
Neural TimeDependent Partial Differential Equation
Partial differential equations (PDEs) play a crucial role in studying a ...
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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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Particles to Partial Differential Equations Parsimoniously
Equations governing physicochemical processes are usually known at micr...
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PDENet 2.0: Learning PDEs from Data with A NumericSymbolic Hybrid Deep Network
Partial differential equations (PDEs) are commonly derived based on empi...
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Deep Learning Models for Global Coordinate Transformations that Linearize PDEs
We develop a deep autoencoder architecture that can be used to find a co...
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Automated design of collective variables using supervised machine learning
Selection of appropriate collective variables for enhancing sampling of ...
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Decision functions from supervised machine learning algorithms as collective variables for accelerating molecular simulations
Selection of appropriate collective variables for enhancing molecular si...
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Learning emergent PDEs in a learned emergent space
We extract datadriven, intrinsic spatial coordinates from observations of the dynamics of large systems of coupled heterogeneous agents. These coordinates then serve as an emergent space in which to learn predictive models in the form of partial differential equations (PDEs) for the collective description of the coupledagent system. They play the role of the independent spatial variables in this PDE (as opposed to the dependent, possibly also datadriven, state variables). This leads to an alternative description of the dynamics, local in these emergent coordinates, thus facilitating an alternative modeling path for complex coupledagent systems. We illustrate this approach on a system where each agent is a limit cycle oscillator (a socalled StuartLandau oscillator); the agents are heterogeneous (they each have a different intrinsic frequency ω) and are coupled through the ensemble average of their respective variables. After fast initial transients, we show that the collective dynamics on a slow manifold can be approximated through a learned model based on local "spatial" partial derivatives in the emergent coordinates. The model is then used for prediction in time, as well as to capture collective bifurcations when system parameters vary. The proposed approach thus integrates the automatic, datadriven extraction of emergent space coordinates parametrizing the agent dynamics, with machinelearning assisted identification of an "emergent PDE" description of the dynamics in this parametrization.
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