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A Sparse Bayesian Learning Algorithm for Estimation of Interaction Kernels in Motsch-Tadmor Model

Jinchao Feng, Sui Tang

TL;DR

This work tackles the problem of identifying asymmetric interaction kernels in the $Motsch{-}Tadmor$ model from trajectory data. It introduces a variational framework based on the implicit form of the governing ODE, reducing kernel discovery to a subspace identification problem and solving it with a sparse Bayesian learning (SBL) approach that incorporates hierarchical priors and uncertainty quantification. A key theoretical result provides identifiability up to scale under conditions on the data distribution and basis support, complemented by a model-selection criterion based on weighted total uncertainty (wTU). Numerical experiments on first-order (opinion dynamics) and second-order (Cucker–Smale) systems demonstrate accurate kernel recovery, robustness to noise, interpretable results, and scalable computation, highlighting the practical impact for data-driven learning of non-symmetric interactions in multi-agent systems.

Abstract

In this paper, we investigate the data-driven identification of asymmetric interaction kernels in the Motsch-Tadmor model based on observed trajectory data. The model under consideration is governed by a class of semilinear evolution equations, where the interaction kernel defines a normalized, state-dependent Laplacian operator that governs collective dynamics. To address the resulting nonlinear inverse problem, we propose a variational framework that reformulates kernel identification using the implicit form of the governing equations, reducing it to a subspace identification problem. We establish an identifiability result that characterizes conditions under which the interaction kernel can be uniquely recovered up to scale. To solve the inverse problem robustly, we develop a sparse Bayesian learning algorithm that incorporates informative priors for regularization, quantifies uncertainty, and enables principled model selection. Extensive numerical experiments on representative interacting particle systems demonstrate the accuracy, robustness, and interpretability of the proposed framework across a range of noise levels and data regimes.

A Sparse Bayesian Learning Algorithm for Estimation of Interaction Kernels in Motsch-Tadmor Model

TL;DR

This work tackles the problem of identifying asymmetric interaction kernels in the model from trajectory data. It introduces a variational framework based on the implicit form of the governing ODE, reducing kernel discovery to a subspace identification problem and solving it with a sparse Bayesian learning (SBL) approach that incorporates hierarchical priors and uncertainty quantification. A key theoretical result provides identifiability up to scale under conditions on the data distribution and basis support, complemented by a model-selection criterion based on weighted total uncertainty (wTU). Numerical experiments on first-order (opinion dynamics) and second-order (Cucker–Smale) systems demonstrate accurate kernel recovery, robustness to noise, interpretable results, and scalable computation, highlighting the practical impact for data-driven learning of non-symmetric interactions in multi-agent systems.

Abstract

In this paper, we investigate the data-driven identification of asymmetric interaction kernels in the Motsch-Tadmor model based on observed trajectory data. The model under consideration is governed by a class of semilinear evolution equations, where the interaction kernel defines a normalized, state-dependent Laplacian operator that governs collective dynamics. To address the resulting nonlinear inverse problem, we propose a variational framework that reformulates kernel identification using the implicit form of the governing equations, reducing it to a subspace identification problem. We establish an identifiability result that characterizes conditions under which the interaction kernel can be uniquely recovered up to scale. To solve the inverse problem robustly, we develop a sparse Bayesian learning algorithm that incorporates informative priors for regularization, quantifies uncertainty, and enables principled model selection. Extensive numerical experiments on representative interacting particle systems demonstrate the accuracy, robustness, and interpretability of the proposed framework across a range of noise levels and data regimes.
Paper Structure (23 sections, 2 theorems, 49 equations, 10 figures, 3 tables, 1 algorithm)

This paper contains 23 sections, 2 theorems, 49 equations, 10 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Works
  3. Contributions
  4. Problem Formulation
  5. Identifiability.
  6. Algorithm
  7. Challenges in Estimation
  8. A Sparse Bayesian Learning (SBL) Approach
  9. Data and Observation Models
  10. The Likelihood
  11. The Sparse Prior and Hyperprior
  12. Bayesian Inference
  13. Posterior of $\boldsymbol{w}$ given $\boldsymbol{\Theta}$
  14. Optimization of hyperparameters $\boldsymbol{\Theta}$
  15. Model Selection
  16. ...and 8 more sections

Key Result

Proposition 1

Let $\hat{\boldsymbol{c}}_{M,L} = \underset{\|\boldsymbol{c}\|_2=1}{\operatorname{arg}\operatorname{min}}\; \mathcal{E}_{M,L}(\boldsymbol{c})$. Then: where $\mathbb{A}_{M,L} = \in \mathbb{R}^{dNML \times K}$ with $\boldsymbol{A}^{(m,l)} = [\boldsymbol{A}_{ik}^{(m,l)}] \in \mathbb{R}^{dN \times K}$, and for $i=1,\cdots,N$ and $k=1,\cdots,K$.

Figures (10)

  • Figure 1: Trajectory profiles for the opinion dynamics for 1D and 2D examples
  • Figure 2: Estimated interaction kernels $\hat{\phi}$ for 1D (left) and 2D (right) opinion dynamics under increasing noise levels ($(M,L) = (3,6)$).
  • Figure 3: Prediction error of $\phi$ for (a) 1D and (b) 2D systems with different $M$ and noise levels ($L=6$ fixed).
  • Figure 4: Average running time for learning opinion dynamics models with two different dimensions.
  • Figure 5: Trajectory profiles for the Cucker-Smale system under different interaction kernels.
  • ...and 5 more figures

Theorems & Definitions (10)

  • Proposition 1
  • Proof 1
  • Remark 1
  • Proposition 2
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7