PACMANN: Point Adaptive Collocation Method for Artificial Neural Networks

Physics-Informed Neural Networks (PINNs) are an emerging tool for approximating the solution of Partial Differential Equations (PDEs) in both forward and inverse problems. PINNs minimize a loss function which includes the PDE residual determined for a set of collocation points. Previous work has shown that the number and distribution of these collocation points have a significant influence on the accuracy of the PINN solution. Therefore, the effective placement of these collocation points is an active area of research. Specifically, adaptive collocation point sampling methods have been proposed, which have been reported to scale poorly to higher dimensions. In this work, we address this issue and present the Point Adaptive Collocation Method for Artificial Neural Networks (PACMANN). Inspired by classic optimization problems, this approach incrementally moves collocation points toward regions of higher residuals using gradient-based optimization algorithms guided by the gradient of the squared residual. We apply PACMANN for forward and inverse problems, and demonstrate that this method matches the performance of state-of-the-art methods in terms of the accuracy/efficiency tradeoff for the low-dimensional problems, while outperforming available approaches for high-dimensional problems; the best performance is observed for the Adam optimizer. Key features of the method include its low computational cost and simplicity of integration in existing physics-informed neural network pipelines.