Predicting the evolution of complex systems governed by partial differential equations remains challenging, especially for nonlinear, chaotic behaviors. This study introduces Koopman-inspired Fourier neural operators and convolutional neural networks 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.