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

The Neural Tangent Kernel (NTK) viewpoint represents a valuable approach to examine the training dynamics of Physics-Informed Neural Networks (PINNs) in the infinite width limit. We leverage this perspective and focus on the case of nonlinear Partial Differential Equations (PDEs) solved by PINNs. We provide theoretical results on the different behaviors of the NTK depending on the linearity of the differential operator. Moreover, inspired by our theoretical results, we emphasize the advantage of employing second-order methods for training PINNs. Additionally, we explore the convergence capabilities of second-order methods and address the challenges of spectral bias and slow convergence. Every theoretical result is supported by numerical examples with both linear and nonlinear PDEs, and we validate our training method on benchmark test cases.

On the Hyperparameters influencing a PINN's generalization beyond the training domain

Physics-Informed Neural Networks (PINNs) are Neural Network architectures trained to emulate solutions of differential equations without the necessity of solution data. They are currently ubiquitous in the scientific literature due to their flexible and promising settings. However, very little of the available research provides practical studies that aim for a better quantitative understanding of such architecture and its functioning. In this paper, we analyze the performance of PINNs for various architectural hyperparameters and algorithmic settings based on a novel error metric and other factors such as training time. The proposed metric and approach are tailored to evaluate how well a PINN generalizes to points outside its training domain. Besides, we investigate the effect of the algorithmic setup on the outcome prediction of a PINN, inside and outside its training domain, to explore the effect of each hyperparameter. Through our study, we assess how the algorithmic setup of PINNs influences their potential for generalization and deduce the settings which maximize the potential of a PINN for accurate generalization. The study that we present returns insightful and at times counterintuitive results on PINNs. These results can be useful in PINN applications when defining the model and evaluating it.