Abstract
Systematic discovery of reduced-order closure models for multi-scale processes remains an important open problem in complex dynamical systems. Even when an effective lower-dimensional representation exists, reduced models are difficult to obtain using solely analytical methods. Rigorous methodologies for finding such coarse-grained representations of multi-scale phenomena would enable accelerated computational simulations and provide fundamental insights into the complex dynamics of interest. We focus on a heterogeneous population of oscillators of Kuramoto type as a canonical model of complex dynamics, and develop a data-driven approach for inferring its coarse-grained description. Our method is based on a numerical optimization of the coefficients in a general equation of motion informed by analytical derivations in the thermodynamic limit. We show that certain assumptions are required to obtain an autonomous coarse-grained equation of motion. However, optimizing coefficient values enables coarse-grained models with conceptually disparate functional forms, yet comparable quality of representation, to provide accurate reduced-order descriptions of the underlying system.
Highlights
Numerical simulations of complex multiscale phenomena are fundamental to modern science, for which commonly sought goals involve the development of tractable yet accurate reduced-order models
We evaluate the performance of the above-described coarse-grained models by applying them to a range of Kuramoto oscillator systems for which we have fine-grained trajectories
We demonstrate the necessity of both Conditions 1 and 2 by showing that coarse graining in a way that satisfies only one or the other Condition gives a worse model than coarse-graining in a way that satisfies both
Summary
Numerical simulations of complex multiscale phenomena are fundamental to modern science, for which commonly sought goals involve the development of tractable yet accurate reduced-order models. In turbulence modeling this is known as the closure problem for the Reynolds-averaged Navier-Stokes equation (RANS) [1,2,3,4] In molecular dynamics it is known as coarse graining [5,6,7]. Perhaps the best-studied case of the Kuramoto model is that of mean-field coupling, where Ki j = K/N for all i, j In this case it is well known that in the limit N → ∞, the system (1) exhibits a phase transition with respect to K: if ωi are sampled from a symmetric, unimodal distribution, there exists Kc such that for K < Kc, the oscillators behave mostly independently, while for K > Kc a subset of oscillators spontaneously locks to a single frequency [36]
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