# Physics-constrained 3D Convolutional Neural Networks for Electrodynamics

arXiv: https://arxiv.org/abs/2301.13715 · HTML: https://mlsanity.com/paper/2301.13715/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 68 references that are in the map.
- First posted on arXiv 31 Jan 2023.

## Builds on (most cited first, 9 of 9)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [On the influence of over-parameterization in manifold based surrogates and deep neural operators](https://mlsanity.com/paper/2203.05071/) (2022)
- [Metasurface-Enabled On-Chip Multiplexed Diffractive Neural Networks in the Visible](https://mlsanity.com/paper/2107.07873/) (2021)
- [Adaptive 3D convolutional neural network-based reconstruction method for 3D coherent diffraction imaging](https://mlsanity.com/paper/2008.10094/) (2020)
- [Physics-Based Deep Neural Networks for Beam Dynamics in Charged Particle Accelerators](https://mlsanity.com/paper/2007.03555/) (2020)
- [High-fidelity Prediction of Megapixel Longitudinal Phase-space Images of Electron Beams using Encoder-Decoder Neural Networks](https://mlsanity.com/paper/2101.10437/) (2021)
- [Adaptive Machine Learning for Time-Varying Systems: Low Dimensional Latent Space Tuning](https://mlsanity.com/paper/2107.06207/) (2021)
- [Mixed Diagnostics for Longitudinal Properties of Electron Bunches in a Free-Electron Laser](https://mlsanity.com/paper/2201.05769/) (2022)

## Built on by (most cited first, 3 of 3)

- [A conditional latent autoregressive recurrent model for generation and forecasting of beam dynamics in particle accelerators](https://mlsanity.com/paper/2403.13858/) (2024)
- [Bidirectional Autoregressive Latent Diffusion for Forward and Inverse Magnetohydrodynamics](https://mlsanity.com/paper/2606.29620/) (2026)
- [Fast Reconstruction of Exact Maxwell Dynamics from Sparse Data](https://mlsanity.com/paper/2605.20514/) (2026)

## Abstract (arXiv)

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 p(r,t) to vector and scalar potentials A(r,t) and V(r,t) from which we generate electromagnetic fields according to Maxwell's equations: B=curl(A), E=-div(V)-dA/dt. Our PCNNs satisfy hard constraints, such as div(B)=0, by construction. Soft constraints push A and V towards satisfying the Lorenz gauge.