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Weak form of a PDE #174

@LoveFrootLoops

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@LoveFrootLoops

Hi again :)

I'd like to bring up a topic that's been keeping me busy lately in the field of Physics-Informed Neural Networks (PINNS). During research I've come across a number of papers that discuss how accurate and effective PINNS are, particularly when dealing with the strong form of a Partial Differential Equation (PDE). Interestingly, these papers reveal that in most cases PINNS don't perform well and might even give us incorrect results. However, they also highlight that PINNS work really well when it comes to weak forms of PDEs.

Just as an example take a look at page 6 of Paper1

Another example can be seen here Paper2 where they had to add an integration formulation of the pde to the strong form to keep the global consistency.

In light of these observations, I would like to propose the incorporation of an integration method. Such an addition would enable the formulation of the loss function in weak form, facilitating integration across the entire domain (effectively summing over collection points). By doing this, you could really enhance the performance of PINNS, bringing them up to date with the latest techniques, particularly in accurately solving PDEs using Neural Networks.

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