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.

翻译：预测由偏微分方程（PDE）支配的复杂系统的演化仍然具有挑战性，特别是对于非线性、混沌行为。本研究引入了受Koopman理论启发的傅里叶神经算子（kFNO）和卷积神经网络（kCNN），以学习火焰锋不稳定性的解推进算子。通过将数据转换到高维潜在空间，这些模型相较于传统方法实现了更精确的多步预测。在一维和二维火焰锋场景中的基准测试表明，所提出的方法在短期预测精度和长期统计特性再现方面均表现出优越性能，为复杂动力系统建模提供了一个有前景的框架。