Enhancing PINN Accuracy for the RLW Equation: Adaptive and Conservative Approaches
This work addresses accuracy issues in PINNs for solving partial differential equations like the RLW equation, offering incremental improvements with guidelines for problem-specific design.
The study tackled the problem of high error rates in physics-informed neural networks (PINNs) for solving the regularized long wave (RLW) equation by developing adaptive and conservative approaches, achieving results within O(10^-5) of established numerical solutions. It found that adaptive PINNs excel in complex nonlinear interactions, while conservative PINNs are better for long-term behavior, challenging the assumption that conservation enforcement always improves performance.
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.