Discovery and inversion of the viscoelastic wave equation in inhomogeneous media

2024-09-27

Computer Physics Communications

AI Summary

Identifying the governing viscoelastic wave equation in heterogeneous media from sparse, noisy observations remains challenging, particularly due to weak modeling capability for spatially varying coefficients. Method: This paper proposes a novel “discover-embed” alternating optimization framework that integrates sparse identification of nonlinear dynamics (SINDy) with recurrent convolutional neural networks (RCNNs) for the first time, enabling robust discovery of partial differential equations (PDEs) under severe data limitations. The framework jointly learns both the PDE structure and spatially varying coefficients via dual-stage alternating optimization over discrete spatiotemporal iterations. Contribution/Results: RCNNs significantly enhance robustness and generalization under high noise (SNR ≤ 5 dB) and extreme subsampling (70% reduction in spatiotemporal resolution). Extensive validation across elastic/viscoelastic and homogeneous/heterogeneous media yields consistent term identification accuracy exceeding 92%, establishing a new paradigm for data-driven PDE discovery.

Technology Category

• Search and Optimization: Learning to Search
• Computer Vision: Learning & Optimization for CV
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Interdisciplinary science discovery with web data mining
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Identify PDEs from sparse noisy data accurately
• Improve sparse regression for varying coefficients
• Robustly discover wave equations in inhomogeneous media

Innovation

• Hybrid framework combines discovery and embedding phases
• Uses recurrent convolutional neural network for optimization
• Robust identification from noisy, low-resolution data