Learning Interaction Kernels in Mean-Field Equations of First-Order Systems of Interacting Particles

Abstract

In this paper, we tackle a critical issue in nonparametric inference for systems of interacting particles on Riemannian manifolds: the identifiability of the interaction functions. Specifically, we define the function spaces on which the interaction kernels can be identified given infinite i.i.d observational derivative data sampled from a distribution. Our methodology involves casting the learning problem as a linear statistical inverse problem using a operator theoretical framework. We prove the well-posedness of inverse problem by establishing the strict positivity of a related integral operator and our analysis allows us to refine the results on specific manifolds such as the sphere and Hyperbolic space. Our findings indicate that a numerically stable procedure exists to recover the interaction kernel from finite (noisy) data, and the estimator will be convergent to the ground truth. This also answers an open question in [MMQZ21] and demonstrate that least square estimators can be statistically optimal in certain scenarios. Finally, our theoretical analysis could be extended to the mean-field case, revealing that the corresponding nonparametric inverse problem is ill-posed in general and necessitates effective regularization techniques.

Learning interaction kernels in mean-field equations of 1st-order systems of interacting particles

“…Here S d´1 denotes the unit sphere in R d . We prove in [18] that G T is symmetric positive semipositive definite. Thus, ¨, ¨ G T is the inner product of the reproducing kernel Hilbert space (RKHS), denoted by H G T with G T as reproducing kernel.…”

Data adaptive RKHS Tikhonov regularization for learning kernels in operators

“…Such nonlinear operators arise in the mean-field equations of interaction particles (see e.g., [18,22,26,28]), and the function φ is called an interaction kernel. More precisely, the mean-field equations are of the form B t u " ν∆u `divpu ˚Kφ uq on R d , where K φ pyq " φp|y|q y |y| .…”

“…The learning of kernel functions in operators is such a problem: given data tpu k , f k qu N k"1 in suitable function spaces, we would like to learn an optimal kernel function φ fitting the operator R φ puq " f to the data. Such a need for learning operators between function spaces has become vital in applications ranging from integral operators solving PDEs and image processing (see e.g., [12,20,24,29]), nonlinear operators in mean-field equation of interacting particle systems in [22,26], homogenized nonlocal operators (see e.g., [25,38,39]), just to name a few. Since there is often limited information to derive a parametric form, the kernel has to be learnt in a nonparametric fashion.…”

Nonparametric learning of kernels in nonlocal operators

“…overcome this curse-of-dimensionality, one can consider other settings with mesh-free representation of data by random samples and a loss functional based on expectations (see, e.g. [27]), and this is beyond the scope of the current study.…”

“…with a force loading term g(x, t), proper boundary conditions and initial conditions û(x, 0) = 0, ∂ t û(x, 0) = 0. Considering the heterogeneous bar of two materials depicted in Figure 3, (27) describes the stress wave propagating with speed c 1 = E 1 /ρ in material 1 and speed c 2 = E 2 /ρ in material 2. We solve the HF-model (27) by the direct numerical solver (DNS) introduced in [44].…”