论文标题
贝叶斯隐藏物理模型：从数据中发现非线性部分差分运算符的不确定性量化
Bayesian Hidden Physics Models: Uncertainty Quantification for Discovery of Nonlinear Partial Differential Operators from Data
论文作者
论文摘要
数据告诉我们有关物理学以及他们不告诉我们什么的数据？人们对使用机器学习模型发现有关物理定律（例如数据的微分方程）的兴趣激增，但是当前的方法缺乏不确定性量化来传达其信誉。这项工作从贝叶斯的角度解决了这一缺点。我们引入了一个包括“叶”模块的新型模型，该模型学会了将不同的实验作为神经网络和单个“ root”模块表示，该模块在其管理非线性差异操作员上表示非参数分布作为高斯工艺。自动分化用于计算从叶子函数的所需部分衍生物作为根的输入。我们的方法量化了学习物理学的可靠性，该物理学的后验分布在操作员上，并传播了新型初始有限价值问题实例解决方案的不确定性。数值实验证明了几个非线性PDE的方法。
What do data tell us about physics-and what don't they tell us? There has been a surge of interest in using machine learning models to discover governing physical laws such as differential equations from data, but current methods lack uncertainty quantification to communicate their credibility. This work addresses this shortcoming from a Bayesian perspective. We introduce a novel model comprising "leaf" modules that learn to represent distinct experiments' spatiotemporal functional data as neural networks and a single "root" module that expresses a nonparametric distribution over their governing nonlinear differential operator as a Gaussian process. Automatic differentiation is used to compute the required partial derivatives from the leaf functions as inputs to the root. Our approach quantifies the reliability of the learned physics in terms of a posterior distribution over operators and propagates this uncertainty to solutions of novel initial-boundary value problem instances. Numerical experiments demonstrate the method on several nonlinear PDEs.
