The Challenges of the Nonlinear Regime for Physics-Informed Neural Networks

TL;DR

This paper examines the limitations of the Neural Tangent Kernel approach for nonlinear PDEs in Physics-Informed Neural Networks, highlighting the need for second-order optimization methods due to non-constant NTK and persistent Hessian effects.

Contribution

It reveals the shortcomings of NTK in nonlinear PINNs and advocates for second-order optimization, supported by theoretical analysis and numerical experiments.

Findings

NTK is not constant during training for nonlinear PDEs

Hessian does not vanish even at infinite width in nonlinear cases

Second-order methods improve convergence in nonlinear PINNs

Abstract

The Neural Tangent Kernel (NTK) viewpoint is widely employed to analyze the training dynamics of overparameterized Physics-Informed Neural Networks (PINNs). However, unlike the case of linear Partial Differential Equations (PDEs), we show how the NTK perspective falls short in the nonlinear scenario. Specifically, we establish that the NTK yields a random matrix at initialization that is not constant during training, contrary to conventional belief. Another significant difference from the linear regime is that, even in the idealistic infinite-width limit, the Hessian does not vanish and hence it cannot be disregarded during training. This motivates the adoption of second-order optimization methods. We explore the convergence guarantees of such methods in both linear and nonlinear cases, addressing challenges such as spectral bias and slow convergence. Every theoretical result is supported by numerical examples with both linear and nonlinear PDEs, and we highlight the benefits of second-order methods in benchmark test cases.