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

2025-05-11

AI Summary

This work addresses the data-driven identification of asymmetric interaction kernels in the Motsch–Tadmor model from observed trajectories. We formulate the nonlinear inverse problem as a subspace identification task under implicit dynamical constraints and establish theoretical identifiability of the kernel up to scale. Methodologically, we develop a physics-informed sparse Bayesian learning algorithm that enables automatic model selection and uncertainty quantification; additionally, we introduce a state-dependent normalized Laplacian operator to better capture heterogeneous interactions. Experiments across diverse multi-particle systems demonstrate that the proposed framework achieves high accuracy, strong robustness to varying noise levels and data sizes, and enhanced interpretability. Our approach thus advances both the theoretical foundations and practical applicability of data-driven collective behavior modeling.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Relational Probabilistic Models
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: Psychology-informed user models and recommender systems
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• Estimating asymmetric interaction kernels in Motsch-Tadmor model from trajectory data
• Developing sparse Bayesian learning algorithm for robust inverse problem solving
• Characterizing identifiability conditions for unique kernel recovery up to scale

Innovation

• Variational framework for kernel identification
• Sparse Bayesian learning with informative priors
• Identifiability result for unique kernel recovery