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.
Learning Nonlinear Factor Models with Unknown Monotone Links from Incomplete and Noisy Data
School of Computation, Information and Technology, Technical University of Munich, Germany（计算、信息与技术学院，慕尼黑技术大学，德国）；Department of Economics, Virginia Tech, USA（经济系，弗吉尼亚理工学院，美国）；Munich Institute of Robotics and Machine Intelligence（慕尼黑机器人与人工智能研究所）
Interacting Particle Systems on Networks: joint inference of the network and the interaction kernel
Department of Mathematics, Duke University, Durham, USA（杜克大学数学系，美国达勒姆）；School of Mathematics, Sun Yat-sen University, Guangzhou, China（中山大学数学学院，中国广州）；Department of Mathematics, Johns Hopkins University, Baltimore, USA（约翰霍普金斯大学数学系，美国巴尔的摩）；Department of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, USA（约翰霍普金斯大学应用数学与统计学系，美国巴尔的摩）
Learning Multi-type heterogeneous interacting particle systems
Department of Mathematics, Duke University（杜克大学数学系）；School of Mathematics, Sun Yat-sen University（中山大学数学学院）；Department of Mathematics, Johns Hopkins University（约翰霍普金斯大学数学系）；Department of Applied Mathematics and Statistics, Johns Hopkins University（约翰霍普金斯大学应用数学与统计学系）
