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

1. Problem Statement

The paper addresses the inverse problem of learning the governing laws (interaction and diffusion kernels) of stochastic multi-agent systems directly from trajectory data.

• Context: Multi-agent systems (MAS) are used to model phenomena in biology, sociology, and epidemiology. These systems often evolve via stochastic pairwise interactions.
• Challenge: In many practical scenarios, the pairwise interaction structure is latent (unobserved). Researchers only have access to the aggregate trajectories of agents ($X^n$) over time, not the specific pairing matrices ($S^n$) or the noise realizations ($\Xi^n$) that drove the dynamics.
• Goal: To reconstruct the functional forms of the drift kernel $P(r)$ (governing deterministic compromise/attraction) and the diffusion kernel $D(r)$ (governing stochastic fluctuations) without prior knowledge of the interaction topology, using limited and potentially noisy trajectory data.

2. Methodology

The authors propose a data-driven framework that bridges microscopic discrete models and macroscopic mean-field descriptions. The approach involves three main stages:

A. Modeling Hierarchy

The system is modeled starting from a discrete-time stochastic binary-interaction model:
$$x_i^{n+1} = x_i^n + \Delta t P(x_i^n, x_j^n)(x_j^n - x_i^n) + \sqrt{\Delta t} D(x_i^n, x_j^n) \xi_{ij}^n$$
where agents interact pairwise. The paper establishes the connection between this microscopic rule and the macroscopic Fokker-Planck equation (mean-field limit) via random-batch approximations and kinetic theory.

B. Regression Formulation

The inverse problem is formulated as a sparse regression task in finite-dimensional spaces.

• Basis Functions: The unknown kernels $P(r)$ and $D(r)$ are approximated using linear combinations of compactly supported, piecewise linear basis functions (e.g., hat functions).
• Objective: Minimize the discrepancy between observed trajectory increments and the model predictions using a least-squares loss function.
• Constraints: Physical constraints (e.g., non-negativity of diffusion, monotonicity of interaction) are enforced via constrained optimization.

C. Two Identification Strategies for Latent Interactions

Since the pairing matrix $S^n$ is unknown, the authors propose two complementary strategies to estimate the conditional expectations required for the regression:

1. Random-Batch Sampling (Microscopic Approach):

   • Instead of knowing the true partners, the algorithm samples $N_p \ll N$ random pairs at each time step to approximate the interaction terms.
   • Ensemble Learning: To reduce variance, the regression is repeated $K$ times with different random batch realizations.
   • Weighting: The resulting $K$ kernel estimates are combined using two weighting schemes:
     • Averaging Rule: Weights are inversely proportional to the error of the reconstructed trajectory.
     • Best-Fit Rule: Selects the single best-performing realization.
   • This approach mimics the "random batch" method used in particle simulations to reduce computational cost.

2. Mean-Field Approximation (Macroscopic Approach):

   • This strategy bypasses pairwise sampling entirely. It reconstructs the empirical particle density $f(x, t)$ from the trajectory data using a histogram estimator.
   • The interaction terms are approximated as nonlocal integrals over the reconstructed density field.
   • This transforms the problem into a continuous nonlocal regression, leveraging the law of large numbers to smooth out stochastic noise.

D. Theoretical Analysis

The paper provides a priori error estimates (Theorem 4.1) for the reconstructed trajectories. It proves that the mean-square error between the true and reconstructed dynamics is bounded by:

• The error in the kernel approximation ($\delta_P, \delta_D$).
• The error in the interaction matrix reconstruction ($\eta_S$).
• The time horizon $T$.
  This establishes the stability and consistency of the proposed framework.

3. Key Results

The methods were validated on benchmark models in both 1D and 2D settings, including:

• Bounded Confidence Model: Opinion dynamics with a threshold-based interaction.
• Attraction-Repulsion Models: Flocking-like behavior with short-range repulsion and long-range attraction.
• Non-local Diffusion: Cases where diffusion depends on relative distances.

Performance Findings:

• Accuracy: Both strategies successfully reconstructed complex, non-smooth kernels (e.g., step functions in bounded confidence) and diffusion profiles.
• Robustness: The framework performed well even with limited data (using only a fraction of the time horizon for training) and partial observability (unknown interaction pairs).
• Comparison:
  • The Mean-Field approach generally yielded slightly lower errors, particularly for diffusion terms in sparse regimes, as it effectively utilized the global density structure.
  • The Random-Batch approach was effective but showed higher sensitivity to data sparsity at large distances (where random pairs are rarely sampled).
• Generalization: The reconstructed kernels were able to predict system behavior accurately beyond the training time window.

4. Significance and Contributions

• Handling Latent Data: The primary contribution is a robust framework for learning interaction laws when the underlying interaction graph is unobserved, a common real-world constraint.
• Dual Strategy: By offering both a microscopic (batch-sampling) and macroscopic (density-based) approach, the method provides flexibility depending on data availability and system scale.
• Interpretability: Unlike "black-box" neural network approaches (e.g., PINNs), this method yields analytical, interpretable functions (piecewise linear approximations) for the kernels, allowing for direct physical insight into the interaction mechanisms.
• Theoretical Guarantees: The derivation of error bounds provides a theoretical foundation for the stability of the learned dynamics, ensuring that small errors in kernel estimation do not lead to catastrophic divergence in trajectory prediction.
• Scalability: The use of sparse regression and random batching makes the method computationally feasible for large systems ($N=10^5$ agents) where full pairwise observation is impossible.

In conclusion, this work advances the field of scientific machine learning for multi-agent systems by providing a rigorous, data-driven tool to uncover hidden interaction laws from incomplete trajectory data, bridging the gap between microscopic stochastic rules and macroscopic collective behavior.