minor typo fixes
This commit is contained in:
Aman Desai authored and 
GitHub committed
1 parent
590a930246
commit
01fde18831
2 files changed
+6
-6
No files matched your search
+5
-5
@@ -15,7 +15,7 @@
|
|||||||
"source": [
|
"source": [
|
||||||
"Let's start with a very reduced example that highlights some of the key capabilities of physics-based learning approaches. Let's assume our physical model is a very simple equation: a parabola along the positive x-axis.\n",
|
"Let's start with a very reduced example that highlights some of the key capabilities of physics-based learning approaches. Let's assume our physical model is a very simple equation: a parabola along the positive x-axis.\n",
|
||||||
"\n",
|
"\n",
|
||||||
"Despite being very simple, there are two solutions for every point along x, i.e. we have two modes, one above the other one below the x-axis, as shown on the left below. If we don't take care a conventional learning approach will give us an approximation like the red one shown in the middle, which is completely off. With an improved learning setup, ideally, by using a discretized numerical solver, we can at least accurately represent one of the modes of the solution (shown in green on the right).\n",
|
"Despite being very simple, there are two solutions for every point along x, i.e. we have two modes, one above the other one below the x-axis, as shown on the left below. If we don't take care, a conventional learning approach will give us an approximation like the red one shown in the middle, which is completely off. With an improved learning setup, ideally, by using a discretized numerical solver, we can at least accurately represent one of the modes of the solution (shown in green on the right).\n",
|
||||||
"\n",
|
"\n",
|
||||||
"```{figure} resources/intro-teaser-side-by-side.png\n",
|
"```{figure} resources/intro-teaser-side-by-side.png\n",
|
||||||
"---\n",
|
"---\n",
|
||||||
@@ -198,7 +198,7 @@
|
|||||||
"outputs": [
|
"outputs": [
|
||||||
{
|
{
|
||||||
"data": {
|
"data": {
|
||||||
"image/png": "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 truncated
|
"image/png": "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 truncated
|
||||||
"text/plain": [
|
"text/plain": [
|
||||||
"<Figure size 432x288 with 1 Axes>"
|
"<Figure size 432x288 with 1 Axes>"
|
||||||
]
|
]
|
||||||
@@ -274,7 +274,7 @@
|
|||||||
"id": "conscious-budapest",
|
"id": "conscious-budapest",
|
||||||
"metadata": {},
|
"metadata": {},
|
||||||
"source": [
|
"source": [
|
||||||
"The loss function is the crucial point for training: we directly incorporate the function f into the loss. In this simple case, the `loss_dp` function simply computes the square of the prediction `y_pred`. \n",
|
"The loss function is the crucial point for training: we directly incorporate the function $f$ into the loss. In this simple case, the `loss_dp` function simply computes the square of the prediction `y_pred`. \n",
|
||||||
"\n",
|
"\n",
|
||||||
"Later on, a lot more could happen here: we could evaluate finite-difference stencils on the predicted solution, or compute a whole implicit time-integration step of a solver. Here we have a simple _mean-squared error_ term of the form $|y_{\\text{pred}}^2 - y_{\\text{true}}|^2$, which we are minimizing during training. It's not necessary to make it so simple: the more knowledge and numerical methods we can incorporate, the better we can guide the training process."
|
"Later on, a lot more could happen here: we could evaluate finite-difference stencils on the predicted solution, or compute a whole implicit time-integration step of a solver. Here we have a simple _mean-squared error_ term of the form $|y_{\\text{pred}}^2 - y_{\\text{true}}|^2$, which we are minimizing during training. It's not necessary to make it so simple: the more knowledge and numerical methods we can incorporate, the better we can guide the training process."
|
||||||
]
|
]
|
||||||
@@ -339,7 +339,7 @@
|
|||||||
"outputs": [
|
"outputs": [
|
||||||
{
|
{
|
||||||
"data": {
|
"data": {
|
||||||
"image/png": "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 truncated
|
"image/png": "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 truncated
|
||||||
"text/plain": [
|
"text/plain": [
|
||||||
"<Figure size 432x288 with 1 Axes>"
|
"<Figure size 432x288 with 1 Axes>"
|
||||||
]
|
]
|
||||||
@@ -375,7 +375,7 @@
|
|||||||
"\n",
|
"\n",
|
||||||
"- We're still only getting one side of the curve! This is to be expected because we're representing the solutions with a deterministic function. Hence, we can only represent a single mode. Interestingly, whether it's the top or bottom mode is determined by the random initialization of the weights in $f$ - run the example a couple of times to see this effect in action. To capture multiple modes we'd need to extend the NN to capture the full distribution of the outputs and parametrize it with additional dimensions.\n",
|
"- We're still only getting one side of the curve! This is to be expected because we're representing the solutions with a deterministic function. Hence, we can only represent a single mode. Interestingly, whether it's the top or bottom mode is determined by the random initialization of the weights in $f$ - run the example a couple of times to see this effect in action. To capture multiple modes we'd need to extend the NN to capture the full distribution of the outputs and parametrize it with additional dimensions.\n",
|
||||||
"\n",
|
"\n",
|
||||||
"- The region with $x$ near zero is typically still off in this example. The network essentially learns a linear approximation of one half of the parabola here. This is partially caused by the weak neural network: it is very small and shallow. In addition, the evenly spread of sample points along the x-axis bias the NN towards the larger y values. These contribute more to the loss, and hence the network invests most of its resources to reduce the error in this region.\n"
|
"- The region with $x$ near zero is typically still off in this example. The network essentially learns a linear approximation of one half of the parabola here. This is partially caused by the weak neural network: it is very small and shallow. In addition, the evenly spread of sample points along the x-axis bias the NN towards the larger $y$ values. These contribute more to the loss, and hence the network invests most of its resources to reduce the error in this region.\n"
|
||||||
]
|
]
|
||||||
},
|
},
|
||||||
{
|
{
|
||||||
|
|||||||
+1
-1
@@ -119,7 +119,7 @@ where a solver targets cases from a certain well-defined problem
|
|||||||
domain repeatedly, it can for instance make a lot of sense to once invest
|
domain repeatedly, it can for instance make a lot of sense to once invest
|
||||||
significant resources to train
|
significant resources to train
|
||||||
a neural network that supports the repeated solves. Based on the
|
a neural network that supports the repeated solves. Based on the
|
||||||
domain-specific specialization of this network, such a hybrid
|
domain-specific specialization of this network, such a hybrid solver
|
||||||
could vastly outperform traditional, generic solvers. And despite
|
could vastly outperform traditional, generic solvers. And despite
|
||||||
the many open questions, first publications have demonstrated
|
the many open questions, first publications have demonstrated
|
||||||
that this goal is not overly far away {cite}`um2020sol,kochkov2021`.
|
that this goal is not overly far away {cite}`um2020sol,kochkov2021`.
|
||||||
|
|||||||
Reference in new issue
Block a user