An Energy Stable Approach for Learning Derivative Operators from Noisy Data for Maxwells Equations

We develop a structure-preserving ADMM method, denoted SP-ADMM, for learning energy-stable spatial derivative stencils for Maxwell equations from noisy data. Starting from the source-free Maxwell system, we focus on a one-dimensional reduction whose energy conservation depends on the skew-adjointness of the spatial derivative operator. The learned derivative is represented by a compact periodic convolution stencil. Unlike standard constrained ADMM, which learns the full stencil and imposes skew-adjointness through equality constraints, SP-ADMM enforces skew-adjointness by construction through a reduced parameterization using only the independent positive-side stencil coefficients. Numerical experiments show that SP-ADMM is especially effective in hidden operator and noisy-data regimes. Across clean data, noisy derivative data, multiple initial conditions, different hidden skew-adjoint operators, training-set sizes, regularization parameters, constraint ablations, and long-time simulations, SP-ADMM achieves the smallest final-time electric-field error while preserving energy to roundoff accuracy. A layered-medium Maxwell propagation test further shows that the learned structure-preserving stencil remains competitive with classical finite differences in a physical reflection/transmission setting. Overall, SP-ADMM provides a data-driven way to learn accurate Maxwell stencils while retaining the energy-conserving structure of the underlying equations.