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

TL;DR

This work addresses the high error rates of standard PINNs for the RLW equation by introducing Adaptive PINN and Conservative PINN architectures, coupled with a two-stage curriculum and causal time-marching. The adaptive method dynamically balances PDE, initial, and boundary residuals, while the conservative method adds explicit mass, momentum, and energy conservation penalties. Across three RLW-driven phenomena—single soliton propagation, two-soliton interaction, and long-time undular bore development—the study demonstrates that performance is problem-dependent: Adaptive PINN excels in nonlinear interactions, while Conservative PINN provides stability for long-time evolution, though conservation enforcement can hinder optimization in highly nonlinear regimes. The results, achieving errors on the order of $O(10^{-5})$ relative to reference solutions, establish mesh-free PINN solvers as viable for complex PDEs and offer practical guidelines for selecting PINN configurations based on the physical dynamics involved.

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.

Figures (19)

• Figure 1: Wave profile comparison at different times for the Adaptive PINN solution versus the exact analytical solution for the single soliton case.
• Figure 2: Wave profile comparison at different times for the Conservative PINN solution versus the exact analytical solution.
• Figure 3: Pointwise absolute error at $t=20$ for (a) Adaptive PINN and (b) Conservative PINN.
• Figure 4: Evolution of the peak absolute error ($L_{\infty}$) over time for the Adaptive and Conservative PINN models.
• Figure 5: Temporal evolution of the conservation laws ($I_1$: Mass, $I_2$: Momentum, $I_3$: Energy) for both enhanced PINN approaches, compared to the constant reference value.