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

Partial differential equations (PDEs) are fundamental mathematical frameworks used to describe complex physical phenomena across diverse scientific and engineering domains. From fluid dynamics and climate modeling to quantum mechanics and biological systems, PDEs encapsulate intricate interactions and dynamical behaviors derived from underlying physical principles. However, solving PDEs, particularly nonlinear equations with complex boundary conditions, poses significant computational challenges, historically limiting our ability to simulate and predict such systems accurately. The computational landscape for solving PDEs has been transformed by the integration of machine learning (ML) and artificial intelligence (AI) techniques. Recent advancements have introduced a proliferation of operator learning methods, each contributing unique insights and capabilities for tackling complex mathematical problems.