Time Dependent Partial Differential Equation

Time-dependent partial differential equations (PDEs) describe how systems evolve over time, and solving them is crucial across numerous scientific and engineering disciplines. Current research focuses on developing efficient and accurate data-driven methods, employing neural operator architectures like Fourier Neural Operators and Multigrid Neural Operators, as well as physics-informed neural networks (PINNs) and graph neural networks (GNNs), often incorporating techniques like temporal decomposition and multi-scale processing to handle complex dynamics. These advancements aim to improve the speed and accuracy of PDE solutions, particularly for problems involving high dimensionality, complex geometries, and long-term predictions, impacting fields ranging from fluid dynamics and climate modeling to materials science and medical imaging.