Koopman Theory-Inspired Method for Learning Time Advancement Operators in Unstable Flame Front Evolution
About
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.
Related benchmarks
| Task | Dataset | Result | Rank | |
|---|---|---|---|---|
| Short-term PDE solution prediction | 1d-KDV (val) | Relative L2 Error7.10e-4 | 9 | |
| Short-term PDE solution prediction | 1d-KDV (train) | Relative L2 Error7.30e-4 | 9 | |
| Short-term PDE solution prediction | 1d-MS beta=40 (train) | Relative L2 Error0.0037 | 7 | |
| Short-term PDE solution prediction | 1d-MS beta=10 (train) | Relative L2 Error4.10e-4 | 7 | |
| Short-term PDE solution prediction | 1d-MS (beta=10) (val) | Rel L2 Error4.60e-4 | 7 | |
| Short-term PDE solution prediction | 1d-KS beta=40 (train) | Relative L2 Error4.00e-4 | 7 | |
| Short-term PDE solution prediction | 1d-KS beta=40 (val) | Relative L2 Error (%)0.043 | 7 | |
| Short-term PDE solution prediction | 1d-KS beta=10 (train) | Rel L2 Error3.90e-4 | 7 | |
| Short-term PDE solution prediction | 1d-KS beta=10 (val) | Relative L2 Error4.20e-4 | 7 | |
| Short-term PDE solution prediction | 1d-MS beta=40 (val) | Relative L2 Error0.01 | 7 |