Energy Dissipation Rate Guided Adaptive Sampling for Physics-Informed Neural Networks: Resolving Surface-Bulk Dynamics in Allen-Cahn Systems

TL;DR

This paper presents EDRAS, an adaptive sampling method guided by energy dissipation rates, significantly improving PINN accuracy in solving Allen-Cahn PDEs in complex geometries by focusing on physically critical regions.

Contribution

The paper introduces EDRAS, a novel energy dissipation guided adaptive sampling strategy that enhances PINN performance for thermodynamically consistent PDEs in irregular domains.

Findings

EDRAS reduces relative mean square error by up to six times compared to traditional methods.

EDRAS is more computationally efficient than other residual-based adaptive sampling approaches.

Numerical experiments reveal insights into phase evolution under dynamic boundary conditions.

Abstract

We introduce the Energy Dissipation Rate guided Adaptive Sampling (EDRAS) strategy, a novel method that substantially enhances the performance of Physics-Informed Neural Networks (PINNs) in solving thermodynamically consistent partial differential equations (PDEs) over arbitrary domains. EDRAS leverages the local energy dissipation rate density as a guiding metric to identify and adaptively re-sample critical collocation points from both the interior and boundary of the computational domain. This dynamical sampling approach improves the accuracy of residual-based PINNs by aligning the training process with the underlying physical structure of the system. In this study, we demonstrate the effectiveness of EDRAS using the Allen-Cahn phase field model in irregular geometries, achieving up to a sixfold reduction in the relative mean square error compared to traditional residual-based adaptive refinement (RAR) methods. Moreover, we compare EDRAS with other residual-based adaptive sampling approaches and show that EDRAS is not only computationally more efficient but also more likely to identify high-impact collocation points. Through numerical solutions of the Allen-Cahn equation with both static (Neumann) and dynamic boundary conditions in 2D disk- and ellipse-shaped domains solved using PINN coupled with EDRAS, we gain significant insights into how dynamic boundary conditions influence bulk phase evolution and thermodynamic behavior. The proposed approach offers an effective, physically informed enhancement to PINN frameworks for solving thermodynamically consistent models, making PINN a robust and versatile computational tool for investigating complex thermodynamic processes in arbitrary geometries.