论文标题
神经EIKONAL求解器：提高物理信息的神经网络的准确性，以求解焦点方程
Neural Eikonal Solver: improving accuracy of physics-informed neural networks for solving eikonal equation in case of caustics
论文作者
论文摘要
由于其灵活性，物理知识神经网络的概念已成为解决微分方程的有用工具。有几种方法使用此概念来求解Eikonal方程，该方程描述了光滑的异质速度模型中声学和弹性波的第一个到达旅行时间。但是，产生苛刻的速度模型会加剧艾科纳尔的挑战，从而导致不足的溶液行为导致不稳定性和准确性的恶化。在本文中，我们回顾了使用神经网络来解决苛性病变的问题的问题。我们介绍了新型的神经Eikonal求解器（NES），用于以两种制定方式求解各向同性二基型方程：单点问题是针对固定源位置；两点问题是针对任意源接收器对。我们提出了几种在产生苛刻速度的速度模型中提供稳定性的技术：改善分解；基于哈密顿的非对称损失函数；高斯激活；对称。在我们的测试中，NES显示了相对均值的误差误差约为0.2-0.4％的二阶货运快速行进方法，并且表现优于现有的神经网络求解器，误差降低了10-60倍，并且训练速度更快2-30倍。 NES的推理时间与快速游行相当。单点NES提供了最精确的解决方案，而两点NES的精度略低，但具有非常紧凑的表示。它在需要大规模计算的各种地震应用中很有用（数百万个源接收器对）：射线建模，旅行时间断层扫描，降压器定位和基尔乔夫迁移。
The concept of physics-informed neural networks has become a useful tool for solving differential equations due to its flexibility. There are a few approaches using this concept to solve the eikonal equation which describes the first-arrival traveltimes of acoustic and elastic waves in smooth heterogeneous velocity models. However, the challenge of the eikonal is exacerbated by the velocity models producing caustics, resulting in instabilities and deterioration of accuracy due to the non-smooth solution behaviour. In this paper, we revisit the problem of solving the eikonal equation using neural networks to tackle the caustic pathologies. We introduce the novel Neural Eikonal Solver (NES) for solving the isotropic eikonal equation in two formulations: the one-point problem is for a fixed source location; the two-point problem is for an arbitrary source-receiver pair. We present several techniques which provide stability in velocity models producing caustics: improved factorization; non-symmetric loss function based on Hamiltonian; gaussian activation; symmetrization. In our tests, NES showed the relative-mean-absolute error of about 0.2-0.4% from the second-order factored Fast Marching Method, and outperformed existing neural-network solvers giving 10-60 times lower errors and 2-30 times faster training. The inference time of NES is comparable with the Fast Marching. The one-point NES provides the most accurate solution, whereas the two-point NES provides slightly lower accuracy but gives an extremely compact representation. It can be useful in various seismic applications where massive computations are required (millions of source-receiver pairs): ray modeling, traveltime tomography, hypocenter localization, and Kirchhoff migration.
