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Einstein Fields: A Neural Perspective To Computational General Relativity

Sandeep Suresh Cranganore, Andrei Bodnar, Arturs Berzins, Johannes Brandstetter

TL;DR

Einstein Fields, a neural representation designed to compress computationally intensive four-dimensional numerical relativity simulations into compact implicit neural network weights, is introduced, taking the first steps to studying the potential of machine learning in numerical relativity.

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 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: \href{https://github.com/AndreiB137/EinFields}{https://github.com/AndreiB137/EinFields}, taking the first steps to studying the potential of machine learning in numerical relativity.

Einstein Fields: A Neural Perspective To Computational General Relativity

TL;DR

Einstein Fields, a neural representation designed to compress computationally intensive four-dimensional numerical relativity simulations into compact implicit neural network weights, is introduced, taking the first steps to studying the potential of machine learning in numerical relativity.

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 -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: \href{https://github.com/AndreiB137/EinFields}{https://github.com/AndreiB137/EinFields}, taking the first steps to studying the potential of machine learning in numerical relativity.

Paper Structure

This paper contains 103 sections, 2 theorems, 138 equations, 23 figures, 14 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Method -- Parametrizing Tensor fields via EinFields
  4. Distortion
  5. Retrieving physics via neural tensor field derivatives
  6. Experimental Verifications
  7. Accuracy and storage efficiency of EinFields: metric and its derived quantities.
  8. Ablation studies
  9. Conlusion
  10. Introduction to General Relativity
  11. Fundamental concepts of differential geometry & and tensor calculus
  12. Contravariant and covariant components
  13. Dual space
  14. Tangent and cotangent spaces
  15. Tangent vectors as vector fields.
  16. ...and 88 more sections

Key Result

Theorem 1

(Cotangent space): Let $\mathcal{M}$ be a smooth ($C^{\infty}$)-manifold (differentiable). The cotangent space $T^{*}_{p} \mathcal{M} := \{(p, v^*|_p) | \langle v^*|_p, v|_p \rangle = \kappa, \ \ \forall \ p \in \mathcal{M}, v|_p \in T_{p} \mathcal{M}, \kappa \in \mathbb{R}\}$ at point $p \in \mathc

Figures (23)

  • Figure 1: A conceptual overview of EinFields training and downstream pipeline. (i) Premise: The Einstein field equations (EFEs) in Eq. \ref{['eq:EFEs']} are highly non-linear PDEs defined on a 4D spacetime manifold, describing the geometric nature of gravitation. Their solutions define the metric tensor field $g_{\alpha \beta}(x^\mu)$, which encodes the full spacetime geometry and serves as a tensorial generalization of the gravitational potential. In this work, we parametrize $g_{\alpha \beta}(x^\mu)$ using a neural network. (ii) Training: The training is conducted on the metric tensor fields defined on 4D spacetime points, such as uniform or hierarchical grids. EinFields instead fit a continuous signal on these discrete representations, thus modeling 4D spacetime as a continuum, and returning the metric tensor field for a 4D spacetime query coordinate $p \equiv (t, x) \in \mathscr{M}$ at arbitrary resolution. (iii) Sobolev supervision: The reconstruction quality of the metric and its derivatives is improved by augmenting Sobolev losses, i.e., metric Jacobian (neighborhood structure) and Hessian (curvatures). (iv) Validation and downstream tasks: Sobolev improved EinFields' AD-based derivatives enable accurate point-wise retrieval of differential geometric quantities, such as the Levi-Civita connection (covariant derivative), geodesics, curvature tensors, and their invariants.
  • Figure 2: The directed-acyclic graph (DAG) for computing the differential geometric quantities from the metric tensor $\mathtt{g}$ in analogy to Figure \ref{['fig:diffgeo_gr']} and Eq. \ref{['eq:simple_dag']} . The transformations include repeated differentiation implemented via forward-mode Jacobian$\mathtt{jacfwd}$ operations and tensor index manipulation using $\mathtt{einsum}$. Tensors are in depicted in teal blue, connection in light-blue, tensor derivatives in green and conservation laws (Bianchi identities $( \mathtt{jacfwd + \Gamma} ) \mathtt{Riem} = 0$) in red.
  • Figure 3: Trendlines of accuracy versus storage memory (KiB) requirement for the metric tensor and Christoffel symbols. For the explicit grid storage this is computed as $\mathtt{num \ of\ grid\ collocation\ points \times 4}$, with 4 bytes for single precision ($\mathtt{FLOAT32}$). For the NeFs, this corresponds to the storage memory of the compact implicit NN weights.
  • Figure 4: Row 1: Geodesics in Schwarzschild spacetime simulated in spherical coordinates -- Eq. \ref{['eq:schwarzschild_spherical']}. Row 2: Geodesics in Kerr spacetime simulated in Boyer–Lindquist coordinates -- Eq. \ref{['eq:kerr_boyer_linduist']}. Distinct regions of the geometry are indicated in solid colors. Green solid lines represent ground-truth geodesics, while the red dotted lines represent our NeFs reconstructed orbits.
  • Figure 5: Spatial deformations (stretching and squeezing) of a circular ring of test particles due to "+" polarized gravitational wave -- Eq. \ref{['eq:gws_distortion']}. The NeF-reconstructed $h_{+} \cos(\omega(t - z))$ and $h_{\times} \cos(\omega(t - z))$ show excellent agreement with the analytic geodesic deviation for the linearized gravity use case. See Table \ref{['tab:grav_waves_results']} for a quantitative evaluation.
  • ...and 18 more figures

Theorems & Definitions (14)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Definition 9
  • ...and 4 more