updated physical loss chapter
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@@ -92,7 +92,6 @@
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"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",
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"Instructions for updating:\n",
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"Instructions for updating:\n",
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"This op will be removed after the deprecation date. Please switch to tf.sets.difference().\n",
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"This op will be removed after the deprecation date. Please switch to tf.sets.difference().\n",
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"Loss: 0.382915\n"
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"Loss: 0.382915\n"
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"nbformat": 4,
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- _Loss-terms_: the physical dynamics (or parts thereof) are encoded in the
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- _Loss-terms_: the physical dynamics (or parts thereof) are encoded in the
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loss function, typically in the form of differentiable operations. The
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loss function, typically in the form of differentiable operations. The
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learning process can repeatedly evaluate the loss, and usually receives
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learning process can repeatedly evaluate the loss, and usually receives
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gradients from a PDE-based formulation. These soft-constraints sometimes also go
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gradients from a PDE-based formulation. These soft constraints sometimes also go
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under the name "physics-informed" training.
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under the name "physics-informed" training.
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- _Interleaved_: the full physical simulation is interleaved and combined with
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- _Interleaved_: the full physical simulation is interleaved and combined with
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"source": [
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"source": [
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"# Burgers Optimization with a Physics-Informed NN\n",
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"# Burgers Optimization with a Physics-Informed NN\n",
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"\n",
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"\n",
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"To illustrate how the physics-informed losses work, let's consider a reconstruction task\n",
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"To illustrate how the physics-informed losses work for variant 2, let's consider a reconstruction task\n",
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"as an inverse problem example.\n",
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"as an inverse problem example.\n",
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"We'll use Burgers equation $\\frac{\\partial u}{\\partial{t}} + u \\nabla u = \\nu \\nabla \\cdot \\nabla u$ as a simple yet non-linear equation in 1D, for which we have a series of _observations_ at time $t=0.5$. \n",
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"We'll use Burgers equation $\\frac{\\partial u}{\\partial{t}} + u \\nabla u = \\nu \\nabla \\cdot \\nabla u$ as a simple yet non-linear equation in 1D, for which we have a series of _observations_ at time $t=0.5$. \n",
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"The solution should fulfill the residual formulation for Burgers equation and match the observations.\n",
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"The solution should fulfill the residual formulation for Burgers equation and match the observations.\n",
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@@ -40,7 +40,7 @@
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"They both represent two-dimensional solutions\n",
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"They both represent two-dimensional solutions\n",
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"$x(p_i,t_i)$ and $y^*(p_i,t_i)$ for a spatial coordinate $p_i$ and a time $t_i$, where the index $i$ sums over a set of chosen $p_i,t_i$ locations at which we evaluate the PDE and the approximated solutions. Thus $y^*$ denotes a reference $u$ for $\\mathcal{P}$ being Burgers equation, which $x$ should approximate as closely as possible. Thus our neural network representation of $x$ will receive $p,t$ as input to produce a velocity solution at the specified position.\n",
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"$x(p_i,t_i)$ and $y^*(p_i,t_i)$ for a spatial coordinate $p_i$ and a time $t_i$, where the index $i$ sums over a set of chosen $p_i,t_i$ locations at which we evaluate the PDE and the approximated solutions. Thus $y^*$ denotes a reference $u$ for $\\mathcal{P}$ being Burgers equation, which $x$ should approximate as closely as possible. Thus our neural network representation of $x$ will receive $p,t$ as input to produce a velocity solution at the specified position.\n",
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"\n",
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"\n",
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"The residual function $R$ above collects additional evaluations of $f(;\\theta)$ and its derivatives to formulate the residual for $\\mathcal{P}$. This approach -- using derivatives of a neural network to compute a PDE residual -- is typically called a _physics-informed_ approach, yielding a _physics-informed neural network_ (PINN) to represent a solution for the inverse reconstruction problem.\n",
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"The residual function $R$ above collects additional evaluations of $f(;\\theta)$ and its derivatives to formulate the residual for $\\mathcal{P}$. This approach -- using derivatives of a neural network to compute a PDE residual -- is typically called a _physics-informed_ approach, yielding a _physics-informed neural network_ (PINN) {cite}`raissi2019pinn` to represent a solution for the inverse reconstruction problem.\n",
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"\n",
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"\n",
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"Thus, in the formulation above, $R$ should simply converge to zero above. We've omitted scaling factors in the objective function for simplicity. Note that, effectively, we're only dealing with individual point samples of a single solution $u$ for $\\mathcal{P}$ here.\n",
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"Thus, in the formulation above, $R$ should simply converge to zero above. We've omitted scaling factors in the objective function for simplicity. Note that, effectively, we're only dealing with individual point samples of a single solution $u$ for $\\mathcal{P}$ here.\n",
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"\n"
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"\n"
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"outputId": "bcceefc0-fb33-44c7-959c-0b742e5a1606"
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"text": [
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"/bin/bash: pip: command not found\n",
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"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"
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"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",
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"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",
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"\n",
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"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",
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"\n",
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"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",
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"\n"
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"/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",
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" for key in self._mapping:\n",
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" yield (key, self._mapping[key])\n",
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"$ python setup.py tf_cuda\n",
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"$ python setup.py tf_cuda\n",
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"before reinstalling phiflow.\n",
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"before reinstalling phiflow.\n"
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" 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"
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]
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"text": [
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"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",
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"Instructions for updating:\n",
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"Instructions for updating:\n",
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"Use keras.layers.Dense instead.\n",
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"Use keras.layers.Dense instead.\n",
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"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",
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"Instructions for updating:\n",
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"Instructions for updating:\n",
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"Please use `layer.__call__` method instead.\n"
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"Please use `layer.__call__` method instead.\n"
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]
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"text": [
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"Size of grid_u: (1, 128, 33, 1)\n",
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"Size of grid_u: (1, 128, 33, 1)\n",
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"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",
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"\n",
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"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",
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"\n",
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"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",
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"\n",
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"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",
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"\n",
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"Randomly initialized network state:\n"
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"Randomly initialized network state:\n"
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]
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},
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{
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"data": {
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|
"image/png": "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 truncated
|
||||||
"text/plain": [
|
"text/plain": [
|
||||||
"<Figure size 1152x360 with 2 Axes>"
|
"<Figure size 1152x360 with 2 Axes>"
|
||||||
]
|
]
|
||||||
@@ -337,17 +325,7 @@
|
|||||||
"id": "bfZvwnFooXa1",
|
"id": "bfZvwnFooXa1",
|
||||||
"outputId": "f0188a14-5922-4faa-b1aa-fb6482bd81f3"
|
"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": [
|
"source": [
|
||||||
"# Boundary loss\n",
|
"# Boundary loss\n",
|
||||||
"N_SAMPLE_POINTS_BND = 100\n",
|
"N_SAMPLE_POINTS_BND = 100\n",
|
||||||
@@ -395,20 +373,20 @@
|
|||||||
"name": "stdout",
|
"name": "stdout",
|
||||||
"output_type": "stream",
|
"output_type": "stream",
|
||||||
"text": [
|
"text": [
|
||||||
"Step 0, loss: 0.095709\n",
|
"Step 0, loss: 0.155002\n",
|
||||||
"Step 1, loss: 0.093502\n",
|
"Step 1, loss: 0.139890\n",
|
||||||
"Step 2, loss: 0.091750\n",
|
"Step 2, loss: 0.128770\n",
|
||||||
"Step 1000, loss: 0.050177\n",
|
"Step 1000, loss: 0.058176\n",
|
||||||
"Step 2000, loss: 0.044144\n",
|
"Step 2000, loss: 0.047925\n",
|
||||||
"Step 3000, loss: 0.039773\n",
|
"Step 3000, loss: 0.043175\n",
|
||||||
"Step 4000, loss: 0.034952\n",
|
"Step 4000, loss: 0.039472\n",
|
||||||
"Step 5000, loss: 0.032054\n",
|
"Step 5000, loss: 0.038181\n",
|
||||||
"Step 6000, loss: 0.030908\n",
|
"Step 6000, loss: 0.035628\n",
|
||||||
"Step 7000, loss: 0.029630\n",
|
"Step 7000, loss: 0.026915\n",
|
||||||
"Step 8000, loss: 0.028747\n",
|
"Step 8000, loss: 0.021984\n",
|
||||||
"Step 9000, loss: 0.028692\n",
|
"Step 9000, loss: 0.018708\n",
|
||||||
"Step 10000, loss: 0.020781\n",
|
"Step 10000, loss: 0.016682\n",
|
||||||
"Runtime 88.29s\n"
|
"Runtime 104.06s\n"
|
||||||
]
|
]
|
||||||
}
|
}
|
||||||
],
|
],
|
||||||
@@ -456,7 +434,7 @@
|
|||||||
"outputs": [
|
"outputs": [
|
||||||
{
|
{
|
||||||
"data": {
|
"data": {
|
||||||
"image/png": 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truncated
|
||||||
"text/plain": [
|
"text/plain": [
|
||||||
"<Figure size 1152x360 with 2 Axes>"
|
"<Figure size 1152x360 with 2 Axes>"
|
||||||
]
|
]
|
||||||
@@ -499,7 +477,7 @@
|
|||||||
{
|
{
|
||||||
"data": {
|
"data": {
|
||||||
"text/plain": [
|
"text/plain": [
|
||||||
"<matplotlib.legend.Legend at 0x11f9b5b90>"
|
"<matplotlib.legend.Legend at 0x7f947c590e20>"
|
||||||
]
|
]
|
||||||
},
|
},
|
||||||
"execution_count": 9,
|
"execution_count": 9,
|
||||||
@@ -508,7 +486,7 @@
|
|||||||
},
|
},
|
||||||
{
|
{
|
||||||
"data": {
|
"data": {
|
||||||
"image/png": 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truncated
|
||||||
"text/plain": [
|
"text/plain": [
|
||||||
"<Figure size 432x288 with 1 Axes>"
|
"<Figure size 432x288 with 1 Axes>"
|
||||||
]
|
]
|
||||||
@@ -561,7 +539,7 @@
|
|||||||
{
|
{
|
||||||
"data": {
|
"data": {
|
||||||
"text/plain": [
|
"text/plain": [
|
||||||
"<matplotlib.legend.Legend at 0x11fa4db90>"
|
"<matplotlib.legend.Legend at 0x7f947cc9cca0>"
|
||||||
]
|
]
|
||||||
},
|
},
|
||||||
"execution_count": 10,
|
"execution_count": 10,
|
||||||
@@ -570,7 +548,7 @@
|
|||||||
},
|
},
|
||||||
{
|
{
|
||||||
"data": {
|
"data": {
|
||||||
"image/png": 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truncated
|
||||||
"text/plain": [
|
"text/plain": [
|
||||||
"<Figure size 432x288 with 1 Axes>"
|
"<Figure size 432x288 with 1 Axes>"
|
||||||
]
|
]
|
||||||
@@ -631,7 +609,7 @@
|
|||||||
{
|
{
|
||||||
"data": {
|
"data": {
|
||||||
"text/plain": [
|
"text/plain": [
|
||||||
"<matplotlib.legend.Legend at 0x11faef650>"
|
"<matplotlib.legend.Legend at 0x7f9474d7b130>"
|
||||||
]
|
]
|
||||||
},
|
},
|
||||||
"execution_count": 11,
|
"execution_count": 11,
|
||||||
@@ -640,7 +618,7 @@
|
|||||||
},
|
},
|
||||||
{
|
{
|
||||||
"data": {
|
"data": {
|
||||||
"image/png": 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truncated
|
||||||
"text/plain": [
|
"text/plain": [
|
||||||
"<Figure size 432x288 with 1 Axes>"
|
"<Figure size 432x288 with 1 Axes>"
|
||||||
]
|
]
|
||||||
@@ -710,7 +688,7 @@
|
|||||||
{
|
{
|
||||||
"data": {
|
"data": {
|
||||||
"text/plain": [
|
"text/plain": [
|
||||||
"<matplotlib.legend.Legend at 0x11fb76c50>"
|
"<matplotlib.legend.Legend at 0x7f947cd89ee0>"
|
||||||
]
|
]
|
||||||
},
|
},
|
||||||
"execution_count": 12,
|
"execution_count": 12,
|
||||||
@@ -719,7 +697,7 @@
|
|||||||
},
|
},
|
||||||
{
|
{
|
||||||
"data": {
|
"data": {
|
||||||
"image/png": 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truncated
|
"image/png": 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truncated
|
||||||
"text/plain": [
|
"text/plain": [
|
||||||
"<Figure size 432x288 with 1 Axes>"
|
"<Figure size 432x288 with 1 Axes>"
|
||||||
]
|
]
|
||||||
@@ -768,13 +746,13 @@
|
|||||||
"name": "stdout",
|
"name": "stdout",
|
||||||
"output_type": "stream",
|
"output_type": "stream",
|
||||||
"text": [
|
"text": [
|
||||||
"Mean absolute error for re-simulation across 16 steps: 0.02124\n"
|
"Mean absolute error for re-simulation across 16 steps: 0.01372\n"
|
||||||
]
|
]
|
||||||
},
|
},
|
||||||
{
|
{
|
||||||
"data": {
|
"data": {
|
||||||
"text/plain": [
|
"text/plain": [
|
||||||
"<matplotlib.legend.Legend at 0x11fc06910>"
|
"<matplotlib.legend.Legend at 0x7f947c5d91f0>"
|
||||||
]
|
]
|
||||||
},
|
},
|
||||||
"execution_count": 13,
|
"execution_count": 13,
|
||||||
@@ -783,7 +761,7 @@
|
|||||||
},
|
},
|
||||||
{
|
{
|
||||||
"data": {
|
"data": {
|
||||||
"image/png": 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|
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truncated
|
||||||
"text/plain": [
|
"text/plain": [
|
||||||
"<Figure size 432x288 with 1 Axes>"
|
"<Figure size 432x288 with 1 Axes>"
|
||||||
]
|
]
|
||||||
@@ -833,7 +811,7 @@
|
|||||||
"outputs": [
|
"outputs": [
|
||||||
{
|
{
|
||||||
"data": {
|
"data": {
|
||||||
"image/png": 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truncated
|
"image/png": 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truncated
|
||||||
"text/plain": [
|
"text/plain": [
|
||||||
"<Figure size 1152x360 with 2 Axes>"
|
"<Figure size 1152x360 with 2 Axes>"
|
||||||
]
|
]
|
||||||
@@ -931,9 +909,9 @@
|
|||||||
"name": "python",
|
"name": "python",
|
||||||
"nbconvert_exporter": "python",
|
"nbconvert_exporter": "python",
|
||||||
"pygments_lexer": "ipython3",
|
"pygments_lexer": "ipython3",
|
||||||
"version": "3.7.6"
|
"version": "3.8.5"
|
||||||
}
|
}
|
||||||
},
|
},
|
||||||
"nbformat": 4,
|
"nbformat": 4,
|
||||||
"nbformat_minor": 1
|
"nbformat_minor": 1
|
||||||
}
|
}
|
||||||
@@ -1,4 +1,4 @@
|
|||||||
Discussion of Physical Soft-Constraints
|
Discussion of Physical Losses
|
||||||
=======================
|
=======================
|
||||||
|
|
||||||
The good news so far is - we have a DL method that can include
|
The good news so far is - we have a DL method that can include
|
||||||
@@ -9,7 +9,7 @@ starting point.
|
|||||||
On the positive side, we can leverage DL frameworks with backpropagation to compute
|
On the positive side, we can leverage DL frameworks with backpropagation to compute
|
||||||
the derivatives of the model. At the same time, this puts us at the mercy of the learned
|
the derivatives of the model. At the same time, this puts us at the mercy of the learned
|
||||||
representation regarding the reliability of these derivatives. Also, each derivative
|
representation regarding the reliability of these derivatives. Also, each derivative
|
||||||
requires backpropagation through the full network, which can be very expensive. Especially so
|
requires backpropagation through the full network. This can be very expensive, especially
|
||||||
for higher-order derivatives.
|
for higher-order derivatives.
|
||||||
|
|
||||||
And while the setup is relatively simple, it is generally difficult to control. The NN
|
And while the setup is relatively simple, it is generally difficult to control. The NN
|
||||||
@@ -23,23 +23,23 @@ Of course, such denomination questions are superficial - if an algorithm is usef
|
|||||||
what name it has. However, here the question helps to highlight some important properties
|
what name it has. However, here the question helps to highlight some important properties
|
||||||
that are typically associated with algorithms from fields like machine learning or optimization.
|
that are typically associated with algorithms from fields like machine learning or optimization.
|
||||||
|
|
||||||
One main reason _not_ to call these physical constraints machine learning (ML), is that the
|
One main reason _not_ to call the optimization of the previous notebook machine learning (ML), is that the
|
||||||
positions where we test and constrain the solution are the final positions we are interested in.
|
positions where we test and constrain the solution are the final positions we are interested in.
|
||||||
As such, there is no real distinction between training, validation and (out of distribution) test sets.
|
As such, there is no real distinction between training, validation and test sets.
|
||||||
Computing the solution for a known and given set of samples is much more akin to classical optimization,
|
Computing the solution for a known and given set of samples is much more akin to classical optimization,
|
||||||
where inverse problems like the previous Burgers example stem from.
|
where inverse problems like the previous Burgers example stem from.
|
||||||
|
|
||||||
For machine learning, we typically work under the assumption that the final performance of our
|
For machine learning, we typically work under the assumption that the final performance of our
|
||||||
model will be evaluated on a different, potentially unknown set of inputs. The _test data_
|
model will be evaluated on a different, potentially unknown set of inputs. The _test data_
|
||||||
should usually capture such out of distribution (OOD) behavior, so that we can make estimates
|
should usually capture such _out of distribution_ (OOD) behavior, so that we can make estimates
|
||||||
about how well our model will generalize to "real-world" cases that we will encounter when
|
about how well our model will generalize to "real-world" cases that we will encounter when
|
||||||
we deploy it into an application.
|
we deploy it in an application.
|
||||||
|
|
||||||
In contrast, for the PINN training as described here, we reconstruct a single solution in a known
|
In contrast, for the PINN training as described here, we reconstruct a single solution in a known
|
||||||
and given space-time region. As such, any samples from this domain follow the same distribution
|
and given space-time region. As such, any samples from this domain follow the same distribution
|
||||||
and hence don't really represent test or OOD sampes. As the NN directly encodes the solution,
|
and hence don't really represent test or OOD samples. As the NN directly encodes the solution,
|
||||||
there is also little hope that it will yield different solutions, or perform well outside
|
there is also little hope that it will yield different solutions, or perform well outside
|
||||||
of the training distribution. If we're interested in a different solution, we most likely
|
of the training range. If we're interested in a different solution, we
|
||||||
have to start training the NN from scratch.
|
have to start training the NN from scratch.
|
||||||
|
|
||||||
|
|
||||||
@@ -48,7 +48,7 @@ have to start training the NN from scratch.
|
|||||||
|
|
||||||
Thus, the physical soft constraints allow us to encode solutions to
|
Thus, the physical soft constraints allow us to encode solutions to
|
||||||
PDEs with the tools of NNs.
|
PDEs with the tools of NNs.
|
||||||
An inherent drawback of this approach is that it yields single solutions,
|
An inherent drawback of this variant 2 is that it yields single solutions,
|
||||||
and that it does not combine with traditional numerical techniques well.
|
and that it does not combine with traditional numerical techniques well.
|
||||||
E.g., the learned representation is not suitable to be refined with
|
E.g., the learned representation is not suitable to be refined with
|
||||||
a classical iterative solver such as the conjugate gradient method.
|
a classical iterative solver such as the conjugate gradient method.
|
||||||
|
|||||||
+58
-15
@@ -10,13 +10,8 @@ We as humans have a lot of knowledge about how to describe physical processes
|
|||||||
mathematically. As the following chapters will show, we can improve the
|
mathematically. As the following chapters will show, we can improve the
|
||||||
training process by guiding it with our human knowledge of physics.
|
training process by guiding it with our human knowledge of physics.
|
||||||
|
|
||||||
```{figure} resources/physloss-overview.jpg
|
|
||||||
---
|
|
||||||
height: 220px
|
|
||||||
name: physloss-overview
|
|
||||||
---
|
|
||||||
Physical losses typically combine a supervised loss with a combination of derivatives from the neural network.
|
|
||||||
```
|
|
||||||
|
|
||||||
## Using physical models
|
## Using physical models
|
||||||
|
|
||||||
@@ -63,17 +58,57 @@ Note that, similar to the data samples used for supervised training, we have no
|
|||||||
residual terms $R$ will actually reach zero during training. The non-linear optimization of the training process
|
residual terms $R$ will actually reach zero during training. The non-linear optimization of the training process
|
||||||
will minimize the supervised and residual terms as much as possible, but there is no guarantee. Large, non-zero residual
|
will minimize the supervised and residual terms as much as possible, but there is no guarantee. Large, non-zero residual
|
||||||
contributions can remain. We'll look at this in more detail in the upcoming code example, for now it's important
|
contributions can remain. We'll look at this in more detail in the upcoming code example, for now it's important
|
||||||
to remember that physical constraints in this way only represent _soft-constraints_, without guarantees
|
to remember that physical constraints in this way only represent _soft constraints_, without guarantees
|
||||||
of minimizing these constraints.
|
of minimizing these constraints.
|
||||||
|
|
||||||
## Neural network derivatives
|
The previous overview did not really make clear how an NN produces $\mathbf{u}$.
|
||||||
|
We can distinguish two different approaches here:
|
||||||
|
via a chosen explicit representation of the target function (v1 in the following), or via using fully-connected neural networks to represent the solution (v2).
|
||||||
|
E.g., for v1 we could set up a _spatial_ grid (or graph, or a set of sample points), while in the second case no explicit representation exists, and the NN instead receives the _spatial coordinate_ to produce the solution at a query position.
|
||||||
|
We'll outline these two variants in more detail the following.
|
||||||
|
|
||||||
In order to compute the residuals at training time, it would be possible to store
|
---
|
||||||
the unknowns of $\mathbf{u}$ on a computational mesh, e.g., a grid, and discretize the equations of
|
|
||||||
$R$ there. This has a fairly long "tradition" in DL, and was proposed by Tompson et al. {cite}`tompson2017` early on.
|
## Variant 1: Residual derivatives for explicit representations
|
||||||
|
|
||||||
|
For variant 1, we choose the discretization and set up a computational mesh that covers our target domain. Without loss of generality, let's assume this is a Cartesian grid that samples the space with positions $\mathbf{p}$. Now, an NN is trained to produce the solution on the grid: $\mathbf{u}(\mathbf{p}) = f(\mathbf{x} ; \theta)$. For a regular grid, a CNN would be a good choice for $f$, while for triangle meshes we could use a graph-network, or a network with point-convolutions for particles.
|
||||||
|
|
||||||
|
```{figure} resources/physloss-overview-v1.jpg
|
||||||
|
---
|
||||||
|
height: 220px
|
||||||
|
name: physloss-overview-v1
|
||||||
|
---
|
||||||
|
Variant 1: the solution is represented by a chosen computational mesh, and produced by an NN. The residual is discretized there, and can be combined with supervised terms.
|
||||||
|
```
|
||||||
|
|
||||||
|
Now, we can discretize the equations of
|
||||||
|
$R$ on our computational mesh, and compute derivatives with our method of choice. Only caveat: to incorporate the residual
|
||||||
|
into training, we have to formulate the evaluation such that a deep learning framework can backpropagate through the
|
||||||
|
calculations. As our network $f()$ produces the solution $\mathbf{u}$, and the residual depends on it ($R(\mathbf{u})$), we at least need $\partial R / \partial \mathbf u$, such that the gradient can be backpropagated for the weights $\theta$. Luckily, if we formulate $R$ in terms of operations of a DL framework, this will be taken care of by the backpropagation functionality of the framework.
|
||||||
|
|
||||||
|
This variant has a fairly long "tradition" in DL, and was, e.g., proposed by Tompson et al. {cite}`tompson2017` early on to learn
|
||||||
|
divergence free motions. To give a specific example: if our goal is to learn velocities $\mathbf u(t)$ which are divergence free $\nabla \cdot \mathbf u=0$, we can employ this training approach to train an NN without having to pre-compute divergence free velocity fields as training data. For brevity, we will drop the spatial index ($\mathbf p$) here, and focus on $t$, which we can likewise simplify: divergence-freeness has to hold at all times, and hence we can consider a single step from $t=0$ with $\Delta t=1$, i.e., a normalized step from a divergent $\mathbf u(0)$ to a divergence-free $\mathbf u(1)$. For a normal solver, we'd have to compute a pressure
|
||||||
|
$p=\nabla^{-2} \mathbf{u}(0)$, such that $\mathbf{u}(1) = \mathbf{u}(0) - \nabla p$. This is the famous fundamental
|
||||||
|
theorem of vector calculus, or
|
||||||
|
[Helmholtz decomposition](https://en.wikipedia.org/wiki/Helmholtz_decomposition), splitting a vector field into a _solenoidal_ (divergence-free) and irrotational part (the pressure gradient).
|
||||||
|
|
||||||
|
To learn this decomposition, we can approximate $p$ with a CNN on our computational mesh: $p = f(\mathbf{u}(0) ; \theta)$. The learning objective becomes minimizing the divergence of $\mathbf u(0)$, which means minimizing
|
||||||
|
$\nabla \cdot \big( \mathbf{u}(0) - \nabla f(\mathbf{u}(0);\theta) \big)$.
|
||||||
|
To implement this residual, all we need to do is provide the divergence operator $(\nabla \cdot)$ of $\mathbf u$ on our computational mesh. This is typically easy to do via
|
||||||
|
a convolutional layer in the DL framework that contains the finite difference weights for the divergence.
|
||||||
|
Nicely enough, in this case we don't even need additional supervised samples, and can typically purely train with this residual formulation. Also, in contrast to variant 2 below, we can directly handle fairly large spaces of solutions here (we're not restricted to learning single solutions)
|
||||||
|
An example implementation can be found in this [code repository](https://github.com/tum-pbs/CG-Solver-in-the-Loop).
|
||||||
|
|
||||||
|
Overall, this variant 1 has a lot in common with _differentiable physics_ training (it's basically a subset). As we'll discuss differentiable physics in a lot more detail
|
||||||
|
in {doc}`diffphys` and after, we'll focus on the direct NN representation (variant 2) from now on.
|
||||||
|
|
||||||
|
---
|
||||||
|
|
||||||
|
## Variant 2: Derivatives from a neural network representation
|
||||||
|
|
||||||
|
The second variant of employing physical residuals as soft constraints
|
||||||
|
instead uses fully connected NNs to represent $\mathbf{u}$. This _physics-informed_ approach was popularized by Raissi et al. {cite}`raissi2019pinn`, and has some interesting pros and cons that we'll outline in the following. We will target this physics-informed version (variant 2) in the following code examples and discussions.
|
||||||
|
|
||||||
A popular variant of employing physical soft-constraints {cite}`raissi2019pinn`
|
|
||||||
instead uses fully connected NNs to represent $\mathbf{u}$. This has some interesting pros and cons that we'll outline in the following, and we will also focus on it in the following code examples and comparisons.
|
|
||||||
|
|
||||||
The central idea here is that the aforementioned general function $f$ that we're after in our learning problems
|
The central idea here is that the aforementioned general function $f$ that we're after in our learning problems
|
||||||
can also be used to obtain a representation of a physical field, e.g., a field $\mathbf{u}$ that satisfies $R=0$. This means $\mathbf{u}(\mathbf{x})$ will
|
can also be used to obtain a representation of a physical field, e.g., a field $\mathbf{u}$ that satisfies $R=0$. This means $\mathbf{u}(\mathbf{x})$ will
|
||||||
@@ -86,6 +121,14 @@ in {doc}`overview`. Now, we can use the same tools to compute spatial derivative
|
|||||||
Note that above for $R$ we've written this derivative in the shortened notation as $\mathbf{u}_{x}$.
|
Note that above for $R$ we've written this derivative in the shortened notation as $\mathbf{u}_{x}$.
|
||||||
For functions over time this of course also works for $\partial \mathbf{u} / \partial t$, i.e. $\mathbf{u}_{t}$ in the notation above.
|
For functions over time this of course also works for $\partial \mathbf{u} / \partial t$, i.e. $\mathbf{u}_{t}$ in the notation above.
|
||||||
|
|
||||||
|
```{figure} resources/physloss-overview-v2.jpg
|
||||||
|
---
|
||||||
|
height: 220px
|
||||||
|
name: physloss-overview-v2
|
||||||
|
---
|
||||||
|
Variant 2: the solution is produced by a fully-connected network, typically requiring a supervised loss with a combination of derivatives from the neural network for the residual. These NN derivatives have their own share of advantages and disadvantages.
|
||||||
|
```
|
||||||
|
|
||||||
Thus, for some generic $R$, made up of $\mathbf{u}_t$ and $\mathbf{u}_{x}$ terms, we can rely on the backpropagation algorithm
|
Thus, for some generic $R$, made up of $\mathbf{u}_t$ and $\mathbf{u}_{x}$ terms, we can rely on the backpropagation algorithm
|
||||||
of DL frameworks to compute these derivatives once we have a NN that represents $\mathbf{u}$. Essentially, this gives us a
|
of DL frameworks to compute these derivatives once we have a NN that represents $\mathbf{u}$. Essentially, this gives us a
|
||||||
function (the NN) that receives space and time coordinates to produce a solution for $\mathbf{u}$. Hence, the input is typically
|
function (the NN) that receives space and time coordinates to produce a solution for $\mathbf{u}$. Hence, the input is typically
|
||||||
@@ -101,7 +144,7 @@ For higher order derivatives, such as $\frac{\partial^2 u}{\partial x^2}$, we ca
|
|||||||
|
|
||||||
## Summary so far
|
## Summary so far
|
||||||
|
|
||||||
The approach above gives us a method to include physical equations into DL learning as a soft-constraint: the residual loss.
|
The approach above gives us a method to include physical equations into DL learning as a soft constraint: the residual loss.
|
||||||
Typically, this setup is suitable for _inverse problems_, where we have certain measurements or observations
|
Typically, this setup is suitable for _inverse problems_, where we have certain measurements or observations
|
||||||
for which we want to find a PDE solution. Because of the high cost of the reconstruction (to be
|
for which we want to find a PDE solution. Because of the high cost of the reconstruction (to be
|
||||||
demonstrated in the following), the solution manifold shouldn't be overly complex. E.g., it is not possible
|
demonstrated in the following), the solution manifold shouldn't be overly complex. E.g., it is not possible
|
||||||
|
|||||||
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|
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@@ -158,7 +158,7 @@ To summarize, supervised training has the following properties.
|
|||||||
- Interactions with external "processes" (such as embedding into a solver) are difficult.
|
- Interactions with external "processes" (such as embedding into a solver) are difficult.
|
||||||
|
|
||||||
The next chapters will explain how to alleviate these shortcomings of supervised training.
|
The next chapters will explain how to alleviate these shortcomings of supervised training.
|
||||||
First, we'll look at bringing model equations into the picture via soft-constraints, and afterwards
|
First, we'll look at bringing model equations into the picture via soft constraints, and afterwards
|
||||||
we'll revisit the challenges of bringing together numerical simulations and learned approaches.
|
we'll revisit the challenges of bringing together numerical simulations and learned approaches.
|
||||||
|
|
||||||
|
|
||||||
Reference in new issue
Block a user