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.

翻译：本文基于观测轨迹数据，研究了Motsch-Tadmor模型中非对称交互核的数据驱动辨识问题。所考虑的模型由一类半线性发展方程控制，其中交互核定义了一个标准化的、状态依赖的拉普拉斯算子，该算子支配着集体动力学行为。为求解由此产生的非线性逆问题，我们提出了一种变分框架，通过利用控制方程的隐式形式将核辨识问题重新表述为子空间辨识问题。我们建立了一个可辨识性结果，刻画了交互核可被唯一恢复（至多相差一个缩放因子）的条件。为稳健地解决该逆问题，我们开发了一种稀疏贝叶斯学习算法，该算法结合了用于正则化的先验信息，量化了不确定性，并实现了原则性的模型选择。在代表性交互粒子系统上的大量数值实验表明，所提出的框架在多种噪声水平和数据范围内具有准确性、鲁棒性和可解释性。