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Learning Collective Behaviors from Observation

Jinchao Feng, Ming Zhong

TL;DR

This work presents a comprehensive variational framework for learning the governing equations of collective dynamics from observations, exploiting the fact that the right-hand side can be written as a low-dimensional interaction kernel applied to high-dimensional agent states. By formulating a loss that couples observed trajectories with a parametric or nonparametric representation of the interaction kernel, the authors achieve dimension reduction, provide convergence guarantees, and extend the approach to first- and second-order dynamics, heterogeneous agents, stochastic noise, geometric constraints, feature map learning, coupled systems, and Gaussian priors. Key contributions include a coercivity-based convergence theory, a scalable linear-algebraic learning formulation, and several extensions that accommodate practical complexities in collective behaviors. The framework supports uncertainty quantification via Gaussian processes and enables joint inference of interaction kernels and auxiliary parameters, offering a principled path from data to predictive, interpretable models of self-organization with broad applicability in biology, physics, and social dynamics.

Abstract

We present a comprehensive examination of learning methodologies employed for the structural identification of dynamical systems. These techniques are designed to elucidate emergent phenomena within intricate systems of interacting agents. Our approach not only ensures theoretical convergence guarantees but also exhibits computational efficiency when handling high-dimensional observational data. The methods adeptly reconstruct both first- and second-order dynamical systems, accommodating observation and stochastic noise, intricate interaction rules, absent interaction features, and real-world observations in agent systems. The foundational aspect of our learning methodologies resides in the formulation of tailored loss functions using the variational inverse problem approach, inherently equipping our methods with dimension reduction capabilities.

Learning Collective Behaviors from Observation

TL;DR

This work presents a comprehensive variational framework for learning the governing equations of collective dynamics from observations, exploiting the fact that the right-hand side can be written as a low-dimensional interaction kernel applied to high-dimensional agent states. By formulating a loss that couples observed trajectories with a parametric or nonparametric representation of the interaction kernel, the authors achieve dimension reduction, provide convergence guarantees, and extend the approach to first- and second-order dynamics, heterogeneous agents, stochastic noise, geometric constraints, feature map learning, coupled systems, and Gaussian priors. Key contributions include a coercivity-based convergence theory, a scalable linear-algebraic learning formulation, and several extensions that accommodate practical complexities in collective behaviors. The framework supports uncertainty quantification via Gaussian processes and enables joint inference of interaction kernels and auxiliary parameters, offering a principled path from data to predictive, interpretable models of self-organization with broad applicability in biology, physics, and social dynamics.

Abstract

We present a comprehensive examination of learning methodologies employed for the structural identification of dynamical systems. These techniques are designed to elucidate emergent phenomena within intricate systems of interacting agents. Our approach not only ensures theoretical convergence guarantees but also exhibits computational efficiency when handling high-dimensional observational data. The methods adeptly reconstruct both first- and second-order dynamical systems, accommodating observation and stochastic noise, intricate interaction rules, absent interaction features, and real-world observations in agent systems. The foundational aspect of our learning methodologies resides in the formulation of tailored loss functions using the variational inverse problem approach, inherently equipping our methods with dimension reduction capabilities.
Paper Structure (12 sections, 2 theorems, 75 equations, 4 figures, 1 algorithm)

This paper contains 12 sections, 2 theorems, 75 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model Equation
  3. Learning Framework
  4. First Order
  5. Other System Identification Methods
  6. Heterogeneous Agents
  7. Stochastic Noise
  8. Riemannian Geometry Constraints
  9. Feature Map Learning
  10. Coupled Systems
  11. Learning with Gaussian Priors
  12. Conclusion

Key Result

theorem 1

Assume that $\phi \in \mathcal{K}_{R, S}$ (an admissible set $\mathcal{K}_{R, S} = \{\phi \in C^1(\mathbb{R}_+): \text{supp}(\phi) \subset [0, R], \sup_{r \in [0, R]}|\phi(r)| + |\phi'(r)| \le S\}$ for some $R, S > 0$). Let $\{\mathcal{H}_n\}_n$ be a sequence of subspaces of $L^{\infty}([0, R])$, wi

Figures (4)

  • Figure 1: Opinion Dynamics introduced in Krause2000SIMT2014, with $N = 20$ agents and $\phi(r) = \chi_{[0, \frac{1}{\sqrt{2}})}(r) + 0.1*\chi_{[\frac{1}{\sqrt{2}}, 1]}(r)$, learned on $[0, 10]$, for other parameters see lu2019nonparametric.
  • Figure 2: Predator-Preys Dynamics introduced in CK2013, with $1: \text{prey}$, $2: \text{predator}$, $N_{1} = 19$ preys, $N_2 = 1$ predator, learned on $[0, 5]$, for other parameters see lu2019nonparametric.
  • Figure 3: Power Law Dynamics introduced in KSUB2011, with $N = 3$ agents (learned from an updated algorithm), for other parameters see FENG2022162.
  • Figure 4: Flocking with external potential (FwEP) model shu2020flocking with $\{N,L,M,\sigma\} = \{20,6,3,0.01\}$, and $\phi^E(r) = 1$, $\phi^A(r) = \frac{1}{(1+r^2)^{1/2}}$, for other parameters see miller2023learning. The light blue regions are two-standard-deviation bands around the means which indicate the uncertainty of the estimators.

Theorems & Definitions (8)

  • remark 1
  • remark 2
  • remark 3
  • definition 1: Definition $3.1$ in lu2019nonparametric
  • theorem 1: Theorem $3.1$ in lu2019nonparametric
  • remark 4
  • theorem 2: Theorem 3.2 in feng2023data
  • definition 2: Definition 3.3 in feng2023data