A Neural Operator based on Dynamic Mode Decomposition

TL;DR

This paper introduces a neural operator that combines dynamic mode decomposition with deep learning to efficiently model spatiotemporal processes and solve PDEs with reduced computational costs.

Contribution

It presents a novel neural operator based on DMD that automatically extracts key modes, improving efficiency and accuracy in solving PDEs compared to existing methods.

Findings

Achieves high reconstruction accuracy on heat, Laplace, and Burgers equations.

Reduces computational costs relative to traditional numerical methods.

Demonstrates superior performance over DeepONet and FNO in benchmark tests.

Abstract

The scientific computation methods development in conjunction with artificial intelligence technologies remains a hot research topic. Finding a balance between lightweight and accurate computations is a solid foundation for this direction. The study presents a neural operator based on the dynamic mode decomposition algorithm (DMD), mapping functional spaces, which combines DMD and deep learning (DL) for spatiotemporal processes efficient modeling. Solving PDEs for various initial and boundary conditions requires significant computational resources. The method suggested automatically extracts key modes and system dynamics using them to construct predictions, reducing computational costs compared to traditional numerical methods. The approach has demonstrated its efficiency through comparative analysis of performance with closest analogues DeepONet and FNO in the heat equation, Laplaces equation, and Burgers equation solutions approximation, where it achieves high reconstruction accuracy.