The Challenges of the Nonlinear Regime for Physics-Informed Neural Networks
This addresses challenges in training PINNs for nonlinear PDEs, an incremental advance for computational physics and machine learning applications.
The paper shows that the Neural Tangent Kernel perspective fails for nonlinear PDEs in Physics-Informed Neural Networks, as the NTK is not constant and the Hessian remains significant, motivating second-order optimization methods with demonstrated benefits in benchmarks.
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.