Partial Differential Equation

Partial differential equations (PDEs) describe a vast range of physical phenomena, and solving them efficiently and accurately is crucial across many scientific disciplines. Current research focuses on developing data-driven methods, particularly neural network architectures like Physics-Informed Neural Networks (PINNs), Fourier Neural Operators (FNOs), and graph neural networks, to overcome limitations of traditional numerical techniques, especially in high-dimensional or complex systems. These approaches leverage machine learning to approximate solutions, often incorporating physical constraints and symmetries to improve accuracy and generalization, and are showing promise in diverse applications from fluid dynamics to materials science. The development of robust and efficient neural PDE solvers has the potential to significantly accelerate scientific discovery and engineering design.