Speaker
Description
Solving partial differential equations (PDEs) using distributed Physics-Informed Neural Networks (PINNs) introduces major computational challenges associated with high-dimensional curvature estimation, ill-conditioned optimization landscapes, and communication overhead in federated environments. In this work, we exploit Kronecker structure and Krylov subspace methods to develop a scalable second-order optimization framework for federated PINNs applied to the 2D Poisson equation. The proposed method combines Kronecker-Factored Approximate Curvature (K-FAC) with Conjugate Gradient (CG) iterations within a matrix-free Gauss–Newton formulation, avoiding explicit Hessian or Jacobian construction through automatic differentiation-based Jacobian-vector products. The symmetric positive definite structure induced by discretized Poisson operators supports efficient CG convergence while enabling scalable curvature-aware optimization. Within the federated learning framework, each client independently solves a local Gauss–Newton system using K-FAC approximations and Krylov iterations, transmitting only local parameter corrections to the central server. Experimental results across grid resolutions from $32\times32$ to $128\times128$ demonstrate up to $57.8\%$ relative improvement in test accuracy and $44.4\%$ reduction in required communication rounds compared with first-order Adam-based optimization. These findings illustrate how structured linear algebra techniques, particularly Kronecker factorization and Krylov subspace solvers, provide an efficient foundation for communication-efficient scientific machine learning under distributed computing constraints.