Convegno
“DIFFERENTIAL EVOLUTIVE MODELS IN SPACES WITH SINGULARITIES”

In this talk I consider a surface embedded in a 3D contact sub-Riemannian manifold. Such a surface inherits a field of direction (with norm) from the ambient space. This field of directions is singular at characteristic points (i.e., where the surface is tangent to the set of admissible directions). In this talk we will study when the normed field of directions permits to give to the surface the structure of metric space (of ``SNCF'' type). I will also study how to define the heat and the Schroedinger equation on such a structure and if the singular points are ``accessible'' or not. When the singular points are accessible we will study self-adjoint extensions with Kirchhoff like boundary conditions.

The talk presents a collection of results with Pascal Heid, Giacomo Sodini, and Giuseppe Savaré. We start the talk by presenting general results of strong density of sub-algebras of bounded Lipschitz functions in metric Sobolev spaces. We apply such results to show the density of smooth cylinder functions in Sobolev spaces of functions on the Wasserstein space $\mathcal P_2$ endowed with a finite positive Borel measure. As a byproduct, we obtain the infinitesimal Hilbertianity of Wassertein Sobolev spaces. By taking advantage of these results, we further address the challenging problem of the numerical approximation of Wassertein Sobolev functions defined on probability spaces. Our particular focus centers on the Wasserstein distance function, which serves as a relevant example. In contrast to the existing body of literature focused on approximating efficiently pointwise evaluations, we chart a new course to define functional approximants by adopting three machine learning-based approaches: 1. Solving a finite number of optimal transport problems and computing the corresponding Wasserstein potentials. 2. Employing empirical risk minimization with Tikhonov regularization in Wasserstein Sobolev spaces. 3. Addressing the problem through the saddle point formulation that characterizes the weak form of the Tikhonov functional's Euler-Lagrange equation. As a theoretical contribution, we furnish explicit and quantitative bounds on generalization errors for each of these solutions. In the proofs, we leverage the theory of metric Sobolev spaces introduced above and we combine it with techniques of optimal transport, variational calculus, and large deviation bounds. In our numerical implementation, we harness appropriately designed neural networks to serve as basis functions. Consequently, our constructive solutions significantly enhance at equal accuracy the evaluation speed, surpassing that of state-of-the-art methods by several orders of magnitude.