从轨迹数据中自动学习群体模型的非对称交互核，兼顾精度与不确定性量化。
A Sparse Bayesian Learning Algorithm for Estimation of Interaction Kernels in Motsch-Tadmor Model
- 基于隐式方程重构，将核识别转为子空间识别问题。
- 在不同噪声水平下均能准确恢复交互核，且可唯一确定（至尺度）。
- 结合稀疏贝叶斯学习，实现正则化、不确定度估计与模型选择。
本文研究基于观测轨迹数据的Motsch-Tadmor模型中非对称交互核的数据驱动识别。该模型由一类半线性演化方程控制，交互核定义了一个依赖状态的归一化拉普拉斯算子，决定群体动力学。为解决由此产生的非线性反问题，我们提出一种变分框架，利用控制方程的隐式形式将核识别转化为子空间识别问题。我们建立了可辨识性结果，刻画了交互核在何种条件下可唯一恢复（至尺度）。为鲁棒求解反问题，我们开发了一种稀疏贝叶斯学习算法，引入信息性先验进行正则化，量化不确定性，并支持合理的模型选择。在典型粒子系统上的大量数值实验表明，该框架在多种噪声水平和数据条件下均具有高精度、强鲁棒性与良好可解释性。
原文摘要 · Abstract (English)
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.
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