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Discussion of Physical Soft-Constraints
The good news so far is - we have a DL method that can include physical laws in the form of soft constraints by minimizing residuals. However, as the very simple previous example illustrates, this is just a conceptual starting point.
On the positive side, we can leverage DL frameworks with backpropagation to compute the derivatives of the model. At the same time, this puts us at the mercy of the learned representation regarding the reliability of these derivatives. Also, each derivative requires backpropagation through the full network, which can be very slow. Especially so for higher-order derivatives.
And while the setup is realtively simple, it is generally difficult to control. The NN has flexibility to refine the solution by itself, but at the same time, tricks are necessary when it doesn’t pick the right regions of the solution.
Is it “Machine Learning”
TODO, discuss - more akin to classical optimization: we test for
space/time positions at training time, and are interested in the
solution there afterwards.
hence, no real generalization, or test data with different distribution. more similar to inverse problem that solves single state e.g. via BFGS or Newton.
Summary
In general, a fundamental drawback of this approach is that it does combine with traditional numerical techniques well. E.g., learned representation is not suitable to be refined with a classical iterative solver such as the conjugate gradient method. This means many powerful techniques that were developed in the past decades cannot be used in this context. Bringing these numerical methods back into the picture will be one of the central goals of the next sections.
✅ Pro: - uses physical model - derivatives via backpropagation
❌ Con: - slow … - only soft constraints - largely incompatible classical numerical methods - derivatives rely on learned representation
Next, let’s look at how we can leverage numerical methods to improve the DL accuracy and efficiency by making use of differentiable solvers.