Koopman Theory-Inspired Method for Learning Time Advancement Operators in Unstable Flame Front Evolution

TL;DR

This paper introduces Koopman-inspired neural network models that learn solution operators for unstable flame front evolution, improving multi-step prediction accuracy and long-term statistical reproduction of complex PDE-governed systems.

Contribution

The paper presents novel Koopman-inspired Fourier Neural Operators and CNNs that transform data into a high-dimensional space for better modeling of nonlinear, chaotic PDE systems.

Findings

Enhanced multi-step prediction accuracy

Superior long-term statistical reproduction

Effective modeling of complex dynamical systems

Abstract

Predicting the evolution of complex systems governed by partial differential equations (PDEs) remains challenging, especially for nonlinear, chaotic behaviors. This study introduces Koopman-inspired Fourier Neural Operators (kFNO) and Convolutional Neural Networks (kCNN) to learn solution advancement operators for flame front instabilities. By transforming data into a high-dimensional latent space, these models achieve more accurate multi-step predictions compared to traditional methods. Benchmarking across one- and two-dimensional flame front scenarios demonstrates the proposed approaches' superior performance in short-term accuracy and long-term statistical reproduction, offering a promising framework for modeling complex dynamical systems.