Improving the accuracy of physics-informed neural networks via last-layer retraining

TL;DR

This paper introduces a post-processing method for physics-informed neural networks that significantly enhances their accuracy, enabling better solutions for PDEs and facilitating transfer learning in complex scenarios.

Contribution

The authors propose a novel last-layer retraining technique that improves PINN accuracy and provides a residual-based metric for optimal basis function selection.

Findings

Errors reduced by four to five orders of magnitude.

Method applicable across architectures and dimensions.

Enables transfer learning in complex PDE problems.

Abstract

Physics-informed neural networks (PINNs) are a versatile tool in the burgeoning field of scientific machine learning for solving partial differential equations (PDEs). However, determining suitable training strategies for them is not obvious, with the result that they typically yield moderately accurate solutions. In this article, we propose a method for improving the accuracy of PINNs by coupling them with a post-processing step that seeks the best approximation in a function space associated with the network. We find that our method yields errors four to five orders of magnitude lower than those of the parent PINNs across architectures and dimensions. Moreover, we can reuse the basis functions for the linear space in more complex settings, such as time-dependent and nonlinear problems, allowing for transfer learning. Our approach also provides a residual-based metric that allows us to optimally choose the number of basis functions employed.