We study the identifiability of the interaction kernels in mean-field equations for intreacting particle systems. The key is to identify function spaces on which a probabilistic loss functional has a unique minimizer. We prove that identifiability holds on any subspace of two reproducing kernel Hilbert spaces (RKHS), whose reproducing kernels are intrinsic to the system and are data-adaptive. Furthermore, identifiability holds on two ambient L2 spaces if and only if the integral operators associated with the reproducing kernels are strictly positive. Thus, the inverse problem is ill-posed in general. We also discuss the implications of identifiability in computational practice.

翻译：我们研究中外方程式中相互作用内核的可识别性,用于调节粒子系统。关键在于确定概率损失功能具有独特的最小化作用的功能空间。我们证明,两个复制内核的Hilbert空间(RKHS)的任何子空间都存在可识别性,这两个空间的再生产内核是系统固有的,是数据适应性的。此外,两个周围L2空间的可识别性只有与再生产内核相关的整体操作器是绝对肯定的,而且只有与再生产内核相关的整体操作器是绝对肯定的,才能存在。因此,反面问题一般不正确。我们还讨论了计算实践中的可识别性的影响。