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

TL;DR

This paper introduces SP-ADMM, a structure-preserving method for learning energy-stable derivative operators for Maxwell equations from noisy data, ensuring energy conservation and high accuracy.

Contribution

The paper presents a novel structure-preserving ADMM approach that enforces skew-adjointness by construction, improving learning of Maxwell derivative operators from noisy data.

Findings

SP-ADMM achieves minimal electric-field error in various noisy and hidden operator regimes.

The learned stencils preserve energy to roundoff accuracy across multiple tests.

The method remains competitive with classical finite differences in physical reflection/transmission simulations.

Abstract

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.