Critical Investigation of Failure Modes in Physics-informed Neural Networks

TL;DR

This paper critically examines the failure modes of physics-informed neural networks (PINNs), revealing how their loss landscape complexities and scale issues hinder convergence, especially with challenging PDE solutions.

Contribution

It provides a detailed analysis of loss landscape behaviors in PINNs and compares physics-informed versus purely data-driven loss functions to identify key optimization challenges.

Findings

Incomparable scales between loss terms impair PINN performance.

PINNs exhibit highly non-convex loss surfaces that are difficult to optimize.

Physics-informed loss functions are more prone to vanishing gradients.

Abstract

Several recent works in scientific machine learning have revived interest in the application of neural networks to partial differential equations (PDEs). A popular approach is to aggregate the residual form of the governing PDE and its boundary conditions as soft penalties into a composite objective/loss function for training neural networks, which is commonly referred to as physics-informed neural networks (PINNs). In the present study, we visualize the loss landscapes and distributions of learned parameters and explain the ways this particular formulation of the objective function may hinder or even prevent convergence when dealing with challenging target solutions. We construct a purely data-driven loss function composed of both the boundary loss and the domain loss. Using this data-driven loss function and, separately, a physics-informed loss function, we then train two neural network models with the same architecture. We show that incomparable scales between boundary and domain loss terms are the culprit behind the poor performance. Additionally, we assess the performance of both approaches on two elliptic problems with increasingly complex target solutions. Based on our analysis of their loss landscapes and learned parameter distributions, we observe that a physics-informed neural network with a composite objective function formulation produces highly non-convex loss surfaces that are difficult to optimize and are more prone to the problem of vanishing gradients.