C-PINN: A neural network framework based on the Cord\`{e}s condition for solving linear and fully nonlinear equations in non-divergence form and its applications

TL;DR

This paper introduces C-PINN, a neural network framework leveraging the C40re9s condition to solve complex PDEs more stably and accurately, especially in high dimensions, with broad scientific applications.

Contribution

It develops a new PINN framework based on the C40re9s condition that improves training stability and extends to nonlinear PDEs like Hamilton-Jacobi-Bellman and Monge-Ampe8re equations.

Findings

Enhanced training stability and accuracy demonstrated in numerical experiments.

Effective solution of high-dimensional PDEs, including nonlinear types.

Broad applicability to scientific and engineering problems.

Abstract

In this paper, we propose a novel Physics-Informed Neural Network (PINN) framework based on the Cord\`{e}s condition for solving both linear and fully nonlinear partial differential equations (PDEs) in non-divergence form, together with their applications. By incorporating the operator structure into the loss function, the proposed method improves the conditioning of the associated optimization problem, thereby enhancing training stability and solution accuracy. The framework is further extended to include Hamilton-Jacobi-Bellman and Monge-Amp\`{e}re equations, with applications to optimal transport. Numerical experiments demonstrate the effectiveness and robustness of the method, as well as its capability to address high-dimensional problems, highlighting the promise of learning-based approaches for tackling challenging PDEs. Owing to its generality and simplicity, the proposed method is expected to be of broad interest to the scientific and engineering communities.