Bayesian autoencoders for data-driven discovery of coordinates, governing equations and fundamental constants

IF 2.9 3区 综合性期刊 Q1 MULTIDISCIPLINARY SCIENCES Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences Pub Date : 2024-03-20 DOI:10.1098/rspa.2023.0506
L. Mars Gao, J. Nathan Kutz
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Abstract

Recent progress in autoencoder-based sparse identification of nonlinear dynamics (SINDy) under 1 constraints allows joint discoveries of governing equations and latent coordinate systems from spatio-temporal data, including simulated video frames. However, it is challenging for 1-based sparse inference to perform correct identification for real data due to the noisy measurements and often limited sample sizes. To address the data-driven discovery of physics in the low-data and high-noise regimes, we propose Bayesian SINDy autoencoders, which incorporate a hierarchical Bayesian Spike-and-slab Gaussian Lasso prior. Bayesian SINDy autoencoder enables the joint discovery of governing equations and coordinate systems with uncertainty estimate. To resolve the challenging computational tractability of the Bayesian hierarchical setting, we adapt an adaptive empirical Bayesian method with Stochastic Gradient Langevin Dynamics (SGLD) which gives a computationally tractable way of Bayesian posterior sampling within our framework. Bayesian SINDy autoencoder achieves better physics discovery with lower data and fewer training epochs, along with valid uncertainty quantification suggested by the experimental studies. The Bayesian SINDy autoencoder can be applied to real video data, withaccurate physics discovery which correctly identifies the governing equation and provides a close estimate for standard physics constants like gravity g, for example, in videos of a pendulum.

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用于数据驱动的坐标、控制方程和基本常数发现的贝叶斯自动编码器
在 ℓ1 约束条件下,基于自动编码器的非线性动力学稀疏识别(SINDy)技术取得了最新进展,可从时空数据(包括模拟视频帧)中联合发现控制方程和潜在坐标系。然而,由于噪声测量和有限的样本量,基于 ℓ1 的稀疏推理很难对真实数据进行正确识别。为了解决低数据和高噪声环境下的数据驱动物理学发现问题,我们提出了贝叶斯 SINDy 自编码器,其中包含了分层贝叶斯 Spike-and-slab Gaussian Lasso 先验。贝叶斯 SINDy 自动编码器能够联合发现具有不确定性估计的控制方程和坐标系。为了解决贝叶斯分层设置的计算可操作性难题,我们采用了随机梯度朗格文动力学(SGLD)的自适应经验贝叶斯方法,在我们的框架内提供了一种计算可操作性强的贝叶斯后验采样方法。贝叶斯 SINDy 自动编码器以较少的数据和较少的训练历时实现了更好的物理发现,并根据实验研究的建议进行了有效的不确定性量化。贝叶斯 SINDy 自动编码器可应用于真实视频数据,并能准确地发现物理现象,例如,在钟摆视频中,它能正确识别控制方程,并为重力 g 等标准物理常数提供接近的估计值。
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来源期刊
CiteScore
6.40
自引率
5.70%
发文量
227
审稿时长
3.0 months
期刊介绍: Proceedings A has an illustrious history of publishing pioneering and influential research articles across the entire range of the physical and mathematical sciences. These have included Maxwell"s electromagnetic theory, the Braggs" first account of X-ray crystallography, Dirac"s relativistic theory of the electron, and Watson and Crick"s detailed description of the structure of DNA.
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