
Linking Machine Learning with Multiscale Numerics: DataDriven Discovery of Homogenized Equations
The datadriven discovery of partial differential equations (PDEs) consi...
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Coarsegrained and emergent distributed parameter systems from data
We explore the derivation of distributed parameter system evolution laws...
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DataDriven Discovery of CoarseGrained Equations
A general method for learning probability density function (PDF) equatio...
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Emergent spaces for coupled oscillators
In this paper we present a systematic, datadriven approach to discoveri...
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Coarsescale PDEs from finescale observations via machine learning
Complex spatiotemporal dynamics of physicochemical processes are often m...
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A Numerical Method for the Parametrization of Stable and Unstable Manifolds of Microscopic Simulators
We address a numerical methodology for the computation of coarsegrained...
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Learning emergent PDEs in a learned emergent space
We extract datadriven, intrinsic spatial coordinates from observations ...
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Particles to Partial Differential Equations Parsimoniously
Equations governing physicochemical processes are usually known at microscopic spatial scales, yet one suspects that there exist equations, e.g. in the form of Partial Differential Equations (PDEs), that can explain the system evolution at much coarser, meso or macroscopic length scales. Discovering those coarsegrained effective PDEs can lead to considerable savings in computationintensive tasks like prediction or control. We propose a framework combining artificial neural networks with multiscale computation, in the form of equationfree numerics, for efficient discovery of such macroscale PDEs directly from microscopic simulations. Gathering sufficient microscopic data for training neural networks can be computationally prohibitive; equationfree numerics enable a more parsimonious collection of training data by only operating in a sparse subset of the spacetime domain. We also propose using a datadriven approach, based on manifold learning and unnormalized optimal transport of distributions, to identify macroscale dependent variable(s) suitable for the datadriven discovery of said PDEs. This approach can corroborate physically motivated candidate variables, or introduce new datadriven variables, in terms of which the coarsegrained effective PDE can be formulated. We illustrate our approach by extracting coarsegrained evolution equations from particlebased simulations with a priori unknown macroscale variable(s), while significantly reducing the requisite data collection computational effort.
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