用神经网络压缩引力模拟数据,实现高精度连续时空建模。
Einstein Fields: A Neural Perspective To Computational General Relativity
- 用神经张量场隐式表示时空度规,自动微分推导物理量。
- 存储量减少4000倍,精度达小数点后5~7位,微分精度提升5个数量级。
- 开源工具支持引力模拟研究,适合物理与机器学习交叉方向者使用。
我们提出Einstein Fields,一种神经表示方法,可将计算量巨大的四维数值相对论模拟压缩为紧凑的隐式神经网络权重。该方法通过建模广义相对论的核心张量场——度规,利用自动微分推导物理量。与传统神经场(如符号距离、占据或辐射场)不同,Einstein Fields属于神经张量场,其在编码时空几何时,动力学特性自然涌现。该隐式方法具备连续建模四维时空、无需网格、存储高效、导数精度高和易用等优势。相比离散表示,存储内存最高减少4000倍,数值精度保持在小数点后5至7位。在单精度下,基于Einstein Fields参数化的度规张量微分精度比传统有限差分法高出最多五个数量级。我们在多个经典广义相对论与数值相对论测试场景中验证了这些特性,并开源了基于JAX的库:https://github.com/AndreiB137/EinFields,为机器学习在数值相对论中的应用迈出第一步。
原文摘要 · Abstract (English)
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.
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