Discovery and inversion of the viscoelastic wave equation in inhomogeneous media
This work addresses a key problem in scientific machine learning for researchers dealing with noisy, low-resolution measurements in wave propagation, though it is incremental as it builds on existing sparse regression methods.
The authors tackled the challenge of identifying partial differential equations from sparse and noisy data by proposing a hybrid framework that alternates between sparse regression and a recurrent convolutional neural network, achieving robust and accurate results even with high noise and limited data.
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.