[论文解读] Solving the wave equation with physics-informed deep learning

Ben Moseley, Andrew Markham+1 more|arXiv (Cornell University)|Jun 21, 2020

Model Reduction and Neural Networks被引用 112

一句话总结

论文证明物理信息神经网络（PINNs）能够在均匀、分层和地球真实介质中求解二维声波方程，并在边界数据之外实现泛化，并能够通过对源位置进行条件化来避免重新训练。

ABSTRACT

We investigate the use of Physics-Informed Neural Networks (PINNs) for solving the wave equation. Whilst PINNs have been successfully applied across many physical systems, the wave equation presents unique challenges due to the multi-scale, propagating and oscillatory nature of its solutions, and it is unclear how well they perform in this setting. We use a deep neural network to learn solutions of the wave equation, using the wave equation and a boundary condition as direct constraints in the loss function when training the network. We test the approach by solving the 2D acoustic wave equation for spatially-varying velocity models of increasing complexity, including homogeneous, layered and Earth-realistic models, and find the network is able to accurately simulate the wavefield across these cases. By using the physics constraint in the loss function the network is able to solve for the wavefield far outside of its boundary training data, offering a way to reduce the generalisation issues of existing deep learning approaches. We extend the approach for the Earth-realistic case by conditioning the network on the source location and find that it is able to generalise over this initial condition, removing the need to retrain the network for each solution. In contrast to traditional numerical simulation this approach is very efficient when computing arbitrary space-time points in the wavefield, as once trained the network carries out inference in a single step without needing to compute the entire wavefield. We discuss the potential applications, limitations and further research directions of this work.

研究动机与目标

• 动机并评估 PINNs 在越来越复杂介质中求解波方程。
• 证明损失中的物理约束能够提升对边界数据以外的泛化。
• 通过对源位置进行条件化来扩展 PINN，使其能对初始条件实现泛化。
• 提出课程学习（curriculum learning）以改善波动型问题的收敛性。
• 识别界面以及高频/三维扩展的挑战与未来方向。

提出的方法

• 将波方程表述为一个物理约束的神经网络问题，其中网络近似 u(t,x)。
• 使用基于初始 FD 数据的边界损失，以及通过自动微分强制波方程的物理损失。
• 使用带 softplus 激活、线性输出的 10 层全连接网络（1024 通道）进行训练。
• 通过对源位置进行条件化来扩展 PINNs，以在初始条件上实现泛化。
• 通过先在边界数据上进行训练，然后在时间范围扩展的情况下加入物理损失来实现课程学习。
• 在三个速度模型（均匀、分层、Marmousi）上进行测试，复杂性逐步增加。

实验结果

研究问题

• RQ1 Can PINNs accurately solve the 2D acoustic wave equation in media with spatially varying velocity?
• RQ2 Does incorporating physics loss enable generalisation beyond boundary training data for wave problems?
• RQ3 Does conditioning the PINN on the source location allow generalisation across different initial conditions without retraining?
• RQ4 What training strategies (e.g., curriculum learning) improve convergence for wave equation PINNs?
• RQ5 What are the limitations of PINNs at interfaces and for complex Earth-realistic models?

主要发现

• PINNs 能在均匀和分层介质中准确模拟波场，与 FD 解一致，并在初始训练时间之外实现泛化。
• 物理损失使网络能够在多种介质中再现透射、反射、压缩、膨胀和球面扩张。
• 对源位置进行条件化使在地球真实的 Marmousi 模型中能够对新源实现无重新训练的泛化。
• 课程学习提高了收敛性相比从一开始就使用物理损失。
• Marmousi 情况显示有效的初始波场建模和后期动力学，但在低振幅反射波方面存在困难，表明界面不连续性挑战。
• 在 Titan V GPU 上训练 Marmousi 的 PINN 大约需要 1 天，之后可以快速进行逐查询波场评估。