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

TL;DR

This paper introduces a sparse Bayesian learning method for accurately identifying asymmetric interaction kernels in the Motsch-Tadmor model from trajectory data, addressing nonlinear inverse problems with robustness and interpretability.

Contribution

It proposes a variational framework and a sparse Bayesian algorithm for the unique and robust estimation of interaction kernels in collective dynamics models.

Findings

High accuracy in kernel recovery across noise levels

Robustness demonstrated through extensive numerical experiments

Provides uncertainty quantification and model interpretability

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.