Enhancing PINN Accuracy for the RLW Equation: Adaptive and Conservative Approaches

TL;DR

This paper develops adaptive and conservative PINN methods to improve accuracy in solving the RLW equation, revealing that problem-specific approaches and explicit conservation enforcement significantly impact performance.

Contribution

Introduces adaptive and conservative PINN approaches tailored for the RLW equation, demonstrating their effectiveness and challenging assumptions about conservation enforcement in PINNs.

Findings

Adaptive PINNs excel in complex nonlinear interactions like soliton collisions.

Conservative PINNs perform better for long-term single soliton and undular bore problems.

Explicit conservation enforcement may hinder optimization in highly nonlinear systems.

Abstract

Standard physics-informed neural network implementations have produced large error rates when using these models to solve the regularized long wave (RLW) equation. Two improved PINN approaches were developed in this research: an adaptive approach with self-adaptive loss weighting and a conservative approach enforcing explicit conservation laws. Three benchmark tests were used to demonstrate how effective PINN's are as they relate to the type of problem being solved (i.e., time dependent RLW equation). The first was a single soliton traveling along a line (propagation), the second was the interaction between two solitons, and the third was the evolution of an undular bore over the course of $t=250$. The results demonstrated that the effectiveness of PINNs are problem specific. The adaptive PINN was significantly better than both the conservative PINN and the standard PINN at solving problems involving complex nonlinear interactions such as colliding two solitons. The conservative approach was significantly better at solving problems involving long term behavior of single solitons and undular bores. However, the most important finding from this research is that explicitly enforcing conservation laws may be harmful to optimizing the solution of highly nonlinear systems of equations and therefore requires special training methods. The results from our adaptive and conservative approaches were within $O(10^{-5})$ of established numerical solutions for the same problem, thus demonstrating that PINNs can provide accurate solutions to complex systems of partial differential equations without the need for a discretization of space or time (mesh free). Moreover, the finding from this research challenges the assumptions that conservation enforcement will always improve the performance of a PINN and provides researchers with guidelines for designing PINNs for use on specific types of problems.