Energy-Equidistributed Moving Sampling Physics-informed Neural Networks for Solving Conservative Partial Differential Equations

TL;DR

This paper introduces EEMS-PINNs, a novel energy-adaptive sampling framework that enhances physics-informed neural networks for solving conservative PDEs by dynamically tracking energy evolution and improving long-term solution accuracy.

Contribution

The paper develops a new energy-equidistributed adaptive sampling method integrated with PINNs, enabling better energy conservation and stability in solving conservative PDEs.

Findings

EEMS-PINNs maintain solution accuracy over long simulations.

The framework effectively preserves conserved energy.

Stable performance in non-conservative systems.

Abstract

This paper presents a novel Energy-Equidistributed adaptive sampling framework for multi-dimensional conservative PDEs, introducing both location-based and velocity-based formulations of Energy-Equidistributed moving mesh PDEs (EMMPDEs). The framework utilizes the energy density function as the monitor function, ensuring that mesh adaptation dynamically tracks energy evolution during temporal integration. These theoretical developments are integrated with deep neural networks to establish the Energy-Equidistributed Moving Sampling Physics-Informed Neural Networks (EEMS-PINNs), which integrate physics-informed learning with energy-adaptive mesh optimization. Extensive numerical experiments demonstrate that EEMS-PINNs effectively maintain solution accuracy in long-time simulations while preserving conserved energy. The framework's robustness is further evidenced by its stable performance in non-conservative systems. The code for this paper can be found at https://github.com/sufe-Ran-Zhang/EMMPDE.