Parabolic Partial Differential Equation

Parabolic partial differential equations (PDEs) model time-dependent processes across diverse scientific fields, and current research focuses on efficiently and accurately solving these equations, especially in high dimensions. This involves developing and analyzing novel numerical methods, including neural operators, multilevel Picard approximations, and physics-informed neural networks (PINNs), often leveraging techniques like backstepping and empirical interpolation to improve computational efficiency and accuracy. These advancements are crucial for tackling complex problems in areas such as finance, fluid dynamics, and control theory, where solving high-dimensional parabolic PDEs is computationally challenging.