Recipes for when Physics Fails: Recovering Robust Learning of Physics Informed Neural Networks

TL;DR

This paper introduces Gaussian Process-based smoothing and sparse inducing points to enhance the robustness of Physics-informed Neural Networks against noisy data and errors, improving their stability and accuracy in solving PDEs.

Contribution

It proposes novel GP-based regularization techniques and uncertainty quantification methods to make PINNs more robust to data errors and overfitting, addressing limitations of existing physics-based regularizations.

Findings

GP smoothing improves PINN robustness against noise.

Sparse inducing points maintain accuracy with fewer data points.

Proposed methods outperform existing benchmarks on Schrödinger and Burgers' equations.

Abstract

Physics-informed Neural Networks (PINNs) have been shown to be effective in solving partial differential equations by capturing the physics induced constraints as a part of the training loss function. This paper shows that a PINN can be sensitive to errors in training data and overfit itself in dynamically propagating these errors over the domain of the solution of the PDE. It also shows how physical regularizations based on continuity criteria and conservation laws fail to address this issue and rather introduce problems of their own causing the deep network to converge to a physics-obeying local minimum instead of the global minimum. We introduce Gaussian Process (GP) based smoothing that recovers the performance of a PINN and promises a robust architecture against noise/errors in measurements. Additionally, we illustrate an inexpensive method of quantifying the evolution of uncertainty based on the variance estimation of GPs on boundary data. Robust PINN performance is also shown to be achievable by choice of sparse sets of inducing points based on sparsely induced GPs. We demonstrate the performance of our proposed methods and compare the results from existing benchmark models in literature for time-dependent Schr\"odinger and Burgers' equations.