
Particles to Partial Differential Equations Parsimoniously
Equations governing physicochemical processes are usually known at micr...
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Particlebased simulations of reactiondiffusion processes with Aboria
Mathematical models of transport and reactions in biological systems hav...
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Learning emergent PDEs in a learned emergent space
We extract datadriven, intrinsic spatial coordinates from observations ...
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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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Stability selection enables robust learning of partial differential equations from limited noisy data
We present a statistical learning framework for robust identification of...
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Swarm Intelligence for Morphogenetic Engineering
We argue that embryological morphogenesis provides a model of how massiv...
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Coarsegrained and emergent distributed parameter systems from data
We explore the derivation of distributed parameter system evolution laws (and in particular, partial differential operators and associated partial differential equations, PDEs) from spatiotemporal data. This is, of course, a classical identification problem; our focus here is on the use of manifold learning techniques (and, in particular, variations of Diffusion Maps) in conjunction with neural network learning algorithms that allow us to attempt this task when the dependent variables, and even the independent variables of the PDE are not known a priori and must be themselves derived from the data. The similarity measure used in Diffusion Maps for dependent coarse variable detection involves distances between local particle distribution observations; for independent variable detection we use distances between local shorttime dynamics. We demonstrate each approach through an illustrative established PDE example. Such variablefree, emergent space identification algorithms connect naturally with equationfree multiscale computation tools.
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