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

arXiv: https://arxiv.org/abs/2412.08426 · HTML: https://mlsanity.com/paper/2412.08426/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 47 references that are in the map.
- First posted on arXiv 11 Dec 2024.

## Builds on (most cited first, 5 of 5)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)

## Built on by (most cited first, 3 of 3)

- [Data-driven quantum Koopman method for simulating nonlinear dynamics](https://mlsanity.com/paper/2507.21890/) (2025)
- [An Inverse Scattering Inspired Fourier Neural Operator for Time-Dependent PDE Learning](https://mlsanity.com/paper/2512.19439/) (2025)
- [Optimizing Basis Function Selection in Constructive Wavelet Neural Networks and Its Applications](https://mlsanity.com/paper/2507.09213/) (2025)

## Abstract (arXiv)

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.