Title: Einstein Fields: A Neural Perspective To Computational General Relativity
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
Keywords: neural fields (implicit neural representations), neural compression, tensor fields, differential geometry, general relativity (GR) and numerical relativity (NR), Sobolev training, differential geometry, finite-difference methods

Abstract: 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 seven decimal digit places or more. Moreover, in single precision, differentiation of the Einstein Fields-parameterized metric tensor is up to five orders of magnitude more accurate compared to finite differencing methods. We demonstrate these properties on several canonical test beds of general relativity and release an open-source JAX-based library, paving the way for more scalable and expressive approaches to next-generation numerical relativity.