Data-Driven Discovery of Coarse-Grained Equations
- URL: http://arxiv.org/abs/2002.00790v5
- Date: Mon, 27 Jul 2020 16:57:59 GMT
- Title: Data-Driven Discovery of Coarse-Grained Equations
- Authors: Joseph Bakarji, Daniel M. Tartakovsky
- Abstract summary: Multiscale modeling and simulations are two areas where learning on simulated data can lead to such discovery.
We replace the human discovery of such models with a machine-learning strategy based on sparse regression that can be executed in two modes.
A series of examples demonstrates the accuracy, robustness, and limitations of our approach to equation discovery.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- 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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