Koopman Theory-Inspired Method for Learning Time Advancement Operators in Unstable Flame Front Evolution
This provides a promising framework for modeling complex dynamical systems, but it is incremental as it adapts existing Koopman theory to neural networks for a specific domain.
The study tackled predicting unstable flame front evolution governed by PDEs by introducing Koopman-inspired neural operators, achieving superior multi-step prediction accuracy and statistical reproduction compared to traditional methods.
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.