Numerical Analysis and Scientific Computing Seminar | Justin Sirignano, Scientific Machine Learning: Applications to PDEs and Convergence Analysis

Justin Sirignano, University of Oxford Mathematical Institute

Applications of deep learning to partial differential equations (PDEs) will be presented along with recent mathematical results. Physics-informed neural networks (PINNs) and Deep Galerkin Methods (DGM) directly solve PDEs with neural networks. For linear elliptic PDEs, we prove that DGM/PINNs -- despite the non-convexity of neural networks -- trained with gradient descent globally converge to the PDE solution. In the second half of the presentation, we discuss using deep learning to model unknown terms within a PDE. The neural network terms in the PDE are optimized using adjoint PDEs. Numerical examples from fluid dynamics and convergence results will be discussed.