Physics-constrained machine learning for electrodynamics based on Fourier transformed Maxwell's equations*

Finding the electromagnetic fields generated from intense, relativistic particle beams can be a computationally expensive process. Recent advances in machine learning offer opportunities to create surrogate models that are faster and more flexible than traditional numerical methods. Building hard physics constraints into these models can ensure their outputs are compatible with known physical laws, while also potentially improving the models' ability to generalize beyond their training data. Utilizing a Fourier transformation-based representation of Maxwell's equations, we developed a physics-constrained neural network (PCNN) for electrodynamics without gauge ambiguity, which we label the Fourier-Helmholtz-Maxwell Neural Operator (FoHM-NO) method. In this approach, both of Gauss's laws and Faraday's law are built in as hard constraints, as well as the longitudinal component of Ampère-Maxwell's law in Fourier space (assuming the continuity equation). A convolutional encoder-decoder network acts as a solution operator for the transverse components of the Fourier transformed vector potential, which is used to predict the generated electromagnetic fields. We demonstrate FoHM-NO's abilities using a simulation of two electron beams. The model accurately predicts the magnetic field generated for the test data and performs the task much quicker than traditional simulations. Given the lack of interpretability of neural networks, accurate uncertainty quantification is paramount, thus it is shown that Monte Carlo Dropout can be used to quantify FoHM-NO's model uncertainty.