Pseudo grid-based physics-informed convolutional-recurrent network solving the integrable nonlinear lattice equations

TL;DR

This paper introduces PG-PhyCRNet, a novel physics-informed neural network that effectively models and extrapolates solutions of integrable nonlinear lattice equations, capturing complex soliton behaviors with high accuracy.

Contribution

The paper presents a pseudo grid-based convolutional-recurrent network that improves the modeling and extrapolation of integrable lattice equations, preserving their mathematical structure.

Findings

PG-PhyCRNet outperforms traditional methods in capturing soliton dynamics.

The model demonstrates strong extrapolation capabilities for steep and high-speed waveforms.

Robustness is confirmed across various pseudo grid partitioning scenarios.

Abstract

Traditional discrete learning methods involve discretizing continuous equations using difference schemes, necessitating considerations of stability and convergence. Integrable nonlinear lattice equations possess a profound mathematical structure that enables them to revert to continuous integrable equations in the continuous limit, particularly retaining integrable properties such as conservation laws, Hamiltonian structure, and multiple soliton solutions. The pseudo grid-based physics-informed convolutional-recurrent network (PG-PhyCRNet) is proposed to investigate the localized wave solutions of integrable lattice equations, which significantly enhances the model's extrapolation capability to lattice points beyond the temporal domain. We conduct a comparative analysis of PG-PhyCRNet with and without pseudo grid by investigating the multi-soliton solutions and rational solitons of the Toda lattice and self-dual network equation. The results indicate that the PG-PhyCRNet excels in capturing long-term evolution and enhances the model's extrapolation capability for solitons, particularly those with steep waveforms and high wave speeds. Finally, the robustness of the PG-PhyCRNet method and its effect on the prediction of solutions in different scenarios are confirmed through repeated experiments involving pseudo grid partitioning.