Physics-Informed Neural Networks with Attention Feature Expansion for Monge-Amp\`ere Equations

TL;DR

This paper introduces a novel physics-informed neural network with attention feature expansion (PINN-AFE) for solving Monge-Ampère equations, demonstrating improved accuracy, efficiency, and applicability to image processing tasks.

Contribution

The study develops a new PINN framework incorporating attention feature expansion and input convex neural networks with theoretical guarantees, enhancing solution accuracy and convergence.

Findings

Validated high accuracy and efficiency through numerical experiments.

Extended the framework to image processing with successful results.

Achieved physically consistent outcomes in image enhancement and registration.

Abstract

The Monge-Amp\`ere equation is a fundamental fully nonlinear elliptic partial differential equation that finds extensive applications across multiple disciplines. This study proposes a novel physics-informed neural network integrated with attention feature expansion (PINN-AFE) for its numerical solution. A multi-head attention enhanced feature pool is constructed to enable adaptive nonlinear feature representation, and input convex neural networks are adopted to impose strict convexity of solutions with rigorous theoretical guarantees. Meanwhile, a dynamically weighted loss function combined with hybrid optimization is formulated to accelerate training convergence. Comprehensive numerical experiments validate the accuracy and computational efficiency of the developed framework. The PINN-AFE paradigm is further extended to image processing tasks, delivering high-quality and physically consistent results in both image enhancement and medical image registration scenarios.