This study examines the identifiability of interaction kernels in mean-field equations of interacting particles or agents, an area of growing interest across various scientific and engineering fields. The main focus is identifying data-dependent function spaces where a quadratic loss functional possesses a unique minimizer. We consider two data-adaptive $L^2$ spaces: one weighted by a data-adaptive measure and the other using the Lebesgue measure. In each $L^2$ space, we show that the function space of identifiability is the closure of the RKHS associated with the integral operator of inversion. Alongside prior research, our study completes a full characterization of identifiability in interacting particle systems with either finite or infinite particles, highlighting critical differences between these two settings. Moreover, the identifiability analysis has important implications for computational practice. It shows that the inverse problem is ill-posed, necessitating regularization. Our numerical demonstrations show that the weighted $L^2$ space is preferable over the unweighted $L^2$ space, as it yields more accurate regularized estimators.

翻译：本研究探讨了交互粒子或智能体平均场方程中相互作用核的可辨识性问题，该领域在科学与工程各领域日益受到关注。核心目标是确定使得二次损失泛函具有唯一极小值的数据依赖函数空间。我们考虑两种数据自适应$L^2$空间：一种由数据自适应测度加权，另一种使用勒贝格测度。在每个$L^2$空间中，我们证明可辨识性函数空间是与反演积分算子关联的再生核希尔伯特空间（RKHS）的闭包。结合先前研究，我们的工作完整刻画了有限或无限粒子交互粒子系统的可辨识性，凸显了这两种情形间的关键差异。此外，可辨识性分析对计算实践具有重要启示：它表明该反问题是不适定的，必须引入正则化。数值实验表明，相较于未加权$L^2$空间，加权$L^2$空间更优，因其能获得更精确的正则化估计量。