Statistical (machine learning) tools for equation discovery require large amounts of data that are typically computer generated rather than experimentally observed. Multiscale modeling and stochastic simulations are two areas where learning on simulated data can lead to such discovery. In both, the data are generated with a reliable but impractical model, e.g., molecular dynamics simulations, while a model on the scale of interest is uncertain, requiring phenomenological constitutive relations and ad-hoc approximations. We replace the human discovery of such models, which typically involves spatial/stochastic averaging or coarse-graining, with a machine-learning strategy based on sparse regression that can be executed in two modes. The first, direct equation-learning, discovers a differential operator from the whole dictionary. The second, constrained equation-learning, discovers only those terms in the differential operator that need to be discovered, i.e., learns closure approximations. We illustrate our approach by learning a deterministic equation that governs the spatiotemporal evolution of the probability density function of a system state whose dynamics are described by a nonlinear partial differential equation with random inputs. A series of examples demonstrates the accuracy, robustness, and limitations of our approach to equation discovery.

用于方程式发现的统计（机器学习）工具需要大量数据，这些数据通常是计算机生成而不是通过实验观察到的。多尺度建模和随机模拟是学习模拟数据可以导致这种发现的两个领域。在这两种数据中，均使用可靠但不切实际的模型（例如分子动力学模拟）来生成数据，而感兴趣的尺度上的模型尚不确定，需要现象学本构关系和即席近似。我们用基于稀疏回归的机器学习策略（可以在两种模式下执行）来代替人工发现的模型（通常涉及空间/随机平均或粗粒度）。第一个直接方程式学习，从整个字典中发现微分运算符。第二，约束方程式学习，仅在微分算子中发现需要发现的项，即学习闭合近似。我们通过学习确定性方程来说明我们的方法，该确定性方程控制着系统状态的概率密度函数的时空演化，该系统状态的动力学由具有随机输入的非线性偏微分方程描述。一系列示例演示了我们的方程发现方法的准确性，鲁棒性和局限性。我们通过学习确定性方程式来说明我们的方法，该确定性方程式控制系统状态的概率密度函数的时空演化，该系统状态的动力学由具有随机输入的非线性偏微分方程描述。一系列示例演示了我们的方程发现方法的准确性，鲁棒性和局限性。我们通过学习确定性方程来说明我们的方法，该确定性方程控制着系统状态的概率密度函数的时空演化，该系统状态的动力学由带有随机输入的非线性偏微分方程描述。一系列示例演示了我们的方程发现方法的准确性，鲁棒性和局限性。