Discovery and inversion of the viscoelastic wave equation in inhomogeneous media

TL;DR

The paper tackles inverse PDE discovery for viscoelastic wave propagation in inhomogeneous media from sparse and noisy measurements. It proposes a hybrid framework that alternates between a sparse-regression–based discovery phase and an RCNN-based embedding phase to refine terms and invert spatially varying coefficients, expressing the governing dynamics via $\mathbf{U}_{tt} = \boldsymbol{\Theta}(\mathbf{U})\boldsymbol{\Xi}(x)$. The RCNN employs FD-based CNN filters to implement discretized differential operators and embeds boundary conditions, enabling high-resolution wavefield predictions and physically consistent coefficient inversion. Across elastic/homogeneous and viscoelastic/inhomogeneous scenarios, the method demonstrates robustness to noise and data scarcity, accurately recovering the governing terms and velocity/viscosity fields and producing high-fidelity wavefield predictions.

Abstract

In scientific machine learning, the task of identifying partial differential equations accurately from sparse and noisy data poses a significant challenge. Current sparse regression methods may identify inaccurate equations on sparse and noisy datasets and are not suitable for varying coefficients. To address this issue, we propose a hybrid framework that combines two alternating direction optimization phases: discovery and embedding. The discovery phase employs current well-developed sparse regression techniques to preliminarily identify governing equations from observations. The embedding phase implements a recurrent convolutional neural network (RCNN), enabling efficient processes for time-space iterations involved in discretized forms of wave equation. The RCNN model further optimizes the imperfect sparse regression results to obtain more accurate functional terms and coefficients. Through alternating update of discovery-embedding phases, essential physical equations can be robustly identified from noisy and low-resolution measurements. To assess the performance of proposed framework, numerical experiments are conducted on various scenarios involving wave equation in elastic/viscoelastic and homogeneous/inhomogeneous media. The results demonstrate that the proposed method exhibits excellent robustness and accuracy, even when faced with high levels of noise and limited data availability in both spatial and temporal domains.

Figures (11)

• Figure 1: Schematic diagram of the proposed framework. (a) Discovery phase preliminarily identifies governing equations from observations. (b) Embedding phase use the RCNN model further optimizes the the imperfect function terms $\boldsymbol{\mathcal{S}}(\cdot)$ and coefficients $\boldsymbol{\Xi}(x)$ identified from discovery pahse. Given the measurements $u_m\left(x^\prime,t^\prime\right)\in\mathbb{R}^{n_t^\prime\times n_x^\prime}$ on a coarse grid (with resolution $n_t^\prime\times n_x^\prime)$, RCNN model can provide the discovered equation, inverted coefficients ($c$ and $\eta$), and high-resolution wavefield prediction $\widehat{\mathbf{{U}}} \in \mathbb{R}^{n_t\times n_x}$. The alternating update strategy implies that the high-resolution wavefield predicted by RCNN are provided to the discovery phase.
• Figure 2: The architecture of RCNN model in embedding phase. (a) The directed acycic graph of a RCNN for the forward modelling. (b) The unrolled directed acyclic graph of the RCNN. $\widehat{\mathbf{U}}_{(1)}$ and $\widehat{\mathbf{U}}_{(T)}$ are the predicted wavefield of the RCNN cell at times $t_{1}$ and $t_{T}$, respectively. (c) The single RCNN cell architecture. $\boldsymbol{\mathcal{S}}(\cdot)\boldsymbol{\Xi}(x)$ is provided by the sparse regression from the previous discovery phase.
• Figure 3: Schematic diagram of FD-based filters and hard embedding of boundary conditions. The filter $\mathcal{K}_{x}=\frac{1}{12\delta x}(1,-8,0,8,1)$ implements a 4th-order FD derivative operation in 1D CNN. For fixed boundary condition (upper figure), directly fill constant $0$ on the extended nodes outside the boundary. For free-surface boundary condition (lower figure), determine the filling values of ghost points based on the values of the internal wavefield.
• Figure 4: Schematic diagram of multi-transmitting formula (MTF).
• Figure 5: Case 1: Discovery-embedding alternately updates the identified function terms and corresponding coefficients. In the first execution of sparse regression (SR1), the discovered equation is $u_{tt} = 5.09 u_{xx} - 5.06 u - 0.1 u_t$. After the fifth loop (Loop 5), the correct equation $u_{tt}=6.25 u_{xx}$ is obtained.