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Discovery of interaction and diffusion kernels in particle-to-mean-field multi-agent systems

Giacomo Albi, Alessandro Alla, Elisa Calzola

Abstract

We propose a data-driven framework to learn interaction kernels in stochastic multi-agent systems. Our approach aims at identifying the functional form of nonlocal interaction and diffusion terms directly from trajectory data, without any a priori knowledge of the underlying interaction structure. Starting from a discrete stochastic binary-interaction model, we formulate the inverse problem as a sequence of sparse regression tasks in structured finite-dimensional spaces spanned by compactly supported basis functions, such as piecewise linear polynomials. In particular, we assume that pairwise interactions between agents are not directly observed and that only limited trajectory data are available. To address these challenges, we propose two complementary identification strategies. The first based on random-batch sampling, which compensates for latent interactions while preserving the statistical structure of the full dynamics in expectation. The second based on a mean-field approximation, where the empirical particle density reconstructed from the data defines a continuous nonlocal regression problem. Numerical experiments demonstrate the effectiveness and robustness of the proposed framework, showing accurate reconstruction of both interaction and diffusion kernels even from partially observed. The method is validated on benchmark models, including bounded-confidence and attraction-repulsion dynamics, where the two proposed strategies achieve comparable levels of accuracy.

Discovery of interaction and diffusion kernels in particle-to-mean-field multi-agent systems

Abstract

We propose a data-driven framework to learn interaction kernels in stochastic multi-agent systems. Our approach aims at identifying the functional form of nonlocal interaction and diffusion terms directly from trajectory data, without any a priori knowledge of the underlying interaction structure. Starting from a discrete stochastic binary-interaction model, we formulate the inverse problem as a sequence of sparse regression tasks in structured finite-dimensional spaces spanned by compactly supported basis functions, such as piecewise linear polynomials. In particular, we assume that pairwise interactions between agents are not directly observed and that only limited trajectory data are available. To address these challenges, we propose two complementary identification strategies. The first based on random-batch sampling, which compensates for latent interactions while preserving the statistical structure of the full dynamics in expectation. The second based on a mean-field approximation, where the empirical particle density reconstructed from the data defines a continuous nonlocal regression problem. Numerical experiments demonstrate the effectiveness and robustness of the proposed framework, showing accurate reconstruction of both interaction and diffusion kernels even from partially observed. The method is validated on benchmark models, including bounded-confidence and attraction-repulsion dynamics, where the two proposed strategies achieve comparable levels of accuracy.
Paper Structure (24 sections, 1 theorem, 84 equations, 13 figures, 6 tables, 2 algorithms)

This paper contains 24 sections, 1 theorem, 84 equations, 13 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Model hierarchy for pairwise interacting agent systems
  3. Mean-field dynamics via random-batch approximation
  4. Kinetic description via binary interactions
  5. Quasi-invariant interaction limit and Fokker--Planck equation
  6. Discovery of drift and diffusion interaction kernels
  7. Kernel estimations with known interaction matrices
  8. Test 1: numerical validation for known interaction matrices
  9. Random-batch kernel estimation with unknown interaction pairs
  10. Robust kernel estimation for unknown interaction pairs
  11. Mean-field kernel reconstruction with unknown interaction pairs
  12. A priori error estimates for the reconstructed trajectories
  13. Numerical experiments
  14. One dimensional tests
  15. Test 1: Bounded Confidence model
  16. ...and 9 more sections

Key Result

Theorem 4.1

Suppose that $\Delta t \leq 1$ and that the trajectories of eq:og_dynamics--eq:reconstructed_dynamics are generated using the same Brownian increments, namely $\Xi^n\equiv \widetilde{\Xi}^n$. Assume that $X^n,\widetilde{X}^n, \bar{X}^n \in [-L,L]^{N_d}$ for all $n=0,1,\ldots,M$ with $N_d:=dN$. In ad and $\eta_{S}:=\max_{0\leq n \leq M-1}\|\mathsf{S}^n-\hat{\mathsf{S}^n}\|_2\leq 2$, the following e

Figures (13)

  • Figure 1: Case of domain $[0,2]$, example of $N_b^P=5$ piecewise linear basis functions on a uniform mesh (left) and on a Chebyshev mesh (right).
  • Figure 2: Test 1: Reconstructed kernel $P(r) = (1 + r^2)^{-2}$ (left) using $N^P_b=10$ basis functions, reconstructed radial density $D(r) = 0.25/(1+r)^2$ using $N^D_p=8$ basis functions (center), comparison between the data density $f$ and the reconstructed $\widehat{f}$ at the final time (right).
  • Figure 3: Test 1: Data used for the reconstruction of the kernel when $\Delta = 4$ and $M_P=25$ (left) and data used for the reconstruction of the diffusion with $\Delta = 1$ and $M_D=10$.
  • Figure 4: Test 1. Reconstruction of the kernel $P=\chi(r<0.5)$ using $21$ basis functions, $\Delta = 2$ and $M_P=50$ using approach \ref{['eq:weights1']} (averaging) and \ref{['eq:weights2']} (best) in Alg. \ref{['alg:1']} and also the mean field approach described in Alg. \ref{['alg:2']}. (left), reconstruction of the density function $D(x)$ using $15$ basis functions $\Delta=1$ and $M_D = 10$ (center), comparison between the data and the reconstructed density at the final time (right).
  • Figure 5: Test 2. Data used for the reconstruction of the kernel when $\ell = 4$ and $M_P=25$ (left) and data used for the reconstruction of the diffusion with $\ell = 1$ and $M_D=10$.
  • ...and 8 more figures

Theorems & Definitions (4)

  • Remark 1
  • Remark 2
  • Theorem 4.1
  • Remark 3