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.
Papers
Physics-informed deep learning and compressive collocation for high-dimensional diffusion-reaction equations: practical existence theory and numerics
Simone Brugiapaglia, Nick Dexter, Samir Karam, Weiqi Wang
A hybrid numerical methodology coupling Reduced Order Modeling and Graph Neural Networks for non-parametric geometries: applications to structural dynamics problems
Victor Matray, Faisal Amlani, Frédéric Feyel, David Néron
Symmetry group based domain decomposition to enhance physics-informed neural networks for solving partial differential equations
Ye Liu, Jie-Ying Li, Li-Sheng Zhang, Lei-Lei Guo, Zhi-Yong Zhang
Solving Partial Differential Equations with Equivariant Extreme Learning Machines
Hans Harder, Jean Rabault, Ricardo Vinuesa, Mikael Mortensen, Sebastian Peitz