We propose a new approach to the solution of the wave propagation and full
waveform inversions (FWIs) based on a recent advance in deep learning called
Physics-Informed Neural Networks (PINNs). In this study, we present an
algorithm for PINNs applied to the 2D acoustic wave equation and test the model
with both forward wave propagation and FWIs case studies. These synthetic case
studies are designed to explore the ability of PINNs to handle varying degrees
of structural complexity using both teleseismic plane waves and seismic point
sources. PINNs meshless formalism allows for a flexible implementation of the
wave equation and different types of boundary conditions. For instance, our
models demonstrate that PINN automatically satisfies absorbing boundary
conditions, a serious computational challenge for common wave propagation
solvers. Furthermore, a priori knowledge of the subsurface structure can be
seamlessly encoded in PINNs formulation. We find that the current
state-of-the-art PINNs provide good results for the forward model, even though
spectral element or finite difference methods are more efficient and accurate.
More importantly, our results demonstrate that PINNs yield excellent results
for inversions on all cases considered and with limited computational
complexity. Using PINNs as a geophysical inversion solver offers exciting
perspectives, not only for the full waveform seismic inversions, but also when
dealing with other geophysical datasets (e.g., magnetotellurics, gravity) as
well as joint inversions because of its robust framework and simple
implementation.

通过基于物理学知识的神经网络（PINNs）方法对二维声波方程进行求解和全波形反演（FWI）问题进行研究，证明其在处理不同结构复杂度的情况下表现出良好的结果，不仅适用于地震学领域，而且可以处理其他地球物理学数据集以及联合反演等问题，拓展了地球物理学反演研究的新途径。

用物理信息指导的神经网络（PINNs）进行波传播和完全波形反演

Physics-informed Neural Networks (PINNs) for Wave Propagation and Full  Waveform Inversions

Physics-informed neural networks (PINNs) are demonstrating remarkable promise
in integrating physical models with gappy and noisy observational data, but
they still struggle in cases where the target functions to be approximated
exhibit high-frequency or multi-scale features. In this work we investigate
this limitation through the lens of Neural Tangent Kernel (NTK) theory and
elucidate how PINNs are biased towards learning functions along the dominant
eigen-directions of their limiting NTK. Using this observation, we construct
novel architectures that employ spatio-temporal and multi-scale random Fourier
features, and justify how such coordinate embedding layers can lead to robust
and accurate PINN models. Numerical examples are presented for several
challenging cases where conventional PINN models fail, including wave
propagation and reaction-diffusion dynamics, illustrating how the proposed
methods can be used to effectively tackle both forward and inverse problems
involving partial differential equations with multi-scale behavior. All code an
data accompanying this manuscript will be made publicly available at
https://github.com/PredictiveIntelligenceLab/MultiscalePINNs.

本研究采用神经切向核理论观察了物理知识指导神经网络的局限性，并通过构建新的体系结构使用时空和多尺度随机傅里叶特征，成功解决了波传播、反应扩散动力学等多尺度问题。