We present a physics-constrained neural network (PCNN) approach to solving Maxwell’s equations for the electromagnetic fields of intense relativistic charged particle beams. We create a 3D convolutional PCNN to map time-varying current and charge densities J(r, t) and ρ(r, t) to vector and scalar potentials A(r, t) and φ(r, t) from which we generate electromagnetic fields according to Maxwell’s equations: B = ∇ × A and E = −∇φ − ∂A/∂t. Our PCNNs satisfy hard constraints, such as ∇ · B = 0, by construction. Soft constraints push A and φ toward satisfying the Lorenz gauge.