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arXiv · 2211.15443

Replacing Automatic Differentiation by Sobolev Cubatures fastens Physics Informed Neural Nets and strengthens their Approximation Power

Juan Esteban Suarez Cardona·Michael Hecht
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Abstract

We present a novel class of approximations for variational losses, being applicable for the training of physics-informed neural nets (PINNs). The loss formulation reflects classic Sobolev space theory for partial differential equations and their weak formulations. The loss computation rests on an extension of Gauss-Legendre cubatures, we term Sobolev cubatures, replacing automatic differentiation (A.D.). We prove the runtime complexity of training the resulting Soblev-PINNs (SC-PINNs) to be less than required by PINNs relying on A.D. On top of one-to-two order of magnitude speed-up the SC-PINNs are demonstrated to achieve closer solution approximations for prominent forward and inverse PDE problems than established PINNs achieve.

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Juan Esteban Suarez Cardona, Michael Hecht. 2022-11-23. Replacing Automatic Differentiation by Sobolev Cubatures fastens Physics Informed Neural Nets and strengthens their Approximation Power. https://arxiv.org/abs/2211.15443

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