In this paper we are concerned with the learnability of nonlocal interaction kernels for first order systems modeling certain social interactions, from observations of realizations of their dynamics. This paper is the first of a series on learnability of nonlocal interaction kernels and presents a variational approach to the problem. In particular, we assume here that the kernel to be learned is bounded and locally Lipschitz continuous and that the initial conditions of the systems are drawn identically and independently at random according to a given initial probability distribution. Then the minimization over a rather arbitrary sequence of (finite dimensional) subspaces of a least square functional measuring the discrepancy from observed trajectories produces uniform approximations to the kernel on compact sets. The convergence result is obtained by combining mean-field limits, transport methods, and a Γ-convergence argument. A crucial condition for the learnability is a certain coercivity property of the least square functional, majoring an L 2 -norm discrepancy to the kernel with respect to a probability measure, depending on the given initial probability distribution by suitable push forwards and transport maps. We illustrate the convergence result by means of several numerical experiments.

We consider nonlinear systems of ordinary differential equations (ODEs) on a state space S. We consider the general setting when S is a Banach space over R or C. We assume the right hand side depends affinely linear on a vector y = (y j) j≥1 of possibly countably many parameters, normalized such that |y j | ≤ 1. Under suitable analyticity assumptions on the ODEs, we prove that the parametric solution {X(t; y) : 0 ≤ t ≤ T } ⊂S of the corresponding IVP depends holomorphically on the parameter vector y, as a mapping from the infinite-dimensional parameter domain U = (−1, 1) N into a suitable function space on [0, T ]×S. Such affine parameter dependence of the ODE arises, among others, in mass action models in computational biology (see, e.g. [18]) and in stochiometry with uncertain reaction rate constants. Using our analytic regularity result, we prove summability theorems for coefficient sequences of generalized polynomial chaos (gpc) expansions of the parametric solutions {X(•; y)} y∈U with respect to tensor product orthogonal polynomial bases of L 2 (U). We give sufficient conditions on the ODEs for N-term truncations of these expansions to converge on the entire parameter space with efficiency (i.e. accuracy versus complexity) being independent of the number of parameters viz. the dimension of the parameter space U .

We show that in analogy to the introduction of Poisson structures twisted by a closed 3-form by Park and Klimcik-Strobl, the study of three dimensional sigma models with Wess-Zumino term leads in a likewise way to twisting of Courant structures by closed 4-forms H.The presentation is kept pedagogical and accessible to physicists as well as to mathematicians, explaining in detail in particular the interplay of field transformations in a sigma model with the type of geometrical structures induced on a target. In fact, as we also show, even if one does not know the mathematical concept of a Courant algebroid, the study of a rather general class of 3-dimensional sigma models leads one to that notion by itself.Courant algebroids became of relevance for mathematical physics lately from several perspectives-like for example by means of using generalized complex structures in String Theory. One may expect that their twisting by the curvature H of some 3-form RamondRamond gauge field will become of relevance as well.

We investigate existence and regularity of a class of semilinear, parametric elliptic PDEs with affine dependence of the principal part of the differential operator on countably many parameters. We establish a priori estimates and analyticity of the parametric solutions. We establish summability results of coefficient sequences of polynomial chaos type expansions of the parametric solutions in terms of tensorized Taylor‐, Legendre‐ and Chebyshev polynomials on the infinite‐dimensional parameter domain. We deduce rates of convergence for N term truncated approximations of expansions of the parametric solution. We also deduce spatial regularity of the solution, and establish convergence rates of N‐term discretizations of the parametric solutions with respect to these polynomials in parameter space and with respect to a multilevel hierarchy of finite element spaces in the spatial domain of the PDE.

We investigate the Besov regularity for solutions of elliptic PDEs. This is based on regularity results in Babuska-Kondratiev spaces. Following the argument of Dahlke and DeVore, we first prove an embedding of these spaces into the scale B r τ,τ (D) of Besov spaces with 1 τ = r d + 1 p. This scale is known to be closely related to n-term approximation w.r.to wavelet systems, and also adaptive Finite element approximation. Ultimately this yields the rate n −r/d for u ∈ K m p,a (D) ∩ H s p (D) for r < r * ≤ m. In order to improve this rate to n −m/d we leave the scale B r τ,τ (D) and instead consider the spaces B m τ,∞ (D). We determine conditions under which the space K m p,a (D)∩H s p (D) is embedded into some space B m τ,∞ (D) for some m d + 1 p > 1 τ ≥ 1 p , which in turn indeed yields the desired n-term rate. As an intermediate step we also prove an extension theorem for Kondratiev spaces.