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

TL;DR

This paper introduces Sobolev cubatures as a novel approximation method for variational losses in physics-informed neural nets, replacing automatic differentiation, leading to faster training and improved solution accuracy for PDE problems.

Contribution

It proposes Sobolev cubatures for loss computation in PINNs, reducing runtime complexity and enhancing approximation quality compared to traditional automatic differentiation methods.

Findings

Speed-up of one-to-two orders of magnitude in training time.

Closer approximation of PDE solutions than existing PINNs.

Effective application to forward and inverse PDE problems.

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.