Linear Partial Differential Equation

Linear partial differential equations (PDEs) model numerous physical phenomena, and current research focuses on developing efficient and accurate numerical solvers. This involves exploring novel hybrid approaches combining classical numerical methods (like multigrid and spectral methods) with machine learning techniques, such as neural networks (including U-Nets) and Gaussian processes, to improve accuracy, efficiency, and uncertainty quantification. These advancements are driven by the need for faster and more robust solutions in diverse applications, ranging from antenna design to power system dynamics, and are leading to improved error estimation and the seamless integration of mechanistic models into probabilistic frameworks.