update burgers notebooks
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@@ -12,35 +12,39 @@
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{
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"cell_type": "code",
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"execution_count": 1,
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"metadata": {},
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"metadata": {
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"scrolled": true
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},
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"outputs": [
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{
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"name": "stderr",
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"output_type": "stream",
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"text": [
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"/home/thuerey/phiflow/phi/tf/__init__.py:7: UserWarning: TensorFlow 2 is not fully supported by PhiFlow.\n",
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" warnings.warn('TensorFlow 2 is not fully supported by PhiFlow.')\n"
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]
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},
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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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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"WARNING:tensorflow:From /Users/thuerey/Dropbox/mbaDevelSelected/phiflow-v1.5/phi/tf/tf_backend.py:287: to_complex64 (from tensorflow.python.ops.math_ops) is deprecated and will be removed in a future version.\n",
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"Could not load resample cuda libraries: CUDA binaries not found at /home/thuerey/phiflow/phi/tf/cuda/build/resample.so. Run \"python setup.py cuda\" to compile them\n",
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"WARNING:tensorflow:From /home/thuerey/phiflow/phi/tf/tf_backend.py:287: to_complex64 (from tensorflow.python.ops.math_ops) is deprecated and will be removed in a future version.\n",
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"Instructions for updating:\n",
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"Use `tf.cast` instead.\n",
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"WARNING:tensorflow:From /Users/thuerey/Dropbox/mbaDevelSelected/phiflow-v1.5/phi/tf/tf_backend.py:375: The name tf.fft is deprecated. Please use tf.signal.fft instead.\n",
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"\n",
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"WARNING:tensorflow:From /Users/thuerey/Dropbox/mbaDevelSelected/phiflow-v1.5/phi/tf/tf_backend.py:387: The name tf.ifft is deprecated. Please use tf.signal.ifft instead.\n",
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"\n",
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"WARNING:tensorflow:From /Users/thuerey/Dropbox/mbaDevelSelected/phiflow-v1.5/phi/tf/tf_backend.py:399: The name tf.real is deprecated. Please use tf.math.real instead.\n",
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"\n"
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"Use `tf.cast` instead.\n"
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]
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},
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{
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"name": "stderr",
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"output_type": "stream",
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"text": [
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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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"/home/thuerey/anaconda3/envs/tf/lib/python3.8/_collections_abc.py:743: MatplotlibDeprecationWarning: The global colormaps dictionary is no longer considered public API.\n",
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" for key in self._mapping:\n",
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"/home/thuerey/anaconda3/envs/tf/lib/python3.8/_collections_abc.py:744: MatplotlibDeprecationWarning: The global colormaps dictionary is no longer considered public API.\n",
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" yield (key, self._mapping[key])\n",
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"/home/thuerey/phiflow/phi/viz/display.py:80: UserWarning: GUI is disabled because of missing dependencies: No module named 'imageio'. To install all dependencies, run $ pip install phiflow[gui]\n",
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" warnings.warn('GUI is disabled because of missing dependencies: %s. To install all dependencies, run $ pip install phiflow[gui]' % import_error)\n",
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"/home/thuerey/phiflow/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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"$ python setup.py tf_cuda\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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@@ -59,7 +63,8 @@
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"\n",
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"# run with phiflow\n",
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"n = 128\n",
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"dt = 1./32.\n",
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"steps = 32\n",
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"dt = 1./steps\n",
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"initial = np.zeros([n,1]) # start from 0\n",
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"\n",
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"domain = Domain([n], boundaries=PERIODIC, box=box[-1:1])\n",
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@@ -69,7 +74,7 @@
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"state_in = state0.copied_with(velocity= variable(state0.velocity) ) # <-- main change!\n",
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"states = [state_in]\n",
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"\n",
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"for i in range(32):\n",
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"for i in range(steps):\n",
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" states.append( Burgers().step(states[-1],dt=dt) )\n",
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"\n",
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"print(\"Each velocity is a phiflow grid like this one: \" + format(states[-1].velocity) )\n",
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@@ -80,25 +85,16 @@
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Note, this didnt run anything! let's set up the loss"
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"Now we don't need to keep track and pass along `state_in`, TF knows about the initial state as a variable we'd like to optimize. This is in contrast to the previous version of this optimization, where we explicitly stored and updated a field with the intial state.\n",
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"\n",
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"Nonetheless, this didnt run anything! We only have a graph of operations in tensorflow so far. Let's set up the loss, such that we can start the optimizer. It's actually pretty simple, we only need to compute the L2 error for the constrained state of the simulation (at $t=1/2$):"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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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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]
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}
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],
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"outputs": [],
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"source": [
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"sess = Session(None) \n",
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"\n",
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@@ -112,9 +108,13 @@
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"As the graph encodes the full temporal sequence, and we can compute and propagate derivatives for all intermediates, we can can now back-propagate a gradient for this loss to any earlier variable, and hence also the initial velocity state that we're interested in (and that we already declared as a `variable()` to TF above).\n",
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"\n",
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"Note: because we're only constraining timestep 16, we could actually omit steps 17 to 31 here, but for fairness and comparison with the PINN case, let's include them.\n",
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"\n",
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"Now we don't need to keep track and pass along `state_in`, TF knows about the initial state as a variable we'd like to optimize."
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"Also, we don't need additional terms for enforcing the boundary conditions, these are already present in the simulation, and hence encoded in the execution graph for the `states` sequence above.\n",
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"\n",
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"Now we're ready to run the optimization with one of the optimizers provided in tensorflow, e.g., the simplest starting point is always _gradient descent_ (GD):"
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]
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},
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{
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@@ -126,13 +126,6 @@
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"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",
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"Instructions for updating:\n",
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"Use tf.where in 2.0, which has the same broadcast rule as np.where\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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"Initial loss: 0.382915\n",
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"Step 0, loss: 0.382915\n",
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"Step 5, loss: 0.194040\n",
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@@ -144,7 +137,7 @@
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"Step 35, loss: 0.009447\n",
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"Step 40, loss: 0.006238\n",
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"Step 45, loss: 0.004161\n",
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"Runtime 1.89s\n"
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"Runtime 1.17s\n"
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]
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}
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],
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@@ -179,22 +172,22 @@
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"execution_count": 4,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"[<matplotlib.lines.Line2D at 0x136ef4bd0>]"
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"[<matplotlib.lines.Line2D at 0x7f68682ae0d0>]"
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]
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},
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"execution_count": 5,
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"execution_count": 4,
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"metadata": {},
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"output_type": "execute_result"
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},
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{
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"data": {
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"image/png": 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truncated
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||||
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAYIAAAD4CAYAAADhNOGaAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjMuMywgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy/Il7ecAAAACXBIWXMAAAsTAAALEwEAmpwYAABAAklEQVR4nO3dd3xTVf/A8c9J0kVbaEvLKFAoiuDzOAArKAooogwHbnGBCiI8IBtkK1MEpKKgbLeC4u/xAdniwIUMEREQmYUWCnRQSkfSJOf3x01rgJbVtGna7/v16qvJvecm394m+eacc885SmuNEEKIisvk7QCEEEJ4lyQCIYSo4CQRCCFEBSeJQAghKjhJBEIIUcFZvB3A5YiMjNT16tXzdhhCCOFTtmzZkqK1jjp7u08mgnr16rF582ZvhyGEED5FKZVQ2HZpGhJCiApOEoEQQlRwkgiEEKKC80giUEotVEodV0r9WcR+pZR6Uym1Vyn1h1Kqqdu+rkqpPa6frp6IRwghxMXzVI3gPaD9efZ3ABq4fnoA7wAopSKAl4HmQDPgZaVUuIdiEkIIcRE8kgi01uuBtPMU6QR8oA0bgDClVE2gHbBWa52mtU4H1nL+hCKEEMLDSquPoBZw2O1+omtbUdvPoZTqoZTarJTafOLEiRILVAghKhqfGUegtZ4LzAWIi4uTubNFqbI5bGxJ2sL25O38deIvrHYr9/3rPu644g4sJp95GwlRqNJ6BScBddzu13ZtSwJuO2v7d6UUkxDnlWXLYt2+daz6exU/HvyRrLysM/av2rOKGiE1mNpxKi1iWngpSiGKr7QSwVKgj1JqEUbHcIbW+qhSajUwya2D+C5geCnFJMQ5tNZsStzEoj8WsWbPGnLsOQX7roq8iua1m9OoWiOybdl8vO1jDqYfZPEfiyURCJ/mkUSglPoU45t9pFIqEeNKID8ArfVsYAXQEdgLZAPPuvalKaXGA5tcDzVOa32+TmchSoTVbuW/O//Le1veY0/qnoLtTaKbcE+je7jzijupVeXM7qsrq17Js188S3pOemmHK4RHeSQRaK0fv8B+DfQuYt9CYKEn4hDiUqVlp/Hxto/5cOuHpGanAhAVHMUj1z7Co9c8Sp2wOkUeGxYUBiCJQPg86eUSFVJyZjKzN87m8+2fk2vPBeDf1f9N97judLiqA35mvws+RkRQBCCJQPg+SQSiQjmRdYLZv87mk22fYHPYAGgd25rnb3yem+rchFLqoh8rPMjo2pJEIHydJAJRIaRmpzJ341w++v2jghpAx6s60ufmPjSManhZj1nJrxL+Zn9y7bnk5OUQ5BfkyZCFKDWSCES5ZrVbeXfLu7zz6zuctp0GoO2Vbenfoj9XV7v6oh5Da83hw3ls355DZqYDraFWLT9atgwhPCicY6ePkZ6TLolA+CxJBKJc0lrz1e6vmLp+KkmnkgBoWa8lA28dyHU1rrvg8YcP21i+PIPVq0/x/fenSU93nFOmW7eqVL4pkmMYiSC6crTH/w4hSoMkAlHubEnawqTvJvH70d8BaBjZkOG3DadlvZbnPS4z08Gnn6bz8cdprF9/+ox9Vauauf76SkRFWXA6NcuWZbBgQSqRP/Qg+PHR0k8gfJokAlFuHDp5iKnrp7Li7xUARFaKZOCtA3n4mocxm8xFHrd7dy6zZp3gvfdSycx0AhAYqOjYsQrt21fmrrtCiYnxP6Mj+bffsrn//n0c/jsa27qHSHtMhr8I3yWJQPi83LxcZm+czZyNc7A5bARYAuge150ezXoQ4h9S6DFaa1auPMWMGcdZsyazYHurViF061aVBx4IIzS06OTRtGkl3nyzDg88sB97anVO5pz09J8lRKmRRCB82rf7v2XcunEcyjgEQKerOzG45eAi2+udTs2XX55kwoRktm41po8IClI8+WQEffpEcf31lS76uSMjjbePIyuU9JxjxfxLhPAeSQTCJyVlJDH+2/Gs3bsWgAZVGzCu7Tia1WlWaHmHQ7NkSToTJiTz55/G5aM1algYMKAa3btHEhFx6W+FqlWNGoMzO4T0nL8u8y8RwvskEQifYnPYWLh5IW/98ha59lyC/YLp26IvXZt2LXQ0sNaa//0vg5Ejj7Bzp5EAatf246WXqtOtWyRBQZe/JEdBjSA7lPRc6SwWvksSgfAZGw9vZNTaUexL2wdAx4YdGXnbSGqE1ii0/PffZzJs2BE2bDCmj65b158RI2rQtWsEAQHFX5MpPNx4+zhzgknNOlnsxxPCWyQRiDIv05rJa+tf49NtnwJQL7weY+8Yy631bi20/LZt2QwbdoRVq04BUK2ahdGja9CjRyT+/p5blM9iUYRWgcwMEydScj32uEKUNkkEokxbvWc1r3z9CsezjuNn8qNn8570at6LAEvAOWWTk/MYNeoICxemojWEhpoYMqQ6AwZUIySk6CuAiiMiwkxmhoMTKXkl8vhClAZJBKJMOnb6GGPXjWX1ntUANKnZhEntJnFV5FXnlM3JcRIff5xXX03m9GknFgv85z9RjB5ds6Adv6RERfqRcMBBepqzRJ9HiJIkiUCUKU7t5LPtnzH5+8lkWjMJ9gtmcMvBPNn4yXMGhWmtWbw4nWHDjpCQYMwkeu+9VZg6tRYNGwaWSrzVovyBXHIy/WXiOeGzPLVCWXtgBmAG5mutJ5+1Px643XW3ElBNax3m2ucAtrv2HdJa3+eJmITvScxI5KVVL7Hh8AYA2tRvw9i2YwsdE7BxYxYDBiTy889GR/C11wYyfXpt2ratXKoxV63q6jDOCpWJ54TPKnYiUEqZgVnAnUAisEkptVRrvTO/jNZ6gFv5F4Embg+Ro7VuXNw4hO/SWvPptk+Z/P1ksvKyiAiKYMwdY7in4T3nrA+QkmJn+PAk5s83VhOrVs3ChAnRPPdcVczmi19LwFPyE4HDlQhk4jnhizxRI2gG7NVa7wdwLVDfCdhZRPnHMdY0FoKkjCSGrxnOTwk/AcYaAa+0fYWqlaqeUc7h0Mydm8LIkUdIT3fg56cYMKAaI0fWoHLlkukIvhj/DCoLlYnnhM/yRCKoBRx2u58INC+soFKqLhALfOO2OVAptRmwA5O11l8WcWwPoAdATExM8aMWXqW1ZtEfi5j8/WRO204THhTO2DvGcneju88pu2FDFr17H+K334wpIdq2DeWtt+rQqFHp9AOcT0GNIDuUtByZeE74ptLuLO4MLNFau0/uXldrnaSUqg98o5TarrXed/aBWuu5wFyAuLg4XTrhipJw5NQRhq8ezo8JPwLQvkF7xrYdS2Rw5BnlTpzIY9gw43JQMEYEx8fX5qGHwi5pScmS9M98QyEy8ZzwWZ5IBElAHbf7tV3bCtMZ6O2+QWud5Pq9Xyn1HUb/wTmJQPg+rTWf//k5E7+dyGnbacICw3il7Svn9AU4HJrZs1MYNeoIJ08azUCDBlVj1KgaBAd7rxmoMAWdxdmVSc9J9XI0QlweTySCTUADpVQsRgLoDDxxdiGlVCMgHPjFbVs4kK21tiqlIoFbgCkeiEmUMcdOH2PYqmGsP7gegLsa3MX4tuPPqQX8+msWvXodKpgZ9K67QnnzzTqldjnopcrvI3BkhZCes9fL0QhxeYqdCLTWdqVUH2A1xuWjC7XWO5RS44DNWuulrqKdgUVaa/dmnauBOUopJ2DC6CMoqpNZ+KgVu1cweu1oTuaepEpgFV654xXubXTvGbWAjAwHI0Yk8c47KWgNMTH+xMfX4oEHyk4zUGH+qRHIxHPCd3mkj0BrvQJYcda2MWfdf6WQ434GrvVEDKLsOZV7irHfjOXLnV8C0KpeK15r/xrVQqoVlNFas2TJSfr2PUxysh2LBQYNqs7o0WWvGagwZ3QWZ0siEL5JRhaLEvHLoV8YsnIIRzOPEmgJZMRtI3ji+ifO+HZ/4ICV3r0Ps3KlMTnczTcHM2dODNde6zuDsoKCTAQGQW6OhRPp2d4OR4jLIolAeJTVbmXaD9NYuGUhANfVuI7pHacTGxFbUCYvTxMff4xXXjlKTo6mShUzr70WzfPPR2Iyld1moKJEVDVxJNFJSorN26EIcVkkEQiP2Xl8JwOXD2RP6h7Mykyfm/vQq3mvMxaM+eWX07zwwiG2bzembe7cOZz4+NrUqHHuojK+IrKqH0cSraSlysRzwjdJIhDF5nA6mL9pPvE/xZPnzCM2PJbpHadzXc3rCsqcOuVg2LAkZs82OoNjY/15550Y2rUr3bmBSkJUpB9gJSfTH6vdWugU2UKUZZIIRLEcOXWEQSsGsTFxIwBPN36al1q/dMbka8uXZ9Cz5yESE/OwWGDIkOqMGlWTSpU8t0iMN0VGGrUZZ3YodqedACQRCN8iiUBcthW7VzByzUhOWU8RFRzFa+1fo3Vs64L9J07k0b9/Ip98YlxNc+ONlViwoK5PdQZfjH/GEhiJQAhfI4lAXLIsWxbjvxnP539+DsAdV9zBq+1eLZgoTmvNokXp9O2bSEqKnaAgxYQJ0fTrV80rM4SWNPcZSCURCF8kiUBckj+S/2DA8gEcTD9IgCWAEa1H8GTjJwsuC01MtNGr1yG++sq4JLRNm1Dmzo3hiivKb3OJ+6AySQTCF0kiEBfFqZ3M2zSP6T9Ox+600zCyIW/c80bB0pFOpzFN9NChSWRmOqlSxczrr9fiueeqlumRwZ5QMPFcdgh2hyQC4XskEYgLSs5MZsjKIfx86GcAujbtykutXiq4OmbPnly6dz/E+vWnAbj//irMmlWH6Gh/r8VcmgrWJMgKxa4lEQjfI4lAnNfavWsZvno46TnpRARFMKXDFG6vb6w6ardrpk8/xssvHyU3V1OtmoVZs+qUqWmiS4P7NBNSIxC+SBKBKFROXg6TvpvEJ9s+AYx5gqZ0mEJUcBQAO3bk0LVrAlu2GNMqdO0aweuv1y74UKxI3Nctlj4C4Ysq3rtWXNBfJ/6i37J+7E3bi7/Zn6GthtK1aVdMyoTLine truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
@@ -232,22 +225,22 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 6,
|
||||
"execution_count": 5,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"[<matplotlib.lines.Line2D at 0x136f4b290>]"
|
||||
"[<matplotlib.lines.Line2D at 0x7f68b7dd9e50>]"
|
||||
]
|
||||
},
|
||||
"execution_count": 6,
|
||||
"execution_count": 5,
|
||||
"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>"
|
||||
]
|
||||
@@ -274,24 +267,24 @@
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"And as image over time..."
|
||||
"And as MAE, and image over time..."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 8,
|
||||
"execution_count": 39,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"(1, 128, 33, 1)\n"
|
||||
"Vels array shape: (1, 128, 33, 1)\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": "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 truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
@@ -309,16 +302,23 @@
|
||||
" a = np.concatenate( [a,a] , axis=3)\n",
|
||||
"\n",
|
||||
" a = np.reshape( a, [a.shape[1],a.shape[2]*a.shape[3]] )\n",
|
||||
" plt.imshow(a, origin='lower', cmap='magma')\n",
|
||||
" plt.imshow(a, origin='upper', cmap='magma')\n",
|
||||
" \n",
|
||||
"# run all states through TF to get numpy arrays \n",
|
||||
"vels = sess.run(states) # convert all states in 1 sess.run() call\n",
|
||||
"vels = [ x.velocity.data for x in vels] \n",
|
||||
"#vels = np.asarray( [ sess.run(x, feed_dict={state_in: stateN}).velocity.data for x in states] ) # also works, but slower\n",
|
||||
"sn = np.concatenate(vels, axis=-1)\n",
|
||||
"sn = np.reshape(sn, list(sn.shape)+[1] ) ; print(sn.shape)\n",
|
||||
"vels = np.concatenate(vels, axis=-1)\n",
|
||||
"vels = np.reshape(vels, list(vels.shape)+[1] ) ; print(\"Vels array shape: \"+format(vels.shape))\n",
|
||||
"\n",
|
||||
"show_state(sn)"
|
||||
"# TODO later?\n",
|
||||
"#error = np.sum( np.abs( np.asarray(vels[0:16]).flatten() - np.asarray(velP[bc_tx:bc_tx+steps]).flatten() )) / (steps*n)\n",
|
||||
"#print(\"Mean absolute error across {} steps: {:7.5f}\".format(steps,error))\n",
|
||||
"\n",
|
||||
"# save for comparison with other methods\n",
|
||||
"np.savez_compressed(\"./temp/burgers-diffphys-solution.npz\", np.reshape(vels,[n,steps+1])) # remove batch & channel dimension\n",
|
||||
"\n",
|
||||
"show_state(vels)"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -328,6 +328,117 @@
|
||||
"So far only 1D Burgers, but this works with more complex model equations and higher dimensions in just the same way."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"# Physics-Informed vs. Differentiable Physics Reconstruction\n",
|
||||
"\n",
|
||||
"now we have both versions, let's compare both reconstructions in more detail...\n",
|
||||
"\n",
|
||||
"let's first look at the solutions side by side, from top to bottom: the \"ground truth\" solution as given by the regular forward simulation, in the middle the PINN reconstruction, and at the bottom the differentiable physics version:"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 54,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"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": [
|
||||
"# note, this requires running the forward sim & PINN notebooks beforehand\n",
|
||||
"sol_gt=npfile=np.load(\"./temp/burgers-groundtruth-solution.npz\")[\"arr_0\"] ; #print(format(sol_gt.shape)) \n",
|
||||
"sol_pi=npfile=np.load(\"./temp/burgers-pinn-solution.npz\")[\"arr_0\"] ; #print(format(sol_pi.shape)) \n",
|
||||
"sol_dp=npfile=np.load(\"./temp/burgers-diffphys-solution.npz\")[\"arr_0\"] ; #print(format(sol_dp.shape)) \n",
|
||||
"\n",
|
||||
"divider = np.ones([10,33])*-1. # we'll sneak in a block of -1s to show a black divider in the image\n",
|
||||
"sbs = np.concatenate( [sol_gt, divider, sol_pi, divider, sol_dp], axis=0)\n",
|
||||
"\n",
|
||||
"show_state(np.reshape(sbs,[1,n*3+20,33,1]))"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"It's clealry visible here "
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 53,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"MAE PINN: \t967.83557, \n",
|
||||
"MAE DiffPhys: \t193.25261\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"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": [
|
||||
"err_pi = np.sum( np.abs(sol_pi-sol_gt))\n",
|
||||
"err_dp = np.sum( np.abs(sol_dp-sol_gt))\n",
|
||||
"#error = np.sum( np.abs( np.asarray(vels[0:16]).flatten() - np.asarray(velP[bc_tx:bc_tx+steps]).flatten() )) / (steps*n)\n",
|
||||
"print(\"MAE PINN: \\t{:7.5f} \\nMAE DiffPhys: \\t{:7.5f}\".format(err_pi,err_dp))\n",
|
||||
"\n",
|
||||
"show_state(np.reshape( np.concatenate([sol_pi-sol_gt, divider, sol_dp-sol_gt],axis=0) ,[1,n*2+10,33,1]))"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"That's pretty clear, the PINN error is ca. 5 times higher than the Differentiable Physics (DP) reconstruction.\n",
|
||||
"\n",
|
||||
"That also shows clearly in the jointly visualized image: the magnitudes of the errors of the DP reconstruction are much closer to zero, as indicated by the purple color above."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Now we have a first example to show similarities and differences of the two approaches. In the section, we'll present a more in depth discussion of these findings, before moving to more complex cases."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Exercises:\n",
|
||||
"As with the PINN version, there's variety of things that can be improved and experimented with here:\n",
|
||||
"\n",
|
||||
"* Try to adjust the training parameters to improve the reconstruction.\n",
|
||||
"* As for the PINN case, you can activate a different optimizer, and observe the changing (not necessarily improved) behavior.\n",
|
||||
"* Vary the number of steps, or the resolution of the simulation and reconstruction."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
@@ -352,7 +463,7 @@
|
||||
"name": "python",
|
||||
"nbconvert_exporter": "python",
|
||||
"pygments_lexer": "ipython3",
|
||||
"version": "3.7.6"
|
||||
"version": "3.8.5"
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
|
||||
+50
-12
@@ -6,8 +6,14 @@
|
||||
"source": [
|
||||
"# Simple forward simulation in phiflow\n",
|
||||
"\n",
|
||||
"how to run\n",
|
||||
"simple model equation $f(x) = ...$ "
|
||||
"how to run ...\n",
|
||||
"simple model equation \n",
|
||||
"$\n",
|
||||
" \\frac{\\partial u}{\\partial{t}} + u \\nabla u =\n",
|
||||
" \\nu \\nabla\\cdot \\nabla u\n",
|
||||
"$ \n",
|
||||
"\n",
|
||||
"Note, the first command with a \"!\" prefix installs the [Φ<sub>Flow</sub> Python package from GitHub](https://github.com/tum-pbs/PhiFlow) via `pip` in your python environment. (Skip or modify this command if necessary.)"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -38,14 +44,28 @@
|
||||
" [0.09414805]\n",
|
||||
" [0.10529345]] ...\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"name": "stderr",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"/home/thuerey/anaconda3/envs/tf/lib/python3.8/_collections_abc.py:743: MatplotlibDeprecationWarning: The global colormaps dictionary is no longer considered public API.\n",
|
||||
" for key in self._mapping:\n",
|
||||
"/home/thuerey/anaconda3/envs/tf/lib/python3.8/_collections_abc.py:744: MatplotlibDeprecationWarning: The global colormaps dictionary is no longer considered public API.\n",
|
||||
" yield (key, self._mapping[key])\n",
|
||||
"/home/thuerey/phiflow/phi/viz/display.py:80: UserWarning: GUI is disabled because of missing dependencies: No module named 'imageio'. To install all dependencies, run $ pip install phiflow[gui]\n",
|
||||
" warnings.warn('GUI is disabled because of missing dependencies: %s. To install all dependencies, run $ pip install phiflow[gui]' % import_error)\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"!pip install --upgrade --quiet phiflow\n",
|
||||
"from phi.flow import *\n",
|
||||
"\n",
|
||||
"# run with phiflow\n",
|
||||
"n = 128\n",
|
||||
"dt = 1./32.\n",
|
||||
"steps = 32\n",
|
||||
"dt = 1./steps\n",
|
||||
"viscosity = 0.01/np.pi\n",
|
||||
"initial = np.asarray( [ [-math.sin(np.pi * x) * 1.] for x in np.linspace(-1,1,n)] )\n",
|
||||
"\n",
|
||||
@@ -84,7 +104,7 @@
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"[<matplotlib.lines.Line2D at 0x122bc1550>]"
|
||||
"[<matplotlib.lines.Line2D at 0x7f17fdeb2610>]"
|
||||
]
|
||||
},
|
||||
"execution_count": 2,
|
||||
@@ -93,7 +113,7 @@
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
@@ -107,8 +127,6 @@
|
||||
"source": [
|
||||
"# we only need \"velocity.data\" from each phiflow state\n",
|
||||
"vels = [x.velocity.data for x in state]\n",
|
||||
"sn = np.concatenate(vels, axis=-1)\n",
|
||||
"sn = np.reshape(sn, list(sn.shape)+[1] ) # print(sn.shape)\n",
|
||||
"\n",
|
||||
"# print(vels[0][0,:,0].shape)\n",
|
||||
"# print(vels[0].shape[1])\n",
|
||||
@@ -131,12 +149,19 @@
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 3,
|
||||
"execution_count": 7,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"Vels array shape: (1, 128, 33, 1)\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
@@ -156,10 +181,23 @@
|
||||
"\n",
|
||||
" a = np.reshape( a, [a.shape[1],a.shape[2]*a.shape[3]] )\n",
|
||||
" #print(\"resulting image size\" +format(a.shape))\n",
|
||||
" plt.imshow(a, origin='lower', cmap='magma')\n",
|
||||
" plt.imshow(a, origin='upper', cmap='magma')\n",
|
||||
" \n",
|
||||
"show_state(sn)"
|
||||
"vels_img = np.asarray( np.concatenate(vels, axis=-1), dtype=np.float32 )\n",
|
||||
"vels_img = np.reshape(vels_img, list(vels_img.shape)+[1] ) ; print(\"Vels array shape: \"+format(vels_img.shape))\n",
|
||||
"\n",
|
||||
"# save for comparison with reconstructions later on\n",
|
||||
"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)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": []
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
@@ -178,7 +216,7 @@
|
||||
"name": "python",
|
||||
"nbconvert_exporter": "python",
|
||||
"pygments_lexer": "ipython3",
|
||||
"version": "3.7.6"
|
||||
"version": "3.8.5"
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
|
||||
+375
-341
@@ -1,347 +1,381 @@
|
||||
{
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 0,
|
||||
"metadata": {
|
||||
"colab": {
|
||||
"name": "Forw Simulations with Φ-Flow.ipynb",
|
||||
"provenance": [],
|
||||
"collapsed_sections": []
|
||||
},
|
||||
"kernelspec": {
|
||||
"name": "python3",
|
||||
"display_name": "Python 3"
|
||||
}
|
||||
"cells": [
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "o4JZ84moBKMr"
|
||||
},
|
||||
"source": [
|
||||
"# Navier-Stokes Forward Simulation with Φ<sub>Flow</sub>\n",
|
||||
"\n",
|
||||
"... now a more complex example with fluid simulations (Navier-Stokes) ... still very simple with phiflow (complexity of the differentiable operators hidden).\n",
|
||||
"\n",
|
||||
"As before, the first command with a \"!\" prefix installs the [Φ<sub>Flow</sub> Python package from GitHub](https://github.com/tum-pbs/PhiFlow) via `pip` in your python environment. (Skip or modify this command if necessary.)"
|
||||
]
|
||||
},
|
||||
"cells": [
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "o4JZ84moBKMr"
|
||||
},
|
||||
"source": [
|
||||
"# Navier-Stokes Forward Simulation with Φ<sub>Flow</sub>\n",
|
||||
"\n",
|
||||
"... now a more complex example with fluid simulations (Navier-Stokes) ... still very simple with phiflow (complexity of the differentiable operators hidden).\n",
|
||||
"\n",
|
||||
"Only for colab: \n",
|
||||
"Execute the cell below to install the [Φ<sub>Flow</sub> Python package from GitHub](https://github.com/tum-pbs/PhiFlow)."
|
||||
]
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 1,
|
||||
"metadata": {
|
||||
"colab": {
|
||||
"base_uri": "https://localhost:8080/"
|
||||
},
|
||||
"id": "da1uZcDXdVcF",
|
||||
"outputId": "1082dc87-796c-4b57-e72e-5790fc1444c9"
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"cell_type": "code",
|
||||
"metadata": {
|
||||
"id": "Z6YoAVKebfNV"
|
||||
},
|
||||
"source": [
|
||||
"!pip install --upgrade --quiet phiflow"
|
||||
],
|
||||
"execution_count": 1,
|
||||
"outputs": []
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"metadata": {
|
||||
"id": "da1uZcDXdVcF",
|
||||
"colab": {
|
||||
"base_uri": "https://localhost:8080/"
|
||||
},
|
||||
"outputId": "1082dc87-796c-4b57-e72e-5790fc1444c9"
|
||||
},
|
||||
"source": [
|
||||
"from phi.flow import * # The Dash GUI is not supported on Google Colab, ignore the warning\n",
|
||||
"import pylab"
|
||||
],
|
||||
"execution_count": 2,
|
||||
"outputs": [
|
||||
{
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"/usr/local/lib/python3.6/dist-packages/phi/viz/display.py:80: UserWarning: GUI is disabled because of missing dependencies: No module named 'dash_core_components'. To install all dependencies, run $ pip install phiflow[gui]\n",
|
||||
" warnings.warn('GUI is disabled because of missing dependencies: %s. To install all dependencies, run $ pip install phiflow[gui]' % import_error)\n"
|
||||
],
|
||||
"name": "stderr"
|
||||
}
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "BVV1IKVqDfLl"
|
||||
},
|
||||
"source": [
|
||||
"## Setting up the simulation\n",
|
||||
"\n",
|
||||
"Φ<sub>Flow</sub> is object-oriented, i.e. you assemble your simulation by constructing a number of objects and adding them to the world.\n",
|
||||
"\n",
|
||||
"The following code sets up four fluid simulations that run in parallel (`batch_size=4`). Each fluid simulation has a circular Inflow at a different location."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"metadata": {
|
||||
"id": "WrA3IXDxv31P"
|
||||
},
|
||||
"source": [
|
||||
"world = World()\n",
|
||||
"fluid = world.add(Fluid(Domain([40, 32], boundaries=CLOSED), buoyancy_factor=0.05), physics=IncompressibleFlow())\n",
|
||||
"world.add(Inflow(Sphere(center=[5,14], radius=3), rate=0.2));"
|
||||
],
|
||||
"execution_count": 3,
|
||||
"outputs": []
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "ExA0Pi2sFVka"
|
||||
},
|
||||
"source": [
|
||||
"The inflow affects the fluid's marker density. Because a Boussinesq model with a positive `buoyancy_factor` is active, the marker field `density` creates an upward force. Note that this density is not the density of the fluid, but the amount of marker quantity present in each cell.\n",
|
||||
"\n",
|
||||
"Let's plot the marker density after one simulation frame."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"metadata": {
|
||||
"id": "WmGZdOwswOva",
|
||||
"colab": {
|
||||
"base_uri": "https://localhost:8080/",
|
||||
"height": 282
|
||||
},
|
||||
"outputId": "3ae4d68d-b586-4bbe-eca9-a223d7720949"
|
||||
},
|
||||
"source": [
|
||||
"world.step()\n",
|
||||
"pylab.imshow(fluid.density.data[0,...,0], origin='lower', cmap='magma')"
|
||||
],
|
||||
"execution_count": 4,
|
||||
"outputs": [
|
||||
{
|
||||
"output_type": "execute_result",
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"<matplotlib.image.AxesImage at 0x7fc128aa7ef0>"
|
||||
]
|
||||
},
|
||||
"metadata": {
|
||||
"tags": []
|
||||
},
|
||||
"execution_count": 4
|
||||
},
|
||||
{
|
||||
"output_type": "display_data",
|
||||
"data": {
|
||||
"image/png": "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\n",
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {
|
||||
"tags": [],
|
||||
"needs_background": "light"
|
||||
}
|
||||
}
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "PA-2tGuWGHv2"
|
||||
},
|
||||
"source": [
|
||||
"We can run more steps by repeatedly calling `world.step()`."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"metadata": {
|
||||
"id": "0hZk5HX3w4Or",
|
||||
"colab": {
|
||||
"base_uri": "https://localhost:8080/"
|
||||
},
|
||||
"outputId": "f7811af7-4b58-4ff6-a8b6-6e7bedefaa6e"
|
||||
},
|
||||
"source": [
|
||||
"for frame in range(10):\n",
|
||||
" print('Computing frame %d' % frame)\n",
|
||||
" world.step(dt=1.5)"
|
||||
],
|
||||
"execution_count": 5,
|
||||
"outputs": [
|
||||
{
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"Computing frame 0\n",
|
||||
"Computing frame 1\n",
|
||||
"Computing frame 2\n",
|
||||
"Computing frame 3\n",
|
||||
"Computing frame 4\n",
|
||||
"Computing frame 5\n",
|
||||
"Computing frame 6\n",
|
||||
"Computing frame 7\n",
|
||||
"Computing frame 8\n",
|
||||
"Computing frame 9\n"
|
||||
],
|
||||
"name": "stdout"
|
||||
}
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "GMKKWQBLHIwP"
|
||||
},
|
||||
"source": [
|
||||
"Now the hot plume is starting to rise:"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"metadata": {
|
||||
"id": "Mfl80CjZxZcL",
|
||||
"colab": {
|
||||
"base_uri": "https://localhost:8080/",
|
||||
"height": 282
|
||||
},
|
||||
"outputId": "92f3a9ba-d403-4799-a543-132ee8ed234c"
|
||||
},
|
||||
"source": [
|
||||
"pylab.imshow(fluid.density.data[0,...,0], origin='lower', cmap='magma')"
|
||||
],
|
||||
"execution_count": 6,
|
||||
"outputs": [
|
||||
{
|
||||
"output_type": "execute_result",
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"<matplotlib.image.AxesImage at 0x7fc1265294a8>"
|
||||
]
|
||||
},
|
||||
"metadata": {
|
||||
"tags": []
|
||||
},
|
||||
"execution_count": 6
|
||||
},
|
||||
{
|
||||
"output_type": "display_data",
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {
|
||||
"tags": [],
|
||||
"needs_background": "light"
|
||||
}
|
||||
}
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "wnbQJvA-HPSL"
|
||||
},
|
||||
"source": [
|
||||
"Let's compute and show a few more steps of the simulation."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"metadata": {
|
||||
"colab": {
|
||||
"base_uri": "https://localhost:8080/",
|
||||
"height": 489
|
||||
},
|
||||
"id": "tkhCOzc0ITsj",
|
||||
"outputId": "f6366c12-1eb5-4ff6-e0d7-94b806bfd8e4"
|
||||
},
|
||||
"source": [
|
||||
"frames = [fluid.density.data[0,...,0]]\n",
|
||||
"for frame in range(20):\n",
|
||||
" print('Computing frame %d' % frame)\n",
|
||||
" world.step(dt=1.5)\n",
|
||||
" if frame%5==0:\n",
|
||||
" frames.append(fluid.density.data[0,...,0])\n",
|
||||
"\n",
|
||||
"pylab.imshow(np.concatenate(frames,axis=1), origin='lower', cmap='magma')"
|
||||
],
|
||||
"execution_count": 7,
|
||||
"outputs": [
|
||||
{
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"Computing frame 0\n",
|
||||
"Computing frame 1\n",
|
||||
"Computing frame 2\n",
|
||||
"Computing frame 3\n",
|
||||
"Computing frame 4\n",
|
||||
"Computing frame 5\n",
|
||||
"Computing frame 6\n",
|
||||
"Computing frame 7\n",
|
||||
"Computing frame 8\n",
|
||||
"Computing frame 9\n",
|
||||
"Computing frame 10\n",
|
||||
"Computing frame 11\n",
|
||||
"Computing frame 12\n",
|
||||
"Computing frame 13\n",
|
||||
"Computing frame 14\n",
|
||||
"Computing frame 15\n",
|
||||
"Computing frame 16\n",
|
||||
"Computing frame 17\n",
|
||||
"Computing frame 18\n",
|
||||
"Computing frame 19\n"
|
||||
],
|
||||
"name": "stdout"
|
||||
},
|
||||
{
|
||||
"output_type": "execute_result",
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"<matplotlib.image.AxesImage at 0x7fc1265346a0>"
|
||||
]
|
||||
},
|
||||
"metadata": {
|
||||
"tags": []
|
||||
},
|
||||
"execution_count": 7
|
||||
},
|
||||
{
|
||||
"output_type": "display_data",
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {
|
||||
"tags": [],
|
||||
"needs_background": "light"
|
||||
}
|
||||
}
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "ooqVxCPM8PXl"
|
||||
},
|
||||
"source": [
|
||||
"It looks simple here, but this is a powerful tool. The simulation could easily be extended to more complex cases or 3D, and they're fully compatible with back-propagation pipelines of deep learning frameworks. \n",
|
||||
"\n",
|
||||
"In the next chapters we'll show how to use these simulations for training NNs, and how to steer and modify them via trained NNs. This will illustrate how much we can improve the training process by having a solver in the loop, and especially by having differentiable solvers."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"metadata": {
|
||||
"id": "BL-AOqwMJmkq"
|
||||
},
|
||||
"source": [
|
||||
""
|
||||
],
|
||||
"execution_count": null,
|
||||
"outputs": []
|
||||
"name": "stderr",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"/home/thuerey/anaconda3/envs/tf/lib/python3.8/_collections_abc.py:743: MatplotlibDeprecationWarning: The global colormaps dictionary is no longer considered public API.\n",
|
||||
" for key in self._mapping:\n",
|
||||
"/home/thuerey/anaconda3/envs/tf/lib/python3.8/_collections_abc.py:744: MatplotlibDeprecationWarning: The global colormaps dictionary is no longer considered public API.\n",
|
||||
" yield (key, self._mapping[key])\n",
|
||||
"/home/thuerey/phiflow/phi/viz/display.py:80: UserWarning: GUI is disabled because of missing dependencies: No module named 'imageio'. To install all dependencies, run $ pip install phiflow[gui]\n",
|
||||
" warnings.warn('GUI is disabled because of missing dependencies: %s. To install all dependencies, run $ pip install phiflow[gui]' % import_error)\n"
|
||||
]
|
||||
}
|
||||
]
|
||||
],
|
||||
"source": [
|
||||
"!pip install --upgrade --quiet phiflow\n",
|
||||
"from phi.flow import * # The Dash GUI is not supported on Google Colab, ignore the warning\n",
|
||||
"import pylab"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "BVV1IKVqDfLl"
|
||||
},
|
||||
"source": [
|
||||
"## Setting up the simulation\n",
|
||||
"\n",
|
||||
"Φ<sub>Flow</sub> is object-oriented, i.e. you assemble your simulation by constructing a number of objects and adding them to the world.\n",
|
||||
"\n",
|
||||
"The following code sets up four fluid simulations that run in parallel (`batch_size=4`). Each fluid simulation has a circular Inflow at a different location."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 2,
|
||||
"metadata": {
|
||||
"id": "WrA3IXDxv31P"
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"world = World()\n",
|
||||
"fluid = world.add(Fluid(Domain([40, 32], boundaries=CLOSED), buoyancy_factor=0.05), physics=IncompressibleFlow())\n",
|
||||
"world.add(Inflow(Sphere(center=[5,14], radius=3), rate=0.2));"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "ExA0Pi2sFVka"
|
||||
},
|
||||
"source": [
|
||||
"The inflow affects the fluid's marker density. Because a Boussinesq model with a positive `buoyancy_factor` is active, the marker field `density` creates an upward force. Note that this density is not the density of the fluid, but the amount of marker quantity present in each cell.\n",
|
||||
"\n",
|
||||
"Let's plot the marker density after one simulation frame."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 3,
|
||||
"metadata": {
|
||||
"colab": {
|
||||
"base_uri": "https://localhost:8080/",
|
||||
"height": 282
|
||||
},
|
||||
"id": "WmGZdOwswOva",
|
||||
"outputId": "3ae4d68d-b586-4bbe-eca9-a223d7720949"
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"<matplotlib.image.AxesImage at 0x7f9cf945eee0>"
|
||||
]
|
||||
},
|
||||
"execution_count": 3,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": "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\n",
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
},
|
||||
"metadata": {
|
||||
"needs_background": "light"
|
||||
},
|
||||
"output_type": "display_data"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"world.step()\n",
|
||||
"pylab.imshow(fluid.density.data[0,...,0], origin='lower', cmap='magma')"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"A lot has happened internally in the `world.step()` call, e.g., for fluids we have a more complex `struct` object that stores the different fields of the simulation. For fluids we have the aforementioned marker density in addition to a velocity."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 8,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"Fluid state: Fluid[density: Grid[40x32(1), size=[40. 32.], float32], velocity: StaggeredGrid[40x32, size=[40. 32.], float32]]\n",
|
||||
"\n",
|
||||
" Velocity content:\n",
|
||||
"(1, 41, 32, 1)\n",
|
||||
"(1, 40, 33, 1)\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"print(\"Fluid state: \" + format(fluid.state))\n",
|
||||
"print(\"\\nVelocity content:\")\n",
|
||||
"[print(grid.data.shape) for grid in fluid.velocity.unstack()];"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Note that we actually created two variables, one for each velocity component. If you're interested in how this works, have a look at the [Struct documentation](https://github.com/tum-pbs/PhiFlow/blob/master/documentation/Structs.ipynb).\n",
|
||||
"\n",
|
||||
"If you look closely at the output from the last `print` command, you'll notice that the shapes of the variables differ. This is because the velocity is sampled in [staggered form](https://github.com/tum-pbs/PhiFlow/blob/master/documentation/Staggered_Grids.md).\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "PA-2tGuWGHv2"
|
||||
},
|
||||
"source": [
|
||||
"We can run more steps by repeatedly calling `world.step()`."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 5,
|
||||
"metadata": {
|
||||
"colab": {
|
||||
"base_uri": "https://localhost:8080/"
|
||||
},
|
||||
"id": "0hZk5HX3w4Or",
|
||||
"outputId": "f7811af7-4b58-4ff6-a8b6-6e7bedefaa6e"
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"Computing frame 0\n",
|
||||
"Computing frame 1\n",
|
||||
"Computing frame 2\n",
|
||||
"Computing frame 3\n",
|
||||
"Computing frame 4\n",
|
||||
"Computing frame 5\n",
|
||||
"Computing frame 6\n",
|
||||
"Computing frame 7\n",
|
||||
"Computing frame 8\n",
|
||||
"Computing frame 9\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"for frame in range(10):\n",
|
||||
" print('Computing frame %d' % frame)\n",
|
||||
" world.step(dt=1.5)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "GMKKWQBLHIwP"
|
||||
},
|
||||
"source": [
|
||||
"Now the hot plume is starting to rise:"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 6,
|
||||
"metadata": {
|
||||
"colab": {
|
||||
"base_uri": "https://localhost:8080/",
|
||||
"height": 282
|
||||
},
|
||||
"id": "Mfl80CjZxZcL",
|
||||
"outputId": "92f3a9ba-d403-4799-a543-132ee8ed234c"
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"<matplotlib.image.AxesImage at 0x7fa84d3553d0>"
|
||||
]
|
||||
},
|
||||
"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": [
|
||||
"pylab.imshow(fluid.density.data[0,...,0], origin='lower', cmap='magma')"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "wnbQJvA-HPSL"
|
||||
},
|
||||
"source": [
|
||||
"Let's compute and show a few more steps of the simulation."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 7,
|
||||
"metadata": {
|
||||
"colab": {
|
||||
"base_uri": "https://localhost:8080/",
|
||||
"height": 489
|
||||
},
|
||||
"id": "tkhCOzc0ITsj",
|
||||
"outputId": "f6366c12-1eb5-4ff6-e0d7-94b806bfd8e4"
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"Computing frame 0\n",
|
||||
"Computing frame 1\n",
|
||||
"Computing frame 2\n",
|
||||
"Computing frame 3\n",
|
||||
"Computing frame 4\n",
|
||||
"Computing frame 5\n",
|
||||
"Computing frame 6\n",
|
||||
"Computing frame 7\n",
|
||||
"Computing frame 8\n",
|
||||
"Computing frame 9\n",
|
||||
"Computing frame 10\n",
|
||||
"Computing frame 11\n",
|
||||
"Computing frame 12\n",
|
||||
"Computing frame 13\n",
|
||||
"Computing frame 14\n",
|
||||
"Computing frame 15\n",
|
||||
"Computing frame 16\n",
|
||||
"Computing frame 17\n",
|
||||
"Computing frame 18\n",
|
||||
"Computing frame 19\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"<matplotlib.image.AxesImage at 0x7fa84d2d8790>"
|
||||
]
|
||||
},
|
||||
"execution_count": 7,
|
||||
"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": [
|
||||
"frames = [fluid.density.data[0,...,0]]\n",
|
||||
"for frame in range(20):\n",
|
||||
" print('Computing frame %d' % frame)\n",
|
||||
" world.step(dt=1.5)\n",
|
||||
" if frame%5==0:\n",
|
||||
" frames.append(fluid.density.data[0,...,0])\n",
|
||||
"\n",
|
||||
"pylab.imshow(np.concatenate(frames,axis=1), origin='lower', cmap='magma')"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {
|
||||
"id": "ooqVxCPM8PXl"
|
||||
},
|
||||
"source": [
|
||||
"It looks simple here, but this is a powerful tool. The simulation could easily be extended to more complex cases or 3D, and they're fully compatible with back-propagation pipelines of deep learning frameworks. \n",
|
||||
"\n",
|
||||
"In the next chapters we'll show how to use these simulations for training NNs, and how to steer and modify them via trained NNs. This will illustrate how much we can improve the training process by having a solver in the loop, and especially by having differentiable solvers."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {
|
||||
"id": "BL-AOqwMJmkq"
|
||||
},
|
||||
"outputs": [],
|
||||
"source": []
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
"colab": {
|
||||
"collapsed_sections": [],
|
||||
"name": "Forw Simulations with Φ-Flow.ipynb",
|
||||
"provenance": []
|
||||
},
|
||||
"kernelspec": {
|
||||
"display_name": "Python 3",
|
||||
"language": "python",
|
||||
"name": "python3"
|
||||
},
|
||||
"language_info": {
|
||||
"codemirror_mode": {
|
||||
"name": "ipython",
|
||||
"version": 3
|
||||
},
|
||||
"file_extension": ".py",
|
||||
"mimetype": "text/x-python",
|
||||
"name": "python",
|
||||
"nbconvert_exporter": "python",
|
||||
"pygments_lexer": "ipython3",
|
||||
"version": "3.8.5"
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 1
|
||||
}
|
||||
+180
-270
@@ -7,11 +7,12 @@
|
||||
"# PINN Example \n",
|
||||
"\n",
|
||||
"Burgers example $u=...$\n",
|
||||
"\n",
|
||||
"at $t=0$ , sine function (boundary_t0)\n",
|
||||
"\n",
|
||||
"at sides $t \\in [0,1], u=0$ (open_boundary)\n",
|
||||
"\n",
|
||||
"Note that we're still sampling with 128 points ($n=128), but now we have a discretization via the NN, so we can \n",
|
||||
"also sample points inbetween; similar to linear interpolatoin or basis functions for FEM, just now the NN internally determines how to use its degrees of freedom to construct the basis functions. So we have no direct control over the reconstruction.\n",
|
||||
"\n",
|
||||
"## Preliminaries"
|
||||
]
|
||||
},
|
||||
@@ -20,24 +21,32 @@
|
||||
"execution_count": 1,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stderr",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"/home/thuerey/phiflow/phi/tf/__init__.py:7: UserWarning: TensorFlow 2 is not fully supported by PhiFlow.\n",
|
||||
" warnings.warn('TensorFlow 2 is not fully supported by PhiFlow.')\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"Could not load resample cuda libraries: CUDA binaries not found at /Users/thuerey/Dropbox/mbaDevelSelected/phiflow-v1.5/phi/tf/cuda/build/resample.so. Run \"python setup.py cuda\" to compile them\n",
|
||||
"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"
|
||||
"Could not load resample cuda libraries: CUDA binaries not found at /home/thuerey/phiflow/phi/tf/cuda/build/resample.so. Run \"python setup.py cuda\" to compile them\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",
|
||||
"/home/thuerey/anaconda3/envs/tf/lib/python3.8/_collections_abc.py:743: MatplotlibDeprecationWarning: The global colormaps dictionary is no longer considered public API.\n",
|
||||
" for key in self._mapping:\n",
|
||||
"/home/thuerey/anaconda3/envs/tf/lib/python3.8/_collections_abc.py:744: MatplotlibDeprecationWarning: The global colormaps dictionary is no longer considered public API.\n",
|
||||
" yield (key, self._mapping[key])\n",
|
||||
"/home/thuerey/phiflow/phi/viz/display.py:80: UserWarning: GUI is disabled because of missing dependencies: No module named 'imageio'. To install all dependencies, run $ pip install phiflow[gui]\n",
|
||||
" warnings.warn('GUI is disabled because of missing dependencies: %s. To install all dependencies, run $ pip install phiflow[gui]' % import_error)\n",
|
||||
"/home/thuerey/phiflow/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"
|
||||
@@ -51,6 +60,13 @@
|
||||
"rnd = math.choose_backend(1) # use same random points for all iterations"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"We're important phiflow here, but we won't use it to compute a solution to the PDE. Now we'll instead use the derivatives of an NN (as explained in the previous section) to set up a loss formulation to train an NN."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 2,
|
||||
@@ -116,15 +132,13 @@
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"name": "stderr",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"WARNING:tensorflow:From <ipython-input-2-a9e33b53329c>: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"
|
||||
"/home/thuerey/anaconda3/envs/tf/lib/python3.8/site-packages/tensorflow/python/keras/legacy_tf_layers/core.py:171: UserWarning: `tf.layers.dense` is deprecated and will be removed in a future version. Please use `tf.keras.layers.Dense` instead.\n",
|
||||
" warnings.warn('`tf.layers.dense` is deprecated and '\n",
|
||||
"/home/thuerey/anaconda3/envs/tf/lib/python3.8/site-packages/tensorflow/python/keras/engine/base_layer_v1.py:1719: UserWarning: `layer.apply` is deprecated and will be removed in a future version. Please use `layer.__call__` method instead.\n",
|
||||
" warnings.warn('`layer.apply` is deprecated and '\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
@@ -132,7 +146,8 @@
|
||||
"# generate array with positions: \n",
|
||||
"# -1 to 1 spatial with 128 cells\n",
|
||||
"# 0 to 1 in time with 1+32 timesteps\n",
|
||||
"grids_xt = np.meshgrid(np.linspace(-1, 1, 128), np.linspace(0, 1, 33), indexing='ij')\n",
|
||||
"n=128\n",
|
||||
"grids_xt = np.meshgrid(np.linspace(-1, 1, n), np.linspace(0, 1, 33), indexing='ij')\n",
|
||||
"grid_x, grid_t = [tf.convert_to_tensor(t, tf.float32) for t in grids_xt]\n",
|
||||
"\n",
|
||||
"# # gx ,gt = \n",
|
||||
@@ -148,7 +163,7 @@
|
||||
"# print(format(grid_x.shape))\n",
|
||||
"\n",
|
||||
"# create 4D tensor with batch and channel dimensions in addition to space and time\n",
|
||||
"# in this case gives shape=(1, 128, 33, 1)\n",
|
||||
"# in this case gives shape=(1, n, 33, 1)\n",
|
||||
"grid_u = math.expand_dims(network(grid_x, grid_t))"
|
||||
]
|
||||
},
|
||||
@@ -156,7 +171,7 @@
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Helper to show"
|
||||
"Helper to show (see below)"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -171,20 +186,12 @@
|
||||
"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 432x288 with 1 Axes>"
|
||||
]
|
||||
@@ -213,7 +220,7 @@
|
||||
" a = np.concatenate( [a,a] , axis=3)\n",
|
||||
" a = np.reshape( a, [a.shape[1],a.shape[2]*a.shape[3]] )\n",
|
||||
" #print(a.shape)\n",
|
||||
" plt.imshow(a, origin='lower', cmap='magma')\n",
|
||||
" plt.imshow(a, origin='upper', cmap='magma')\n",
|
||||
" \n",
|
||||
"print(\"Randomly initialized network state:\")\n",
|
||||
"show_state(grid_u)"
|
||||
@@ -236,12 +243,13 @@
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"name": "stderr",
|
||||
"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"
|
||||
"/home/thuerey/anaconda3/envs/tf/lib/python3.8/site-packages/tensorflow/python/keras/legacy_tf_layers/core.py:171: UserWarning: `tf.layers.dense` is deprecated and will be removed in a future version. Please use `tf.keras.layers.Dense` instead.\n",
|
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" warnings.warn('`tf.layers.dense` is deprecated and '\n",
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"/home/thuerey/anaconda3/envs/tf/lib/python3.8/site-packages/tensorflow/python/keras/engine/base_layer_v1.py:1719: UserWarning: `layer.apply` is deprecated and will be removed in a future version. Please use `layer.__call__` method instead.\n",
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" warnings.warn('`layer.apply` is deprecated and '\n"
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]
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}
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],
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"Step 10000, loss: 0.000867\n",
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"Runtime 106.27s\n"
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|
||||
"Runtime 77.39s\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
@@ -501,7 +329,11 @@
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Looks pretty good on first sight!"
|
||||
"This training can take a significant amount of time, around 1 minute on a typical notebook, but at least the error goes down significantly, and the network seems to successfully converge to a solution.\n",
|
||||
"\n",
|
||||
"Let's show the reconstruction of the network, by evaluating the network at the centers of a regular grid, so that we can show the solution as an image. Note that this is actually fairly expensive, we have to run through the whole network with a few thousand weights for all of the $128 \\times 32$ points in the grid.\n",
|
||||
"\n",
|
||||
"It looks pretty good on first sight, though:\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -511,7 +343,7 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": "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 truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
@@ -523,10 +355,7 @@
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"# a = session.run(grid_u)\n",
|
||||
"# a = a[0,:,:,0].transpose() # ; print(a.shape)\n",
|
||||
"# pylab.imshow(a, origin='lower', cmap='magma')\n",
|
||||
"show_state(grid_u)"
|
||||
"show_state(grid_u) # note, this helper internally does a `sess.run()`"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -537,8 +366,7 @@
|
||||
"\n",
|
||||
"## Evaluation\n",
|
||||
"\n",
|
||||
"Let's compare solution at $t=0$\n",
|
||||
"Actually not that great, blue curve from PINN quite far away from constraits (in gray)... Solution get's better with larger number of iterations, but requires a surprisingly large number of them for a fairly simple case. The shock in the middle is not well recovered (the maximum is still much too low).\n"
|
||||
"Let's compare solution in a bit more detail. Here are the actual sample points used for constraining the solution (at time step 16, $t=1/2$) shown in gray, versus the reconstructed solution in blue:\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -549,7 +377,7 @@
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"<matplotlib.collections.PathCollection at 0x13ab749d0>"
|
||||
"<matplotlib.collections.PathCollection at 0x7f31f079da90>"
|
||||
]
|
||||
},
|
||||
"execution_count": 8,
|
||||
@@ -558,7 +386,7 @@
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
@@ -585,7 +413,13 @@
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"Let's check how well the initial state was reconstructed, compared ot the (known) ground truth solution."
|
||||
"Not too bad at the sides of the domain (the Dirichlet boundary conditions $u=0$ are fulfilled), but the shock at $x=0$ is not well represented.\n",
|
||||
"\n",
|
||||
"Let's check how well the initial state at $t=0$ was reconstructed. That's the most interesting, and toughest part of the problem (the rest basically follows from the model equation and boundary conditions, given the first state).\n",
|
||||
"\n",
|
||||
"And turns out, it's actually not that great, blue curve from PINN quite far away from constraits (in gray)... Solution get's better with larger number of iterations, but requires a surprisingly large number of them for a fairly simple case. The shock in the middle is not well recovered (the maximum is still much too low).\n",
|
||||
"\n",
|
||||
"... shown in blue, compared ot the (known) ground truth solution in gray ..."
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -596,7 +430,7 @@
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"[<matplotlib.lines.Line2D at 0x13ac00110>]"
|
||||
"[<matplotlib.lines.Line2D at 0x7f31f0719a00>]"
|
||||
]
|
||||
},
|
||||
"execution_count": 9,
|
||||
@@ -605,7 +439,7 @@
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
@@ -617,8 +451,6 @@
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"n = 128\n",
|
||||
"\n",
|
||||
"# ground truth solution at t0\n",
|
||||
"t0gt = np.asarray( [ [-math.sin(np.pi * x) * 1.] for x in np.linspace(-1,1,n)] )\n",
|
||||
"\n",
|
||||
@@ -634,9 +466,10 @@
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"That's not very close...\n",
|
||||
"especially the maximum / minimum at $x=\\pm 1/2$ are far off, and the boudaries at $x=\\pm 1$ are not fulfilled.\n",
|
||||
"\n",
|
||||
"But we have the forward simulator, so we can use the $t=0$ solution of the network to \n",
|
||||
"evaluate how well the temporal evoluation was reconstructed by the PINN."
|
||||
"We have the forward simulator for this simulation, so we can use the $t=0$ solution of the network to \n",
|
||||
"evaluate how well the temporal evoluation was reconstructed by the PINN. This measures how well the temporal evolution of the model equation was captured via the soft constraints of the PINN loss."
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -654,7 +487,7 @@
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"[<matplotlib.lines.Line2D at 0x13ac91710>]"
|
||||
"[<matplotlib.lines.Line2D at 0x7f31f069f1f0>]"
|
||||
]
|
||||
},
|
||||
"execution_count": 10,
|
||||
@@ -663,7 +496,7 @@
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
@@ -722,7 +555,7 @@
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"[<matplotlib.lines.Line2D at 0x13ad0b910>]"
|
||||
"[<matplotlib.lines.Line2D at 0x7f31f0671e80>]"
|
||||
]
|
||||
},
|
||||
"execution_count": 11,
|
||||
@@ -731,7 +564,7 @@
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": "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 truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
@@ -757,31 +590,35 @@
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"And the error in the PINN reconstruction. Not surprisingly grows over time. \n",
|
||||
"\n",
|
||||
"The shock is not \"undone\" well towards $t=0$.\n",
|
||||
"\n",
|
||||
"(Old: BC at 0, the last state at $t=1$ in purple doesnt capture the sharp shock very well.)"
|
||||
"Not surprisingly, the errors grow over time. Especially the steepening of the solution near the shock at $x=0$ is not \"captured\" well. It's a bit difficult to see in these two graphs, though, let's quantify the error and show the actual difference:"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 12,
|
||||
"execution_count": 20,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"[(2048,), (2048,)]\n",
|
||||
"Mean absolute error across 16 steps: 0.01397\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"[<matplotlib.lines.Line2D at 0x13ad82d10>]"
|
||||
"[<matplotlib.lines.Line2D at 0x7f31f07df610>]"
|
||||
]
|
||||
},
|
||||
"execution_count": 12,
|
||||
"execution_count": 20,
|
||||
"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>"
|
||||
]
|
||||
@@ -793,6 +630,9 @@
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"error = np.sum( np.abs( np.asarray(vels[0:16]).flatten() - np.asarray(velP[bc_tx:bc_tx+steps]).flatten() )) / (steps*n)\n",
|
||||
"print(\"Mean absolute error for re-simulation across {} steps: {:7.5f}\".format(steps,error))\n",
|
||||
"\n",
|
||||
"fig = plt.figure().gca()\n",
|
||||
"fig.plot(pltx, (vels[0 ].flatten()-velP[bc_tx ].flatten()), lw=2, color='blue')\n",
|
||||
"#fig.plot(pltx, (vels[steps//4].flatten()-velP[bc_tx+steps//4].flatten()), lw=2, color='green')\n",
|
||||
@@ -804,17 +644,21 @@
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"All steps over time, with a color map... (note x axis does not show actual simulation time)"
|
||||
"This should show a mean absolute error of ca. $1.4 \\cdot 10^{-2}$, which is significant for the value range of the simulation.\n",
|
||||
"\n",
|
||||
"And for comparison with the forward simulation and following cases, here are also all steps over time with a color map... (note x axis does not show actual simulation time)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 13,
|
||||
"metadata": {},
|
||||
"metadata": {
|
||||
"scrolled": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": "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 truncated
|
||||
"text/plain": [
|
||||
"<Figure size 432x288 with 1 Axes>"
|
||||
]
|
||||
@@ -832,6 +676,72 @@
|
||||
"\n",
|
||||
"show_state(sn, run_session=False)"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"More important, though, the mean absolute error (MAE) for the whole solution, i.e., from start to end:\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"error_full = np.sum( np.abs( np.asarray(vels[0:16]).flatten() - np.asarray(velP[bc_tx:bc_tx+steps]).flatten() )) / (steps*n)\n",
|
||||
"print(\"Overall MAE: {:7.5f}\".format(error_full))"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"... store solution , just for comparison ..."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 29,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"Vels array shape: (128, 33)\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"vels = session.run(grid_u) # special for showing NN results, run through TF \n",
|
||||
"vels = np.reshape( vels, [vels.shape[1],vels.shape[2]] )\n",
|
||||
"\n",
|
||||
"# save for comparison with other methods\n",
|
||||
"np.savez_compressed(\"./temp/burgers-pinn-solution.npz\",vels) ; print(\"Vels array shape: \"+format(vels.shape))"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"## Exercises\n",
|
||||
"\n",
|
||||
"This is just a starting point, the parameters were chosen to compare to run relatively quickly by default. The solution can be improved however. E.g., try to:\n",
|
||||
"\n",
|
||||
"* Adjust parameters of the training to further decrease the error without making the solution diverge\n",
|
||||
"* Adapt the NN architecture for further improvements (keep track of the weight count, though!)\n",
|
||||
"* Activate a different optimizer, and observe the changing behavior (this typically requires adjusting the learning rate). Note that the more complex optimizers don't necessarily do better in this relatively simple example.\n",
|
||||
"* Or modify the setup to make the test case more interesting: e.g., move the boundary conditions further back in time, to let the reconstruction go \"further backward\" in time."
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": null,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": []
|
||||
}
|
||||
],
|
||||
"metadata": {
|
||||
@@ -850,7 +760,7 @@
|
||||
"name": "python",
|
||||
"nbconvert_exporter": "python",
|
||||
"pygments_lexer": "ipython3",
|
||||
"version": "3.7.6"
|
||||
"version": "3.8.5"
|
||||
}
|
||||
},
|
||||
"nbformat": 4,
|
||||
|
||||
Reference in new issue
Block a user