Variational Physics-informed Neural Operator (VINO) for Solving Partial Differential Equations

TL;DR

VINO is a novel deep learning framework that efficiently solves PDEs by minimizing their energy formulations, offering improved accuracy and performance over existing methods without requiring labeled data.

Contribution

This paper introduces VINO, a physics-informed neural operator that leverages a variational approach to enhance PDE solving efficiency and accuracy, especially at higher mesh resolutions.

Findings

VINO outperforms existing deep learning PDE solvers.

VINO's accuracy improves with increased mesh resolution.

VINO requires no labeled data for training.

Abstract

Solving partial differential equations (PDEs) is a required step in the simulation of natural and engineering systems. The associated computational costs significantly increase when exploring various scenarios, such as changes in initial or boundary conditions or different input configurations. This study proposes the Variational Physics-Informed Neural Operator (VINO), a deep learning method designed for solving PDEs by minimizing the energy formulation of PDEs. This framework can be trained without any labeled data, resulting in improved performance and accuracy compared to existing deep learning methods and conventional PDE solvers. By discretizing the domain into elements, the variational format allows VINO to overcome the key challenge in physics-informed neural operators, namely the efficient evaluation of the governing equations for computing the loss. Comparative results demonstrate VINO's superior performance, especially as the mesh resolution increases. As a result, this study suggests a better way to incorporate physical laws into neural operators, opening a new approach for modeling and simulating nonlinear and complex processes in science and engineering.