Scientific machine learning has been successfully applied to inverse problems and PDE discoveries in computational physics. One caveat of current methods however is the need for large amounts of (clean) data in order to recover full system responses or underlying physical models. Bayesian methods may be particularly promising to overcome these challenges as they are naturally less sensitive to sparse and noisy data. In this paper, we propose to use Bayesian neural networks (BNN) in order to: 1) Recover the full system states from measurement data (e.g. temperature, velocity field, etc.). We use Hamiltonian Monte-Carlo to sample the posterior distribution of a deep and dense BNN, and show that it is possible to accurately capture physics of varying complexity without overfitting. 2) Recover the parameters in the underlying partial differential equation (PDE) governing the physical system. Using the trained BNN as a surrogate of the system response, we generate datasets of derivatives potentially comprising the latent PDE of the observed system and perform a Bayesian linear regression (BLR) between the successive derivatives in space and time to recover the original PDE parameters. We take advantage of the confidence intervals on the BNN outputs and introduce the spatial derivative variance into the BLR likelihood to discard the influence of highly uncertain surrogate data points, which allows for more accurate parameter discovery. We demonstrate our approach on a handful of example applied to physics and non-linear dynamics.
翻译:在计算物理学中,对反问题和PDE发现成功地应用了科学机器学习,但目前方法的一个告诫是,需要大量(清洁)数据,以恢复整个系统的反应或基本物理模型。贝叶斯方法可能特别有希望应对这些挑战,因为它们自然对稀疏和繁琐的数据不那么敏感。在本文件中,我们提议利用贝叶西亚神经网络(BNN)来:1) 从测量数据(例如温度、速度场等)中恢复整个系统状态。我们利用汉密尔顿·蒙特-卡洛对深厚的BNNN的后方分布进行取样,并表明有可能准确地捕捉到复杂程度不同的物理学,而不会过度配置。 2)重新利用基本部分差异方程式(PDE)中管辖物理系统的参数。我们利用经过培训的BNNN网络作为系统反应的替代工具,产生衍生物数据集,这些衍生物可能包括观测到的潜伏的PDE,并进行巴伊西亚线回归。我们利用空间连续衍生物和时间来恢复最初的PDE参数,并表明可以准确捕捉到BDR的精确度,我们利用BMR的精确度结果,从而展示了我们对BMR的精确度的概率的概率。