论文标题

数据驱动的粗粒方程发现

Data-Driven Discovery of Coarse-Grained Equations

论文作者

Bakarji, Joseph, Tartakovsky, Daniel M.

论文摘要

方程发现的统计(机器学习)工具需要大量的数据,这些数据通常是计算机生成的,而不是实验观察到的。多尺度建模和随机模拟是两个领域,在模拟数据上学习可以导致这种发现。在这两种情况下,数据都具有可靠但不切实际的模型,例如分子动力学模拟,而感兴趣尺度的模型却不确定,需要现象学本质上的关系和临时近似。我们取代了这种模型的人类发现,这些模型通常涉及空间/随机平均或粗粒,并基于基于稀疏回归的机器学习策略,可以以两种模式执行。第一个直接方程学习是从整个词典中发现的差分运算符。第二个受约束的方程学习,仅发现需要发现的差分运算符中的那些术语,即学习闭合近似值。我们通过学习一个确定性方程来说明我们的方法,该方程控制了系统状态的概率密度函数的时空演化,其动力学由具有随机输入的非线性偏微分方程描述。一系列示例证明了我们方程发现方法的准确性,鲁棒性和局限性。

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.

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