Recent progress in autoencoder-based sparse identification of nonlinear dynamics (SINDy) under $\ell_1$ constraints allows joint discoveries of governing equations and latent coordinate systems from spatio-temporal data, including simulated video frames. However, it is challenging for $\ell_1$-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 sparsifying prior: Spike-and-slab Gaussian Lasso. Bayesian SINDy autoencoder enables the joint discovery of governing equations and coordinate systems with a theoretically guaranteed uncertainty estimate. To resolve the challenging computational tractability of the Bayesian hierarchical setting, we adapt an adaptive empirical Bayesian method with Stochatic 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, with accurate 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.
翻译:以自动编码器为基础的非线性动态(SINDy)的稀疏识别(SINDy)最近取得进展,在美元1美元的制约下,可以通过spatio-时空数据,包括模拟视频框架,联合发现治理方程式和潜在协调系统。然而,对于基于1美元的基于自动编码器的稀释推论,由于测量噪音和抽样规模往往有限,对真实数据进行正确识别,这是具有挑战性的。为了应对低数据和高音制度下的数据驱动物理发现,我们建议Bayesian SINDy自动计算器,其中包含一种对先前的Bayesian精确度进行分级测量的:Spick-slab Gaussian Lasso。Bayesian SINDy自动编码使管理方程式的联合发现并协调系统与理论上保证的不确定性估计。为了解决Bayesian等级设置的具有挑战性的计算性可调控性,我们采用适应性的经验性巴耶斯方法,使Bayesian 后值的精确度测算法方法在我们的框架中可以进行精确的精确的测算方法。Bayesian labsbsbsbs Vedia decal disal disal disal dal disal dreal disal dreal disal disal disal dismisal 和建议, 和Sined disald disald disald disald disald disald dism disal 和Sined disald disald disald disald disald disald disald disald 提供一种精确的精确的精确的精确的精确的Sdiadaldaldaldald disald disal 和Smd 和Smd disald disaldaldaldaldaldaldaldaldaldaldaldaldaldald disaldaldald 和制数据学数据学数据研究, 和S。