Learning microstructure in active matter

TL;DR

This work tackles the nonequilibrium challenge of predicting microstructure in active matter by combining particle-resolved simulations, a deep neural network surrogate, and symbolic regression to derive closed-form expressions for the radial and anisotropic pair-correlation functions. By training on a wide range of packing fractions $\varphi$ and activities $\mathrm{Pe}$, the authors produce compact analytical formulas for $g(r)$ and $g(r,\theta)$ that accurately reproduce simulation data, including near-contact peaks and activity-induced anisotropy. These closed-form expressions offer direct input to nonequilibrium continuum theories and enable efficient design of pattern formation, confinement, and external-potential scenarios beyond low-density limits. Overall, the approach provides a data-driven pathway to structure-based theory in active and passive systems, with broad applicability and potential for inverse design and dynamical-property prediction from structure.

Abstract

Understanding microstructure in terms of closed-form expressions is an open challenge in nonequilibrium statistical physics. We propose a simple and generic method that combines particle-resolved simulations, deep neural networks and symbolic regression to predict the pair-correlation function of passive and active particles. Our analytical closed-form results closely agree with Brownian dynamics simulations, even at relatively large packing fractions and for strong activity. The proposed method is broadly applicable, computationally efficient, and can be used to enhance the predictive power of nonequilibrium continuum theories and for designing pattern formation.

Figures (11)

• Figure 1: Schematic illustration of the proposed method. (a) Brownian dynamics simulations of active Brownian particles as a function of time, packing fraction $\varphi$, and Péclet number $\mathrm{Pe}$. (b) From these snapshots, we compute the radial distribution function, either isotropic $\mathrm{g}(r)$ (passive or angle-averaged) or fully anisotropic $\mathrm{g}(r, \theta)$ (active). (c) A deep neural network learns the mapping $(r, \theta, \varphi, \mathrm{Pe}) \mapsto \mathrm{g}(r, \theta)$, providing a smooth, differentiable surrogate for the simulation-measured microstructure. (d) Symbolic regression converts the learned surrogate into compact, closed-form analytical expressions. (e) These analytical formulas accurately reproduce near-contact peaks, coordination-shell oscillations, and activity-induced microstructure, offering ready-to-use structural input for nonequilibrium theory.
• Figure 2: Equilibrium microstructure learned from data. Radial distribution function $\mathrm{g}(r)$ of passive Brownian particles at two non-trained packing fractions, $\varphi=0.23$ and $\varphi=0.43$. Symbols represent Brownian dynamics simulation data, the blue dashed line shows predictions from the trained deep neural network (learned), and the black solid line corresponds to the Percus–Yevick (PY) reference solution. Red solid lines represent analytical predictions from Eq. (\ref{['eq:pbp_symbolic']}).
• Figure 3: Activity-induced angle-averaged microstructure. Validation of analytical predictions from Eq. (\ref{['eq:abp_symbolic']}) for various Péclet numbers at a fixed area fraction $\varphi=0.45$. Solid lines represent analytical predictions, dashed lines represent learned results from the trained deep neural network, and symbols denote simulation data.
• Figure 4: Anisotropic microstructure of active Brownian particles. Angle-resolved pair correlation function $\mathrm{g}(r,\theta)$ of active Brownian particles (ABPs) obtained from Brownian-dynamics simulations [(a), (d)] for $(\varphi,\mathrm{Pe})=(0.45,15)$ and $(\varphi,\mathrm{Pe})=(0.30,35)$, respectively, and from deep neural network (learned) predictions [(c),(f)] for the same state points. Both simulations and learned predictions reveal a pronounced anisotropic microstructure, characterized by particle accumulation in front of a reference active particle ($\theta=180^\circ$) and depletion in its wake ($\theta=0^\circ$). The central panels [(b), (e)] show radial cuts along the propulsion direction ($\theta=180^\circ$), demonstrating quantitative agreement between simulation data, learned predictions, and the analytical predictions obtained from Eq. (\ref{['eq:abp_angle_symbolic']}) over the full radial range.
• Figure S1: Loss curves in curriculum learning: Training and testing (inset) loss plotted against epoch number for (a) low ($\mathrm{Pe} \le 25$), (b) intermediate ($25 < \mathrm{Pe} \le 35$), and (c) high activity ($\mathrm{Pe} > 35$).