Discovery and inversion of the viscoelastic wave equation in inhomogeneous media

• URL: http://arxiv.org/abs/2409.18370v1
• Date: Fri, 27 Sep 2024 01:05:45 GMT
• Title: Discovery and inversion of the viscoelastic wave equation in inhomogeneous media
• Authors: Su Chen, Yi Ding, Hiroe Miyake, Xiaojun Li,
• Abstract summary: Current sparse regression methods may identify inaccurate equations on sparse and noisy datasets. We propose a hybrid framework that combines two alternating direction optimization phases: discovery and embedding. The proposed method exhibits excellent robustness and accuracy, even when faced with high levels of noise.
• Score: 3.6864706261549127
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

Related papers

• Accelerated zero-order SGD under high-order smoothness and overparameterized regime [79.85163929026146]
  We present a novel gradient-free algorithm to solve convex optimization problems. Such problems are encountered in medicine, physics, and machine learning. We provide convergence guarantees for the proposed algorithm under both types of noise.
  arXiv  Detail & Related papers  (2024-11-21T10:26:17Z)
• Score-based Generative Models with Adaptive Momentum [40.84399531998246]
  We propose an adaptive momentum sampling method to accelerate the transforming process. We show that our method can produce more faithful images/graphs in small sampling steps with 2 to 5 times speed up.
  arXiv  Detail & Related papers  (2024-05-22T15:20:27Z)
• Towards stable real-world equation discovery with assessing differentiating quality influence [52.2980614912553]
  We propose alternatives to the commonly used finite differences-based method. We evaluate these methods in terms of applicability to problems, similar to the real ones, and their ability to ensure the convergence of equation discovery algorithms.
  arXiv  Detail & Related papers  (2023-11-09T23:32:06Z)
• A Geometric Perspective on Diffusion Models [57.27857591493788]
  We inspect the ODE-based sampling of a popular variance-exploding SDE. We establish a theoretical relationship between the optimal ODE-based sampling and the classic mean-shift (mode-seeking) algorithm.
  arXiv  Detail & Related papers  (2023-05-31T15:33:16Z)
• Score-based Diffusion Models in Function Space [140.792362459734]
  Diffusion models have recently emerged as a powerful framework for generative modeling. We introduce a mathematically rigorous framework called Denoising Diffusion Operators (DDOs) for training diffusion models in function space. We show that the corresponding discretized algorithm generates accurate samples at a fixed cost independent of the data resolution.
  arXiv  Detail & Related papers  (2023-02-14T23:50:53Z)
• Score-based Continuous-time Discrete Diffusion Models [102.65769839899315]
  We extend diffusion models to discrete variables by introducing a Markov jump process where the reverse process denoises via a continuous-time Markov chain. We show that an unbiased estimator can be obtained via simple matching the conditional marginal distributions. We demonstrate the effectiveness of the proposed method on a set of synthetic and real-world music and image benchmarks.
  arXiv  Detail & Related papers  (2022-11-30T05:33:29Z)
• A Priori Denoising Strategies for Sparse Identification of Nonlinear Dynamical Systems: A Comparative Study [68.8204255655161]
  We investigate and compare the performance of several local and global smoothing techniques to a priori denoise the state measurements. We show that, in general, global methods, which use the entire measurement data set, outperform local methods, which employ a neighboring data subset around a local point.
  arXiv  Detail & Related papers  (2022-01-29T23:31:25Z)
• Low-Rank Hankel Tensor Completion for Traffic Speed Estimation [7.346671461427793]
  We propose a purely data-driven and model-free solution to the traffic state estimation problem. By imposing a low-rank assumption on this tensor structure, we can approximate characterize both global patterns and the unknown complex local dynamics. We conduct numerical experiments on both synthetic simulation data and real-world high-resolution data, and our results demonstrate the effectiveness and superiority of the proposed model.
  arXiv  Detail & Related papers  (2021-05-21T00:08:06Z)
• Deep-learning of Parametric Partial Differential Equations from Sparse and Noisy Data [2.4431531175170362]
  In this work, a new framework, which combines neural network, genetic algorithm and adaptive methods, is put forward to address all of these challenges simultaneously. A trained neural network is utilized to calculate derivatives and generate a large amount of meta-data, which solves the problem of sparse noisy data. Next, genetic algorithm is utilized to discover the form of PDEs and corresponding coefficients with an incomplete candidate library. A two-step adaptive method is introduced to discover parametric PDEs with spatially- or temporally-varying coefficients.
  arXiv  Detail & Related papers  (2020-05-16T09:09:57Z)
• A Data-Driven Approach for Discovering Stochastic Dynamical Systems with Non-Gaussian Levy Noise [5.17900889163564]
  We develop a new data-driven approach to extract governing laws from noisy data sets. First, we establish a feasible theoretical framework, by expressing the drift coefficient, diffusion coefficient and jump measure. We then design a numerical algorithm to compute the drift, diffusion coefficient and jump measure, and thus extract a governing equation with Gaussian and non-Gaussian noise.
  arXiv  Detail & Related papers  (2020-05-07T21:29:17Z)
• Enhancement of shock-capturing methods via machine learning [0.0]
  We develop an improved finite-volume method for simulating PDEs with discontinuous solutions. We train a neural network to improve the results of a fifth-order WENO method. We find that our method outperforms WENO in simulations where the numerical solution becomes overly diffused.
  arXiv  Detail & Related papers  (2020-02-06T21:51:39Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.