基于物理信息神经网络的穿刺演化

中文摘要 AI 辅助

我们研究了物理信息神经网络（PINNs）在球对称真空中求解爱因斯坦方程的应用。我们的神经网络架构旨在求解1+1维移动穿刺演化，通过最小化一个包含数据项、重建时空的里奇张量以及移动穿刺规范条件的损失函数来实现。该网络表示时移、位移以及Z4c共形变量χ和γ̃_rr，这些变量在穿刺处保持正则。我们考虑了在穿刺规范下单个无自旋黑洞的演化，并展示了PINNs能够插值数值相对论（NR）数据、预测其演化，并在未提供数据的区域重建时空。作为插值器，该网络提供了NR数据的紧凑、可微表示。仅给定早期时间和外边界数据，它就能将穿刺向前推进时间，并再现NR解的主要特征。仅给定外部数据且无规范条件（一个不适定的反问题），它能在其自选坐标系中恢复史瓦西时空的内部，我们通过学习到Kerr-Schild形式的坐标变换验证了这一点。最后，我们评估了PINNs在NR中的优势与局限性，并概述了其未来应用的前景。

英文摘要

We investigate the application of physics-informed neural networks (PINNs) to solve the vacuum Einstein equations in spherical symmetry. Our neural network architecture is designed to solve the 1+1D moving-puncture evolution, by minimizing a loss that combines a data term, the Ricci tensor of the reconstructed spacetime, and the moving-puncture gauge conditions. It represents the lapse, the shift and the Z4c conformal variables $χ$ and $\widetildeγ_{rr}$, which remain regular at the puncture. We consider the evolution of a single, non-spinning black-hole in the puncture gauge and show that PINNs can interpolate numerical relativity (NR) data, predict its evolution, and reconstruct spacetime in regions where no data is provided. As an interpolant, the network provides a compact, differentiable representation of the NR data. Given only early-time and outer-boundary data, it evolves the puncture forward in time and reproduces the main features of the NR solution. Given only exterior data and no gauge condition, an ill-posed inverse problem, it recovers the interior of a Schwarzschild spacetime in coordinates of its own choosing, which we verify by learning the coordinate transformation to the Kerr--Schild form. Finally, we assess the strengths and limitations of PINNs for NR and outline their prospects for future applications.

发表机构

• The Pennsylvania State University（宾夕法尼亚州立大学）
• Purdue University（普渡大学）
• Friedrich-Schiller-Universität Jena（耶拿弗里德里希·席勒大学）

机构由 AI 辅助整理，请以论文原文为准。

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15 pages, 7 figures