CENTER FOR COMPUTATION & TECHNOLOGY

Despite advances in simulating multiphysics problems through numerical discretization of PDEs, mesh-based approximation remains challenging, especially for high-dimensional problems governed by parameterized PDEs. Moreover, other PDE-related problems, such as PDE-constrained shape optimization, introduce additional difficulties including mesh deformation and correction. While Physics-Informed Neural Networks (PINNs) offer an alternative, they often lack the accuracy of traditional methods like finite element methods. Relying solely on a 'black-box' approach may not be the best choice for scientific computing. Inspired by adaptive finite element methods, we propose a deep adaptive sampling approach to solve low-regularity parametric PDEs and high-dimensional committor functions in rare event simulations. Additionally, by integrating the mesh-free nature of neural networks into the direct-adjoint looping (DAL), we achieve fully mesh-independent solutions for PDE-constrained shape optimization problems.