CMX Lunch Seminar

In this talk, I will give an overview of kernel methods and their applications to both the numerical solution of partial differential equations (PDEs) and for surrogate modeling via operator learning. As PDE solvers, kernel-based finite differences enable the high-order accurate solution of PDEs on point clouds. I will show applications of this to PDEs on moving domains and on manifolds. In the context of operator learning, I will show recent work on two neural operators that leverage the strengths of closed-form kernels: (1) the Kernel Neural Operator (KNO), a particular generalization of the well-known Fourier Neural Operator (FNO); and (2) Ensemble and Partition-of-Unity Deep Operator Networks (DeepONets), which leverage localized kernel-based approximation to enhance accuracy near solution features of interest.