,25 +229,18 @@
"\n",
"grads=[]\n",
"for optim_step in range(5):\n",
" velocities = [velocity]\n",
" with math.record_gradients(velocity.values):\n",
" for time_step in range(STEPS):\n",
" v1 = diffuse.explicit(1.0*velocities[-1], NU, DT)\n",
" v2 = advect.semi_lagrangian(v1, v1, DT)\n",
" velocities.append(v2)\n",
" (loss,velocities), grad = gradient_function(velocity)\n",
" print('Optimization step %d, loss: %f' % (optim_step,loss))\n",
" grads.append( grad[0] )\n",
"\n",
" loss = field.l2_loss(velocities[16] - SOLUTION_T16)*2./N # MSE\n",
" print('Optimization step %d, loss: %f' % (optim_step,loss))\n",
"\n",
" grads.append( math.gradients(loss, velocity.values) )\n",
"\n",
" velocity = velocity - LR * grads[-1]\n",
"\n"
" velocity = velocity - LR * grads[-1]\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"metadata": {
"id": "pXTveI7uZKnW"
},
"source": [
"\n",
"Now we'll check well the 16th state of the simulation actually matches the target after the 5 update steps. This is what the loss measures, after all. The next graph shows the constraints (i.e. the solution we'd like to obtain) in green, and the reconstructed state after the initial state `velocity` (which we updated five times via the gradient by now) was updated 16 times by the solver. \n",
@@ -233,11 +250,14 @@
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"metadata": {
"id": "ce-MV-4eZKnX",
"outputId": "1ec5844e-2915-4656-dec3-585260fdfa5c"
},
"outputs": [
{
"data": {
"image/png": 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truncated
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
@@ -262,7 +282,9 @@
},
{
"cell_type": "markdown",
"metadata": {},
"metadata": {
"id": "2r8mVjyrZKnY"
},
"source": [
"This seems to be going in the right direction! It's definitely not perfect, but we've only computed 5 GD update steps so far. The two peaks with a positive velocity on the left side of the shock and the negative peak on the right side are starting to show.\n",
"\n",
@@ -272,7 +294,9 @@
},
{
"cell_type": "markdown",
"metadata": {},
"metadata": {
"id": "23T8WOKUZKnZ"
},
"source": [
"## More optimization steps\n",
"\n",
@@ -283,22 +307,25 @@
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"metadata": {
"id": "JIpn4s4YZKnZ",
"outputId": "85274162-8128-4e2b-a6a6-76a7d3a07c62"
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"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"
"Optimization step 5, loss: 0.183476\n",
"Optimization step 10, loss: 0.096224\n",
"Optimization step 15, loss: 0.054792\n",
"Optimization step 20, loss: 0.032819\n",
"Optimization step 25, loss: 0.020334\n",
"Optimization step 30, loss: 0.012852\n",
"Optimization step 35, loss: 0.008185\n",
"Optimization step 40, loss: 0.005186\n",
"Optimization step 45, loss: 0.003263\n",
"Runtime 130.33s\n"
]
}
],
@@ -306,21 +333,11 @@
"import time\n",
"start = time.time()\n",
"\n",
"for optim_step in range(45):\n",
" velocities = [velocity]\n",
" with math.record_gradients(velocity.values):\n",
" for time_step in range(STEPS):\n",
" v1 = diffuse.explicit(1.0*velocities[-1], NU, DT)\n",
" v2 = advect.semi_lagrangian(v1, v1, DT)\n",
" velocities.append(v2)\n",
"\n",
" loss = field.l2_loss(velocities[16] - SOLUTION_T16)*2./N # MSE\n",
" if optim_step%5==0: \n",
" print('Optimization step %d, loss: %f' % (optim_step,loss))\n",
"\n",
" grad = math.gradients(loss, velocity.values)\n",
"\n",
" velocity = velocity - LR * grad\n",
"for optim_step in range(5,50):\n",
" (loss,velocities), grad = gradient_function(velocity)\n",
" velocity = velocity - LR * grad[0]\n",
" if optim_step%5==0: \n",
" print('Optimization step %d, loss: %f' % (optim_step,loss))\n",
"\n",
"end = time.time()\n",
"print(\"Runtime {:.2f}s\".format(end-start))"
@@ -328,9 +345,11 @@
},
{
"cell_type": "markdown",
"metadata": {},
"metadata": {
"id": "JA07WBp8ZKna"
},
"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 1.5 to 2). This behavior stems from DP providing gradients for the whole solutions with all its discretization points and time steps, rather than localized updates.\n",
"Thinking back to the PINN version from {doc}`diffphys-code-burgers`, the error decreases much more strongly (by ca. two orders of magnitude) with a comparable runtime. This behavior stems from DP providing gradients for the whole solutions with all its discretization points and time steps, rather than localized updates.\n",
"\n",
"Let's plot again how well our solution at $t=0.5$ (blue) matches the constraints (green) now:"
]
@@ -338,11 +357,14 @@
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"metadata": {
"id": "a3BFszmLZKnb",
"outputId": "d341b7dc-0794-43c3-fb1b-a6b9bc2615dd"
},
"outputs": [
{
"data": {
"image/png": 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truncated
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAZAAAAEWCAYAAABIVsEJAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjUuMiwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy8qNh9FAAAACXBIWXMAAAsTAAALEwEAmpwYAABT1UlEQVR4nO3dd3gU1dfA8e9JD4SShF5CQJGiNAURVECKSrEX7KggiiJgQVEUARuKgjRFuh3bqwKiqBQLAgKKBRQpUhJagFBCeva8f8wkvyUklJBkU87nefJkd+bOzMnsZs/ee2fuFVXFGGOMOVV+vg7AGGNM8WQJxBhjTJ5YAjHGGJMnlkCMMcbkiSUQY4wxeWIJxBhjTJ5YAimFRORCEdkgIgkicrWv4/EFEblVRL7J47YXi8j6ohSTMb5gCaQEE5ElIhIvIsHZVo0EJqpqmKp+LiIqImfm87G3iEiSm6QSsn8wishDIrJLRA6JyIwcYvQuGywiL4rINnefG0RksIjIScYS7f6NAZnLVPU9Vb00L3+bqv6oqg3ysm1BxXQSx3tSRP5zX4sYEfnQa90SEelTEMfNKxHpICIxJygzS0Sey+P+I0TkMxE5IiJbReSW45QdLiJpXu/lBBGpl5fjljSWQEooEYkGLgYUuDLb6jrA2nw6TsBxVl/hJqkw7w9GEbkMGAJ0cmOpB4w4zn4+dst2A8oBtwN9gXGnGX6pICK9cM5ZZ1UNA1oCC30blc9NAlKBqsCtwBsicvZxyn/o9V4OU9XNhRJlUaeq9lMCf4BhwFJgDDDPa/kmwAMkAQnAMpwkc8R93tMt1wNYAxwAfgaaeu1jC/A48AeQAgTkcPwtOB9YOcX2PvCC1/NOwK5cynYCkoHa2Za3BjKAM93nS4AXgV+AQ8AXQIS7bpv7Nya4P22AO4GfvPanwP3ABuAw8Cxwhvu3HwI+AoLcsh2AGPdxT6/9JrjnY4m7rjvwm7v9dmC41/FOJqa2wErgoPu7rde6JW6MS914vwEq5XIOJwKv5bLuefc8JrtxTHSXNwS+BfYD64EbvbaZBUx21x8GvgfquOsEGAvscf/uP4Fzcjn2XcDf7j42A/e6y8vivD89XuenRrZt+wJpOEkgAZh7Cv8bZd3tzvJa9g4wKpfyw4F3ff0/XRR/fB6A/RTQCwsb3Q/E89x/tKpe67bg9eHufpCd6fW8hfsB0BrwB3q52wR7bb8GqA2E5nL8LcBuIM79cGvmte533ETlPq/kxhCZw35GAd/ncoytXh86S4BY4Bz3A+LTzH96INrdf4DXtndybAL5AigPnI2TCBbi1I4qAOuAXm7ZDrgJJFs85d0PxHu9yjXBqek3dc/H1ScTExABxOPUHAKAm93nkV5/7ybgLCDUfZ7bB+BtOIlgME7twz/b+iVAH6/nZXES3l3usVsAe4HG7vpZOB/67YBgnJpgZtyXAauBijjJpBFQPZe4uuMkaQHaA4nAucc7x9m2nwU8l23ZPJwvPTn9zPN6fydm2+5RcklCOAnkoHsO1wL9fP3/XVR+rAmrBBKRi3Cahj5S1dU4HzS5tvHmoC/wpqquUNUMVX0L5wP1Aq8y41V1u6om5bKPW3E+JOsAi4EFIlLRXReG8w+ZKfNxuRz2UwnYmcsxdrrrM72jqn+p6hHgaeBGEfHPZducvKyqh1R1LfAX8I2qblbVg8BXOB88ORIRP5ya1RJVfRNAVZeo6p+q6lHVP4APcD4oT0Z3YIOqvqOq6ar6AfAPcIVXmZmq+q/7GnwENM9pR6r6LvAgzof798AeEXn8OMfuAWxR1ZnusX/DScg3eJX5UlV/UNUUYCjQRkRq43xZKYdTgxFV/VtVc3z9VPVLVd2kju9xvmhcfPzTcnyq2kNVK+by08MtFoZTO/J2kJzff+Cc20ZAZeAeYJiI3Hw6cZYUlkBKpl44H3573efvu8tOVh3gERE5kPmDU9uo4VVm+/F2oKpLVTVJVRNV9UWcb4CZHw4JON/WM2U+PpzDrvYC1XM5THV3fU4xbQUCOTrBnMhur8dJOTwPO862z+N8AA3IXCAirUVksYjEichB4L5TiKcGzt/gbStQ0+v5Lq/HiceLT50O+s44NYP7gGfdvqic1AFaZ3v9bwWqeZXJOteqmoDz7byGqi7CaTKbhJOopoiI92udRUS6ishyEdnvHqMbp/Z65VX29x/u85zef6jqOlXd4X6Z+hmnxnV9AcdYLFgCKWFEJBS4EWjvXuW0C3gIaCYizU5yN9uB57N9eyvjfgvOdKrDOCtOUwU4zQDesTQDdqvqvhy2+w7nw6y290IRaY2T1BZ5LfYuE4XzbXhvHmI9JSJyE04T0/Wqmua16n1gDk7/TQWcfoPMc3CimHbgfJB7i8JppsszVU1T1Y9x+q/OySWW7TjNht6vf5iq9vMqk3WuRSQMp8lth3uM8ap6HtAYp4ltcPY43KvuPgVewWlerQjM5+TPT45lROSrbFdLef985Rb7FwgQkfpemzbj5C8s8X4vl2qWQEqeq3E6RRvjNGk0x6l+/wjckcs2u3Ha+jNNBe5zv0GLiJQVke4iklsV/ygiEuXeaxIkIiEiMhjnm+VSt8jbQG8Raew2az2F0559DFX9Dqcv4lMROVtE/EXkAuBd4A1V3eBV/DZ3n2VwLlX+RFUzcPphPNn+xnwhIi2ACTh9G3HZVpcD9qtqsoicz9HNiCeKaT5wlojcIiIBItIT5zWdl4cY78x8/UTET0S64vTzrHCLZH/957nHvl1EAt2fViLSyKtMNxG5SESCcDrzl6vqdrdcaxEJxLkwI9n9O7MLwuk/iQPS3Zi8L2HeDUSKSIXj/GnZ40ZVu+rRV0t5/3R1yxwB/g8Y6b63LwSuwulIz+n8XSUi4e7/wvk4tcwvjhNXqWEJpOTphdM2vk1Vd2X+4DQr3JrLZbfDgbfc5oobVXUVTlvvRJyO2404HbwnqxzwhrttLHA50DWzhqGqXwMv4/SNbMNpmnnmOPu7zi37NU7zw7vAdJx2fW/v4CSiXUAIbnOSqibiNDEtdf/GC8g/VwHhwE85fNO9H+dD6jDOVXEfZW50opjcc9UDeATYBzwG9PBqljwVh4Ancc71AZxz309Vf3LXjwOuF+eeofGqehjnw/wmnFrFLuAlnA/8TO/jvGb7cS7UuM1dXh7nC0g8zuu6DxidPSD3GANwzkk8TnKd47X+H5w+o83u+amRfR8474HG7vrPT+F8gPPahOJcLPIBzvlYC1k3iiZ4lb0J53/gMM6Xn5fcfsFST1RtQilT/InIEpyrrqb5OpaSTkRm4Vwh9ZSvYzG+ZTUQY4wxeWIJxBhjTJ5YE5Yxxpg8sRqIMcaYPDneQHglTqVKlTQ6OtrXYRhjTLGyevXqvapaOfvyUpVAoqOjWbVqla/DMMaYYkVEso+KAFgTljHGmDyyBGKMMSZPLIEYY4zJE5/2gYjIDJzhGvao6jk5rBecYRa64Yw2eqeq/uqu64UzhhI4cwLY0ALGnKa0tDRiYmJITk72dSjGB0JCQqhVqxaBgYEnVd7XneizcMZbejuX9V2B+u5Pa5zxlVqLSATOODwtcUbGXC0ic1Q1vsAjNqYEi4mJoVy5ckRHRyMnN+W8KSFUlX379hETE0PdunVPahufNmGp6g84g7Hl5irgbXfCmeVARRGpjjMxzrequt9NGt/iDNhnjDkNycnJREZGWvIohUSEyMjIU6p9FvU+kJocPUlQjLsst+XHEJG+IrJKRFbFxWUfbdsYk50lj9LrVF97XzdhFThVnQJMAWjZsqWN22IKVWpGKqtjV/Pnrj/5J+4fUtJTuLLxlXQ6oxMBfiX+38+UcEX9HRzL0bPM1XKXxQIdsi1fUmhRGXMcR1KPsHDTQr7+92t+2vITR9KOHLX+6w1fUy2sGqO7jaZtVFsfRVl0+fv706RJE9LT06lbty7vvPMOFStWzLV8XFwcPXr0IDU1lfHjx3Pxxac1rbo5BUU9gcwB+ovIbJxO9IOqulNEFgAviEi4W+5S4AlfBWmMqrIyZiWz/5jNNxu+ISk9KWvdWZXOonWt1jSs0pDE1ETe+/09tsRv4cM/PrQEkoPQ0FDWrFkDQK9evZg0aRJDhw7NtfzChQtp0qQJ06ad/FQwGRkZ+Pv7n26opZ6vL+P9AKcmUUlEYnCurAoEUNXJONN6dsOZDSwRuMtdt19EngVWursaqarH64w3pkCkpKfw2brPmLV6Fhv2/W923RY1WtCjYQ+6nNGFmhWO7p47M/JM7vr0LuKT7KLBE2nTpg1//PEHAJs2beKBBx4gLi6OMmXKMHXqVJKTk3nsscdISkpi1apVLFu2jB9//JFnnnmGlJQUzjjjDGbOnElYWBjR0dH07NmTb7/9lscee4yIiIhcy/Xq1Yu5c+eSlpbGxx9/TMOGDUlISODBBx9k1apViAjPPPMM1113Hd98802O+ykNfJpAVPXmE6xX4IFc1s0AZhREXMacyP7E/bz3+3u889s77EvcB0DlspW5ockN3HjOjdSqUItly44w7vkDfPfd3zRoEMILL9Sgbt1gKoZWBCjyCeSMV84okP1uenTTSZXLyMhg4cKF9O7dG4C+ffsyefJk6tevz4oVK7j//vtZtGgRI0eOZNWqVUycOJG9e/fy3HPP8d1331G2bFleeuklxowZw7BhwwCIjIzk119/Ze/evVx77bW5lqtUqRK//vorr7/+Oq+88grTpk3j2WefpUKFCvz5558AxMfHn/B4JV1Rb8IypkjZdXgXk3+ZzMd/fkxyunO549lVz6ZPyz50Pasrgf6BbN2aQpfrN7Jw4eGs7X77LYnPPjvAkCFV6T3IaXkt6gnEV5KSkmjevDmxsbE0atSILl26kJCQwM8//8wNN9yQVS4lJeWYbZcvX866deu48MILAUhNTaVNmzZZ63v27HlS5a699loAzjvvPP7v//4PgO+++47Zs2dnlQkPD2fevHnH3U9Line truncated
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
@@ -362,7 +384,9 @@
},
{
"cell_type": "markdown",
"metadata": {},
"metadata": {
"id": "BX8hUE0XZKnd"
},
"source": [
"Not bad. But how well is the initial state recovered via backpropagation through the 16 simulation steps? This is what we're changing, and because it's only indirectly constrained via the observation later in time there is more room to deviate from a desired or expected solution.\n",
"\n",
@@ -372,11 +396,14 @@
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"metadata": {
"id": "3oBdj2EaZKnd",
"outputId": "1e7dbd37-8d14-4acc-f526-1bab26c6a3a4"
},
"outputs": [
{
"data": {
"image/png": 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truncated
"image/png": "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 truncated
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
@@ -400,7 +427,9 @@
},
{
"cell_type": "markdown",
"metadata": {},
"metadata": {
"id": "Ug2pTemYZKne"
},
"source": [
"Naturally, this is a tougher task: the optimization receives direct feedback what the state at $t=0.5$ should look like, but due to the non-linear model equation, we typically have a large number of solutions that exactly or numerically very closely satisfy the constraints. Hence, our minimizer does not necessarily find the exact state we started from (we can observe some numerical oscillations from the diffusion operator here with the default settings). However, the solution is still quite close in this Burgers scenario.\n",
"\n",
@@ -410,11 +439,14 @@
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"metadata": {
"id": "HiaIZgUlZKnf",
"outputId": "6fb48869-39bb-422d-a9bd-4fd1e33d57ce"
},
"outputs": [
{
"data": {
"image/png": 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truncated
"image/png": 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truncated
"text/plain": [
"<Figure size 1152x360 with 2 Axes>"
]
@@ -451,7 +483,9 @@
},
{
"cell_type": "markdown",
"metadata": {},
"metadata": {
"id": "ozka-l1UZKng"
},
"source": [
"## Physics-informed vs. differentiable physics reconstruction\n",
"\n",
@@ -463,7 +497,10 @@
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"metadata": {
"id": "1b0CXDEDZKng",
"outputId": "ee4c501b-47a7-4673-9955-0a503f7001bd"
},
"outputs": [
{
"name": "stdout",
@@ -475,7 +512,7 @@
},
{
"data": {
"image/png": 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truncated
"text/plain": [
"<Figure size 1152x360 with 2 Axes>"
]
@@ -501,7 +538,9 @@
},
{
"cell_type": "markdown",
"metadata": {},
"metadata": {
"id": "D8qbYQpHZKni"
},
"source": [
"It's quite clearly visible here that the PINN solution (in the middle) recovers the overall shape of the solution, hence the temporal constraints are at least partially fulfilled. However, it doesn't manage to capture the amplitudes of the GT solution very well.\n",
"\n",
@@ -512,22 +551,24 @@
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"execution_count": 13,
"metadata": {
"id": "VoxYVPwjZKni"
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"MAE PINN: 0.23281 \n",
"MAE PINN: 0.19298 \n",
"MAE DP: 0.06382\n",
"\n",
"Error GT to PINN (top) , DiffPhys (bottom):\n"
"Error GT to PINN (top) , GT to DiffPhys (bottom):\n"
]
},
{
"data": {
"image/png": 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truncated
"text/plain": [
"<Figure size 1152x360 with 2 Axes>"
]
@@ -543,15 +584,17 @@
"err_dp = np.sum( np.abs(sol_dp-sol_gt)) / (STEPS*N)\n",
"print(\"MAE PINN: {:7.5f} \\nMAE DP: {:7.5f}\".format(err_pi,err_dp))\n",
"\n",
"print(\"\\nError GT to PINN (top) , DiffPhys (bottom):\")\n",
"print(\"\\nError GT to PINN (top) , GT to DiffPhys (bottom):\")\n",
"show_state(np.reshape( np.concatenate([sol_pi-sol_gt, divider, sol_dp-sol_gt],axis=0) ,[N*2+10,33,1]))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"metadata": {
"id": "4_zwf8mkZKnj"
},
"source": [
"That's a pretty clear result: the PINN error is almost 4 times higher than the one from the Differentiable Physics (DP) reconstruction.\n",
"That's a pretty clear result: the PINN error is more than 3 times larger than the one from the Differentiable Physics (DP) reconstruction.\n",
"\n",
"This difference also shows clearly in the jointly visualized image at the bottom: the magnitudes of the errors of the DP reconstruction are much closer to zero, as indicated by the purple color above.\n",
"\n",
@@ -590,5 +633,5 @@
}
},
"nbformat": 4,
"nbformat_minor": 4
"nbformat_minor": 1
}
diffphys-code-ns.ipynb
+1 -1
View File
@@ -416,7 +416,7 @@
"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",
"The loss should have gone down significantly, from almost 300 to below 170, 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"
]
overview-burgers-forw.ipynb
+1 -1
View File
@@ -45,7 +45,7 @@
"name": "stdout",
"output_type": "stream",
"text": [
"Using phiflow version: 2.0.3\n"
"Using phiflow version: 2.2.0\n"
]
}
],
physicalloss-code.ipynb
+33 -47
View File
@@ -61,7 +61,7 @@
},
{
"cell_type": "code",
"execution_count": 1,
"execution_count": null,
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/"
@@ -69,13 +69,7 @@
"id": "L30jgXowoXat",
"outputId": "bcceefc0-fb33-44c7-959c-0b742e5a1606"
},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": []
}
],
"outputs": [],
"source": [
"!pip install --quiet phiflow==1.5.1\n",
"\n",
@@ -180,7 +174,7 @@
},
{
"cell_type": "code",
"execution_count": 4,
"execution_count": null,
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/"
@@ -188,15 +182,7 @@
"id": "xxLRbUQzoXay",
"outputId": "a2504aae-26dc-48a3-a5fa-12c6539f3f62"
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n"
]
}
],
"outputs": [],
"source": [
"N=128\n",
"grids_xt = np.meshgrid(np.linspace(-1, 1, N), np.linspace(0, 1, 33), indexing='ij')\n",
@@ -241,7 +227,7 @@
},
{
"data": {
"image/png": 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truncated
"image/png": 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truncated
"text/plain": [
"<Figure size 1152x360 with 2 Axes>"
]
@@ -298,7 +284,7 @@
},
{
"cell_type": "code",
"execution_count": 6,
"execution_count": null,
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/"
@@ -354,20 +340,20 @@
"name": "stdout",
"output_type": "stream",
"text": [
"Step 0, loss: 0.155002\n",
"Step 1, loss: 0.139890\n",
"Step 2, loss: 0.128770\n",
"Step 1000, loss: 0.058176\n",
"Step 2000, loss: 0.047925\n",
"Step 3000, loss: 0.043175\n",
"Step 4000, loss: 0.039472\n",
"Step 5000, loss: 0.038181\n",
"Step 6000, loss: 0.035628\n",
"Step 7000, loss: 0.026915\n",
"Step 8000, loss: 0.021984\n",
"Step 9000, loss: 0.018708\n",
"Step 10000, loss: 0.016682\n",
"Runtime 104.06s\n"
"Step 0, loss: 0.100037\n",
"Step 1, loss: 0.097836\n",
"Step 2, loss: 0.096151\n",
"Step 1000, loss: 0.054487\n",
"Step 2000, loss: 0.049147\n",
"Step 3000, loss: 0.045515\n",
"Step 4000, loss: 0.042890\n",
"Step 5000, loss: 0.039945\n",
"Step 6000, loss: 0.037641\n",
"Step 7000, loss: 0.035227\n",
"Step 8000, loss: 0.033604\n",
"Step 9000, loss: 0.031556\n",
"Step 10000, loss: 0.029434\n",
"Runtime 110.02s\n"
]
}
],
@@ -415,7 +401,7 @@
"outputs": [
{
"data": {
"image/png": 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truncated
"image/png": "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 truncated
"text/plain": [
"<Figure size 1152x360 with 2 Axes>"
]
@@ -458,7 +444,7 @@
{
"data": {
"text/plain": [
"<matplotlib.legend.Legend at 0x7f947c590e20>"
"<matplotlib.legend.Legend at 0x7f8eca2a7a00>"
]
},
"execution_count": 9,
@@ -467,7 +453,7 @@
},
{
"data": {
"image/png": 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truncated
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truncated
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
@@ -520,7 +506,7 @@
{
"data": {
"text/plain": [
"<matplotlib.legend.Legend at 0x7f947cc9cca0>"
"<matplotlib.legend.Legend at 0x7f8eca1b48e0>"
]
},
"execution_count": 10,
@@ -529,7 +515,7 @@
},
{
"data": {
"image/png": 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truncated
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
@@ -590,7 +576,7 @@
{
"data": {
"text/plain": [
"<matplotlib.legend.Legend at 0x7f9474d7b130>"
"<matplotlib.legend.Legend at 0x7f8eca556820>"
]
},
"execution_count": 11,
@@ -599,7 +585,7 @@
},
{
"data": {
"image/png": 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truncated
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truncated
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
@@ -669,7 +655,7 @@
{
"data": {
"text/plain": [
"<matplotlib.legend.Legend at 0x7f947cd89ee0>"
"<matplotlib.legend.Legend at 0x7f8ec93a4310>"
]
},
"execution_count": 12,
@@ -678,7 +664,7 @@
},
{
"data": {
"image/png": 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truncated
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truncated
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
@@ -727,13 +713,13 @@
"name": "stdout",
"output_type": "stream",
"text": [
"Mean absolute error for re-simulation across 16 steps: 0.01372\n"
"Mean absolute error for re-simulation across 16 steps: 0.01136\n"
]
},
{
"data": {
"text/plain": [
"<matplotlib.legend.Legend at 0x7f947c5d91f0>"
"<matplotlib.legend.Legend at 0x7f8eaa76ed90>"
]
},
"execution_count": 13,
@@ -742,7 +728,7 @@
},
{
"data": {
"image/png": 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truncated
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
@@ -792,7 +778,7 @@
"outputs": [
{
"data": {
"image/png": 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truncated
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truncated
"text/plain": [
"<Figure size 1152x360 with 2 Axes>"
]