# Einstein Fields: A Neural Perspective To Computational General Relativity

arXiv: https://arxiv.org/abs/2507.11589 · HTML: https://mlsanity.com/paper/2507.11589/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Jul 2025.

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

- [Finding geodesics with the Deep Ritz method](https://mlsanity.com/paper/2510.15177/) (2025)
- [Lean-Quantum: Toward AI-Assisted Formalization of Quantum Information](https://mlsanity.com/paper/2607.05492/) (2026)
- [Neural surrogates for designing gravitational wave detectors](https://mlsanity.com/paper/2511.19364/) (2025)

## Abstract (arXiv)

We introduce Einstein Fields, a neural representation designed to compress computationally intensive four-dimensional numerical relativity simulations into compact implicit neural network weights. By modeling the metric, the core tensor field of general relativity, Einstein Fields enable the derivation of physical quantities via automatic differentiation. Unlike conventional neural fields (e.g., signed distance, occupancy, or radiance fields), Einstein Fields fall into the class of Neural Tensor Fields with the key difference that, when encoding the spacetime geometry into neural field representations, dynamics emerge naturally as a byproduct. Our novel implicit approach demonstrates remarkable potential, including continuum modeling of four-dimensional spacetime, mesh-agnosticity, storage efficiency, derivative accuracy, and ease of use. It achieves up to a 4,000-fold reduction in storage memory compared to discrete representations while retaining a numerical accuracy of five to seven decimal places. Moreover, in single precision, differentiation of the Einstein Fields-parameterized metric tensor is up to five orders of magnitude more accurate compared to naive finite differencing methods. We demonstrate these properties on several canonical test beds of general relativity and numerical relativity simulation data, while also releasing an open-source JAX-based library: hrefhttps://github.com/AndreiB137/EinFieldshttps://github.com/AndreiB137/EinFields, taking the first steps to studying the potential of machine learning in numerical relativity.