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Physical Loss Terms
The supervised setting of the previous sections can quickly yield approximate solutions with a fairly simple training process, but what’s quite sad to see here is that we only use physical models and numerics as an “external” tool to produce a big pile of data 😢.
We as humans have a lot of knowledge about how to describe physical processes mathematically. As the following chapters will show, we can improve the training process by guiding it with our human knowledge of physics.
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Physical losses typically combine a supervised loss with a combination of derivatives from the neural network. ```
Using physical models
Given a PDE for \mathbf{u}(\mathbf{x},t) with a time evolution, we can typically express it in terms of a function \mathcal F of the derivatives of \mathbf{u} via
\mathbf{u}_t = \mathcal F ( \mathbf{u}_{x}, \mathbf{u}_{xx}, ... \mathbf{u}_{xx...x} ) ,
where the _{\mathbf{x}} subscripts denote spatial derivatives with respect to one of the spatial dimensions of higher and higher order (this can of course also include mixed derivatives with respect to different axes).
In this context we can employ DL by approximating the unknown \mathbf{u} itself with a NN, denoted by \tilde{\mathbf{u}}. If the approximation is accurate, the PDE naturally should be satisfied, i.e., the residual R should be equal to zero:
R = \mathbf{u}_t - \mathcal F ( \mathbf{u}_{x}, \mathbf{u}_{xx}, ... \mathbf{u}_{xx...x} ) = 0 .
This nicely integrates with the objective for training a neural network: similar to before we can collect sample solutions [x_0,y_0], ...[x_n,y_n] for \mathbf{u} with \mathbf{u}(\mathbf{x})=y. This is typically important, as most practical PDEs we encounter do not have unique solutions unless initial and boundary conditions are specified. Hence, if we only consider R we might get solutions with random offset or other undesirable components. Hence the supervised sample points help to pin down the solution in certain places. Now our training objective becomes
\text{arg min}_{\theta} \ \alpha_0 \sum_i (f(x_i ; \theta)-y_i)^2 + \alpha_1 R(x_i) , (physloss-training)
where \alpha_{0,1} denote hyperparameters that scale the contribution of the supervised term and the residual term, respectively. We could of course add additional residual terms with suitable scaling factors here.
Note that, similar to the data samples used for supervised training, we have no guarantees that the residual terms R will actually reach zero during training. The non-linear optimization of the training process will minimize the supervised and residual terms as much as possible, but worst case, large non-zero residual contributions can remain. We’ll look at this in more detail in the upcoming code example, for now it’s important to remember that physical constraints in this way only represent soft-constraints, without guarantees of minimizing these constraints.
Neural network derivatives
In order to compute the residuals at training time, it would be
possible to store the unknowns of \mathbf{u} on a computational mesh, e.g., a
grid, and discretize the equations of R
there. This has a fairly long “tradition” in DL, and was proposed by
Tompson et al. {cite}tompson2017 early on.
A popular variant of employing physical soft-constraints
{cite}raissi2018hiddenphys instead uses fully connected NNs
to represent \mathbf{u}. This has some
interesting pros and cons that we’ll outline in the following, and we
will also focus on it in the following code examples and
comparisons.
The central idea here is that the aforementioned general function f that we’re after in our learning problems can also be used to obtain a representation of a physical field, e.g., a field \mathbf{u} that satisfies R=0. This means \mathbf{u}(\mathbf{x}) will be turned into \mathbf{u}(\mathbf{x}, \theta) where we choose the NN parameters \theta such that a desired \mathbf{u} is represented as precisely as possible.
One nice side effect of this viewpoint is that NN representations
inherently support the calculation of derivatives. The derivative \partial f / \partial \theta was a key
building block for learning via gradient descent, as explained in
{doc}overview. Now, we can use the same tools to compute
spatial derivatives such as \partial
\mathbf{u} / \partial x, Note that above for R we’ve written this derivative in the
shortened notation as \mathbf{u}_{x}.
For functions over time this of course also works for \partial \mathbf{u} / \partial t, i.e. \mathbf{u}_{t} in the notation above.
Thus, for some generic R, made up of \mathbf{u}_t and \mathbf{u}_{x} terms, we can rely on the backpropagation algorithm of DL frameworks to compute these derivatives once we have a NN that represents \mathbf{u}. Essentially, this gives us a function (the NN) that receives space and time coordinates to produce a solution for \mathbf{u}. Hence, the input is typically quite low-dimensional, e.g., 3+1 values for a 3D case over time, and often produces a scalar value or a spatial vector. Due to the lack of explicit spatial sampling points, an MLP, i.e., fully-connected NN is the architecture of choice here.
To pick a simple example, Burgers equation in 1D, $ + u u = u $ , we can directly formulate a loss term R = \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} - \nu \frac{\partial^2 u}{\partial x^2} u that should be minimized as much as possible at training time. For each of the terms, e.g. \frac{\partial u}{\partial x}, we can simply query the DL framework that realizes u to obtain the corresponding derivative. For higher order derivatives, such as \frac{\partial^2 u}{\partial x^2}, we can simply query the derivative function of the framework multiple times. In the following section, we’ll give a specific example of how that works in tensorflow.
Summary so far
The approach above gives us a method to include physical equations into DL learning as a soft-constraint: the residual loss. Typically, this setup is suitable for inverse problems, where we have certain measurements or observations for which we want to find a PDE solution. Because of the high cost of the reconstruction (to be demonstrated in the following), the solution manifold shouldn’t be overly complex. E.g., it is not possible to capture a wide range of solutions, such as with the previous supervised airfoil example, with such a physical residual loss.