Phys. Rev. Research 3, 043074 (2021) - Inferring microscale properties of interacting systems from macroscale observations

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Inferring microscale properties of interacting systems from macroscale observations

Nazareno Campioni, Dirk Husmeier, Juan Morales, Jennifer Gaskell, and Colin J. Torney

Phys. Rev. Research 3, 043074 – Published 26 October 2021

Abstract

Emergent dynamics of complex systems are observed throughout nature and society. The coordinated motion of bird flocks, the spread of opinions, fashions and fads, or the dynamics of an epidemic, are all examples of complex macroscale phenomena that arise from fine-scale interactions at the individual level. In many scenarios, observations of the system can only be made at the macroscale, while we are interested in creating and fitting models of the microscale dynamics. This creates a challenge for inference as a formal mathematical link between the microscale and macroscale is rarely available. Here, we develop an inferential framework that bypasses the need for a formal link between scales and instead uses sparse Gaussian process regression to learn the drift and diffusion terms of an empirical Fokker-Planck equation, which describes the time evolution of the probability density of a macroscale variable. This gives us access to the likelihood of the microscale properties of the physical system and a second Gaussian process is then used to emulate the log-likelihood surface, allowing us to define a fast, adaptive MCMC sampler, which iteratively refines the emulator when needed. We illustrate the performance of our method by applying it to a simple model of collective motion.

Received 1 September 2020
Accepted 25 June 2021

DOI:https://doi.org/10.1103/PhysRevResearch.3.043074

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Diffusion Patterns in complex systems Random walks Stochastic inference Stochastic processes

Self-propelled particles

Coarse graining Fokker–Planck equation Markovian processes Metropolis algorithm Monte Carlo methods

Interdisciplinary PhysicsStatistical Physics & Thermodynamics

Authors & Affiliations

Nazareno Campioni ¹, Dirk Husmeier ¹, Juan Morales², Jennifer Gaskell¹, and Colin J. Torney ¹

¹School of Mathematics & Statistics, University of Glasgow, University Avenue, Glasgow, G12 8QQ, United Kingdom
²Grupo de Ecología Cuantitativa, INIBIOMA-CONICET, Universidad Nacional del Comahue, Quintral 1250, CP: 8400, San Carlos de Bariloche, Río Negro, Argentina

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Vol. 3, Iss. 4 — October - December 2021

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Statistical Physics

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Figure 1
Graphical summary of the proposed inference scheme. If the microstates, corresponding to the particles' positions $x_{i} (t)$ and velocities $u_{i} (t)$ in Eq. (1) were observable, we could infer the parameters of the physical model directly from the corresponding microstate data (arrows 1a and 1b). However, such high-resolution data is usually not available, and the challenge therefore is to infer the physical model parameters from macroscale features—the distribution of average velocities $U$ in our case (arrows 2a and 2b). This distribution is in principle defined by the physical model and its parameters via Eq. (7), giving rise to the likelihood of the physical model given the macroscale data [Eq. (8)]. However, the mathematical expression of this physical model likelihood depends on two functions—the diffusion function D and the drift function F—which are not analytically tractable. We therefore approximate these functions by two Gaussian process models fitted to simulated macroscale output, based on Eqs. ((12, 13, 14)) (arrows 3–5). Note that the inference of these Gaussian processes is based on the probability of the simulation output given the Gaussian process, which is independent of the physical model. Inserting the Gaussian process approximations of F and D back into Eq. (7) then leads to an approximation of the physical model likelihood [Eq. (8)], which is used for inference of the physical model parameters. Note that in order to make the inference computationally efficient, the physical model likelihood is approximated (or emulated) by another Gaussian processes, which is not included in the present figure.
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Figure 2
(a) Sample time series from the simulation model for 20 individuals moving along a line of dimensionless length 36 with parameters $v_{0} = 1$ and $Δ t = 1$ . The switches between the two metastable states $U = 1 and U = - 1$ represent cohesive movement in clockwise or counterclockwise direction. (b) Inferred diffusion function $D (U)$ (dark blue line) and $95 %$ posterior credible interval (light blue shaded region) learnt from simulation results using sparse GP regression, as a function of the average velocity $U$ . (c) Idem for the drift function $F (U)$ . (d) Stationary probability density mean and uncertainty calculated based on 50 000 simulation outputs. (e) Stationary probability density from 500 000 simulation outputs. Note that by increasing the number of simulations the uncertainty has greatly reduced.
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Figure 3
Inference results for $N = 20$ , $α = 0.3$ , and $δ = 2$ . Results shown for 10 000 samples after burn-in of 40 000 (Geweke's diagnostic [45] was used to test convergence: highest absolute z score was 0.73 for $α$ and 0.89 for $δ$ ). (a) Posterior distribution of the weighting given to social cues ( $α$ ). Vertical bar represents the true parameter value. (b) Posterior distribution for the interaction range ( $δ$ ). Vertical bar represents the true parameter value. (c) Refined surrogate log-likelihood; the true parameter value is indicated by the red, full circle. (d) Variance associated with the refined surrogate log-likelihood. Black crosses show the first initial points from our space-filling design; black open circles show every tenth refinement. The more explored regions have lower uncertainty.
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Figure 4
Inference results for $N = 100, α = 0.8, and δ = 1$ . Results shown for 10 000 samples after burn-in of 20 000 (Geweke's diagnostic [45] was used to test convergence: highest absolute z score was 0.43 for $α$ and 0.34 for $δ$ ). (a) Posterior distribution of the weighting given to social cues ( $α$ ). Vertical bar represents the true parameter value. (b) Posterior distribution for the interaction range ( $δ$ ). Vertical bar represents the true parameter value. (c) Refined surrogate log-likelihood; the true parameter value is indicated by the red, full circle. (d) Variance associated with the refined surrogate log-likelihood. Black crosses show the first initial points from our space-filling design; black open circles show every tenth refinement. For larger population size there is lower uncertainty in the simulation output so less refinement is required.
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Figure 5
Inference results for $N = 30, α = 0.6, δ = 1.5$ , and $η = 0.5$ . Results shown for 20 000 samples after burn-in of 30 000 (Geweke's diagnostic [45] was used to test convergence: highest absolute z score was 0.67 for $α$ , 0.59 for $δ$ and 0.39 for $η$ ). (a) Posterior distribution of the weighting given to social cues ( $α$ ). (b) Posterior distribution for the interaction range ( $δ$ ). (c) Posterior distribution for the noise term ( $η)$ . The vertical bars show the true parameter values.
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Physical Review Research

Inferring microscale properties of interacting systems from macroscale observations

Nazareno Campioni, Dirk Husmeier, Juan Morales, Jennifer Gaskell, and Colin J. Torney

Phys. Rev. Research 3, 043074 – Published 26 October 2021

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