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Optimally weighted loss functions for solving PDEs with Neural Networks · Around

2020

Optimally weighted loss functions for solving PDEs with Neural Networks

van der Meer, Remco, Oosterlee, Cornelis, Borovykh, Anastasia

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Recent works have shown that deep neural networks can be employed to solve partial differential equations, giving rise to the framework of physics informed neural networks.

We introduce a generalization for these methods that manifests as a scaling parameter which balances the relative importance of the different constraints imposed by partial differential equations.
A mathematical motivation of these generalized methods is provided, which shows that for linear and well-posed partial differential equations, the functional form is convex.
We then derive a choice for the scaling parameter that is optimal with respect to a measure of relative error.

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