Data-Driven Discovery of Coarse-Grained Equations

Abstract

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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• Nonlinear stochastic modeling with Langevin regression
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• A posteriori learning for quasi-geostrophic turbulence parametrization
• Extended Dynamic Mode Decomposition for Inhomogeneous Problems
• Explicit physics-informed neural networks for non-linear upscaling closure: the case of transport in tissues
• Differentiable Physics: A Position Piece
• Information-geometry of physics-informed statistical manifolds and its use in data assimilation
• Autonomous learning of nonlocal stochastic neuron dynamics
• Statistical Mechanics of Dynamical System Identification
• Robust discovery of partial differential equations in complex situations