Learning Flame Evolution Operator under Hybrid Darrieus Landau and Diffusive Thermal Instability

In recent years, the integration of artificial intelligence (AI) and machine learning (ML) with the natural sciences and physical engineering has led to significant advancements, particularly in addressing the complexities of nonlinear partial differential equations (PDE). These equations are fundamental in understanding various physical phenomena, ranging from turbulent fluid dynamics to complicate physico-chemical processes. Within the domain of nonlinear PDE systems lies a rich tapestry of intricate dynamics, including instabilities, multiscale interactions, and chaotic behaviors. To enhance predictive capabilities and design robust control strategies in engineering applications, computational methods are indispensable. These methods, often in the form of numerical solvers, enable the accurate simulation of PDE solutions across spatial and temporal domains. Implicit in these solvers is the concept of the functional-mapping operator, which could iteratively advances the PDE solution functions in time, providing a pathway to explore the evolution of physical systems over extended durations. A distinctive class of machine learning methods has emerged, capable of learning and replicating the behavior of these PDE operators. Recent advancements have seen the proliferation of operator learning methods, each offering unique insights and capabilities.