preliminary version of DP NS example, not fully working so far
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@@ -8,8 +8,9 @@
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"source": [
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"# Differentiable Fluid Simulations\n",
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"\n",
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"<br>**TODO , NEXT finish opt case, diverges? see colab example , finish texts before end.**\n",
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"<br>**TODO , fix bottom row, crop in display with [1:,0:]**\n",
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"<br>**TODO , NEXT finish opt case, diverges? see colab example**\n",
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"\n",
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"**update from diffphys-code-ns-v1-syncWith2.ipynb, diff display for target, adjust params; run optimization longer so that it properly converges**\n",
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"\n",
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"<br><br>\n",
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"\n",
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@@ -51,7 +52,7 @@
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"sampling locations and time. \n",
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"The index $i$ here runs over all discretized, spatial degrees of freedom in our fluid solver (we'll have $32\\times40$ below).\n",
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"\n",
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"In contrast to before, we are not dealing with pre-computed quantities anymore, but $x_{t_e,i}$ is a complex, non-linear function itself.\n",
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"In contrast to before, we are not dealing with pre-computed quantities anymore, but now $x_{t_e,i}$ is a complex, non-linear function itself.\n",
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"More specifically, the simulator starts with the initial velocity $\\mathbf{u}_0$ and density $d_0$ to compute the $x_{t_e,i}$, by $n$ evaluations of the discretized pde $\\mathcal{P}$.\n",
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"This gives as simulated state and quantitiy to optimize: $x_{t_e,i} = d_{t_e} = \\mathcal{P}^n(\\mathbf{u}_0,d_0)$. We will leave $d_0$ fixed in the following, and focus on $\\mathbf{u}_0$ as our degrees of freedom.\n",
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"\n"
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@@ -69,7 +70,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"execution_count": 1,
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"metadata": {
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"id": "da1uZcDXdVcF"
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},
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@@ -123,7 +124,7 @@
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"The last `print` verifies that our `INFLOW` grid likewise has an `inflow_loc` dimension in addition to the two spatial ones. You can test for the existence of a tensor dimension in phiflow with the `.exists` boolean, which can be evaluated for any dimension name. E.g., above `INFLOW.inflow_loc.exists` will give `True`, while `INFLOW.unknown_dim.exists` will give `False`.\n",
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"The last `print` statement verifies that our `INFLOW` grid likewise has an `inflow_loc` dimension in addition to the two spatial ones. You can test for the existence of a tensor dimension in phiflow with the `.exists` boolean, which can be evaluated for any dimension name. E.g., above `INFLOW.inflow_loc.exists` will give `True`, while `INFLOW.some_unknown_dim.exists` will give `False`.\n",
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"\n",
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"Phiflow tensors are automatically broadcast to new dimensions via their names, and hence typically no-reshaping operations are required. E.g., you can easily add or multiply tensors with differing dimensions. Below we'll multiply a staggered grid with a tensor of ones along the `inflow_loc` dimension to get a staggered velocity that has `x,y,inflow_loc` as dimensions via `DOMAIN.staggered_grid(0) * math.ones(inflow_loc=4)`."
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]
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@@ -142,7 +143,7 @@
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"outputs": [
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{
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"data": {
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truncated
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"image/png": "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 truncated
|
||||
"text/plain": [
|
||||
"<Figure size 1152x360 with 4 Axes>"
|
||||
]
|
||||
@@ -171,8 +172,7 @@
|
||||
"for i in range(INFLOW.shape.inflow_loc):\n",
|
||||
" axes[i].imshow(smoke.values.inflow_loc[i].numpy('y,x'), origin='lower', cmap='magma')\n",
|
||||
" axes[i].set_title(f\"Inflow {math.to_int(INFLOW_LOCATION).inflow_loc[i].vector.unstack_spatial('x,y')}\" + (\", Reference\" if i==3 else \"\"))\n",
|
||||
"pylab.tight_layout()\n",
|
||||
"\n"
|
||||
"pylab.tight_layout()\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -223,7 +223,8 @@
|
||||
" for _ in range(20):\n",
|
||||
" smoke,velocity = step(smoke,velocity)\n",
|
||||
" \n",
|
||||
" loss = field.l2_loss(diffuse.explicit(smoke - field.stop_gradient(smoke.inflow_loc[-1]), 1, 1, 10))\n",
|
||||
" #loss = field.l2_loss(diffuse.explicit(smoke - field.stop_gradient(smoke.inflow_loc[-1]), 1, 1, 10))\n",
|
||||
" loss = field.l2_loss(smoke - field.stop_gradient(smoke.inflow_loc[-1]) )\n",
|
||||
" \n",
|
||||
" return loss, smoke, velocity\n"
|
||||
]
|
||||
@@ -252,7 +253,7 @@
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"(x=33, y=40) float32 -1.1465684 < ... < 1.2370852"
|
||||
"(x=33, y=40) float32 -0.9229158759117126 < ... < 2.964700937271118"
|
||||
]
|
||||
},
|
||||
"execution_count": 5,
|
||||
@@ -291,7 +292,7 @@
|
||||
},
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": "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 truncated
|
||||
"text/plain": [
|
||||
"<Figure size 1152x360 with 8 Axes>"
|
||||
]
|
||||
@@ -342,26 +343,36 @@
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
"Optimization step 0, loss: 22.651701\n",
|
||||
"Optimization step 1, loss: 22.533409\n",
|
||||
"Optimization step 2, loss: 22.277126\n",
|
||||
"Optimization step 3, loss: 22.404198\n",
|
||||
"Optimization step 4, loss: 22.268917\n",
|
||||
"Optimization step 5, loss: 22.391558\n",
|
||||
"Optimization step 6, loss: 22.448061\n",
|
||||
"Optimization step 7, loss: 22.387615\n",
|
||||
"Optimization step 8, loss: 24.646418\n",
|
||||
"Optimization step 9, loss: 26.083023\n"
|
||||
"Optimization step 0, loss: 62.215240\n",
|
||||
"Optimization step 1, loss: 62.544254\n",
|
||||
"Optimization step 2, loss: 62.061436\n",
|
||||
"Optimization step 3, loss: 63.683025\n",
|
||||
"Optimization step 4, loss: 64.774864\n",
|
||||
"Optimization step 5, loss: 71.795349\n",
|
||||
"Optimization step 6, loss: 74.215714\n",
|
||||
"Optimization step 7, loss: 60.461643\n",
|
||||
"Optimization step 8, loss: 65.317680\n",
|
||||
"Optimization step 9, loss: 65.126022\n",
|
||||
"Optimization step 10, loss: 65.193939\n",
|
||||
"Optimization step 11, loss: 65.325432\n",
|
||||
"Optimization step 12, loss: 65.471474\n",
|
||||
"Optimization step 13, loss: 65.525093\n",
|
||||
"Optimization step 14, loss: 64.756439\n",
|
||||
"Optimization step 15, loss: 65.134476\n",
|
||||
"Optimization step 16, loss: 64.554466\n",
|
||||
"Optimization step 17, loss: 65.206245\n",
|
||||
"Optimization step 18, loss: 66.495720\n",
|
||||
"Optimization step 19, loss: 66.141006\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"#initial_velocity = DOMAIN.staggered_grid(0) * math.ones(inflow_loc=4)\n",
|
||||
"initial_velocity = DOMAIN.staggered_grid(0) * math.ones(inflow_loc=4) # reset\n",
|
||||
"LR = 1e-02\n",
|
||||
"\n",
|
||||
"sim_grad_wloss = field.functional_gradient(simulate, wrt=[1], get_output=True)\n",
|
||||
"\n",
|
||||
"for optim_step in range(10):\n",
|
||||
"for optim_step in range(20):\n",
|
||||
" initial_smoke = DOMAIN.grid(math.zeros(inflow_loc=4))\n",
|
||||
" \n",
|
||||
" # sim_grad__wloss returns: [loss,s,v,velocity _grad] \n",
|
||||
@@ -389,7 +400,7 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 864x360 with 6 Axes>"
|
||||
]
|
||||
@@ -416,7 +427,7 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 864x360 with 6 Axes>"
|
||||
]
|
||||
@@ -442,7 +453,9 @@
|
||||
"source": [
|
||||
"## Re-simulation\n",
|
||||
"\n",
|
||||
"Most importantly, we can also visualize how the full simulation over the course of 20 steps turns out given the new initial conditions for each of the inflow locations. This is what happened internally at optimization time, and was measured by our loss function, but it's good to get an intuition for which solution the optimization has found.\n"
|
||||
"Most importantly, we can also visualize how the full simulation over the course of 20 steps turns out given the new initial conditions for each of the inflow locations. This is what happened internally at optimization time, and was measured by our loss function, but it's good to get an intuition for which solution the optimization has found.\n",
|
||||
"\n",
|
||||
"Below, we re-run the forward simulation with the new initial conditions from `initial_velocity`:\n"
|
||||
]
|
||||
},
|
||||
{
|
||||
@@ -452,7 +465,7 @@
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<Figure size 1152x360 with 4 Axes>"
|
||||
]
|
||||
@@ -464,7 +477,6 @@
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"# re-run sim\n",
|
||||
"smoke = initial_smoke \n",
|
||||
"velocity = initial_velocity\n",
|
||||
"\n",
|
||||
|
||||
@@ -300,6 +300,8 @@ By default DL frameworks store the internal states for every differentiable oper
|
||||
If we take a step back and look at $p = (\nabla^2)^{-1} b$, it's gradient $\partial p / \partial b$
|
||||
is just $((\nabla^2)^{-1})^T$. And in this case, $(\nabla^2)$ is a symmetric matrix, and so $((\nabla^2)^{-1})^T=(\nabla^2)^{-1}$. This is the identical inverse matrix that we encountered in the original equation above, and hence we can re-use our unmodified iterative solver to compute the gradient. We don't need to take it apart and slow it down by storing intermediate states. However, the iterative solver computes the matrix-vector-products for $(\nabla^2)^{-1} b$. So what is $b$ during back-propagation? In an optimization setting we'll always have our loss function $L$ at the end of the forward chain. The back-propagation step will then give a gradient for the output, let's assume it is $\partial L/\partial p$ here, which needs to be propagated to the earlier operations of the forward pass. Thus, we can simply invoke our iterative solve during the backward pass to compute $\partial p / \partial b = S(\partial L/\partial p)$. And assuming that we've chosen a good solver as $S$ for the forward pass, we get exactly the same performance and accuracy in the backwards pass.
|
||||
|
||||
If you're interested in a code example, the [differentiate-pressure example]( https://github.com/tum-pbs/PhiFlow/blob/master/demos/differentiate_pressure.py) of phiflow uses exactly this process for an optimization through a pressure projection step: a flow field that is constrained on the right side, is optimized for the content on the left, such that it matches the target on the right after a pressure projection step.
|
||||
|
||||
The main take-away here is: it is important _not to blindly back-propagate_ through the forward computation, but to think about which steps of the analytic equations for the forward pass to compute gradients for. In cases like the above, we can often find improved analytic expressions for the gradients, which we can then compute numerically.
|
||||
|
||||
```{admonition} Implicit Function Theorem & Time
|
||||
|
||||
Reference in new issue
Block a user