The eikonal equation is utilized across a wide spectrum of science and engineering disciplines. In seismology, it regulates seismic wave traveltimes needed for applications like source localization, imaging, and inversion. Several numerical algorithms have been developed over the years to solve the eikonal equation. However, these methods require considerable modifications to incorporate additional physics, such as anisotropy, and may even breakdown for certain complex forms of the eikonal equation, requiring approximation methods. Moreover, they suffer from computational bottleneck when repeated computations are needed for perturbations in the velocity model and/or the source location, particularly in large 3D models. Here, we propose an algorithm to solve the eikonal equation based on the emerging paradigm of physics-informed neural networks (PINNs). By minimizing a loss function formed by imposing the eikonal equation, we train a neural network to output traveltimes that are consistent with the underlying partial differential equation. We observe sufficiently high traveltime accuracy for most applications of interest. We also demonstrate how the proposed algorithm harnesses machine learning techniques like transfer learning and surrogate modeling to speed up traveltime computations for updated velocity models and source locations. Furthermore, we use a locally adaptive activation function and adaptive weighting of the terms in the loss function to improve convergence rate and solution accuracy. We also show the flexibility of the method in incorporating medium anisotropy and free-surface topography compared to conventional methods that require significant algorithmic modifications. These properties of the proposed PINN eikonal solver are highly desirable in obtaining a flexible and efficient forward modeling engine for seismological applications.

eikonal 方程在广泛的科学和工程学科中被使用。在地震学中，它调节震源定位、成像和反演等应用所需的地震波走时。多年来，已经开发了几种数值算法来求解特征方程。但是，这些方法需要进行大量修改才能加入其他物理特性，例如各向异性，甚至可能会分解某些复杂形式的 eikonal 方程，需要使用近似方法。此外，当速度模型和/或源位置中的扰动需要重复计算时，它们会遇到计算瓶颈，特别是在大型 3D 模型中。这里，我们提出了一种基于物理信息神经网络 (PINN) 的新兴范式来求解特征方程的算法。通过最小化通过施加特征方程形成的损失函数，我们训练神经网络输出与底层偏微分方程一致的旅行时间。对于大多数感兴趣的应用，我们观察到足够高的走时精度。我们还演示了所提出的算法如何利用机器学习技术（如转移学习和代理建模）来加速更新速度模型和源位置的旅行时间计算。此外，我们使用局部自适应激活函数和损失函数中项的自适应加权来提高收敛速度和求解精度。与需要大量算法修改的传统方法相比，我们还展示了该方法在结合中等各向异性和自由表面形貌方面的灵活性。所提出的 PINN eikonal 求解器的这些特性对于获得用于地震学应用的灵活高效的前向建模引擎非常理想。