Energy Conserving Data Driven Discretizations for Maxwells Equations

Abstract: We study data-driven construction of spatial discretizations for the one-dimensional Maxwell system. Given high-fidelity training data generated by a spectral discretization, we learn a linear convolution stencil that approximates the spatial derivative operator appearing in Maxwell's equations. The stencil is obtained by solving a convex quadratic optimization problem, subject to linear constraints that enforce skew-adjointness of the discrete derivative. These constraints guarantee a semi-discrete energy identity for the resulting Maxwell system. We prove that our constraints characterize the class of skew-symmetric convolution operators and express the associated numerical wave speed and CFL restriction for the classical leapfrog scheme in terms of the learned stencil's Fourier symbol. We then compare several convex solvers for the resulting quadratic program -- projected gradient, Nesterov-accelerated gradient, ADMM, and an interior-point reference implemented in CVXPY -- and evaluate the learned schemes in time-dependent one-dimensional Maxwell simulations using a Crank--Nicolson (CN) time discretization. Our numerical experiments show that (i) energy-constrained learned stencils achieve accuracy comparable to standard central differences while exactly preserving the discrete electromagnetic energy under CN time-stepping, and (ii) ADMM and interior-point methods produce nearly identical operators, with ADMM offering a favorable tradeoff between accuracy, constraint satisfaction, and runtime.