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Given only a finite collection of points sampled from a Riemannian manifold embedded in a Euclidean space, in this paper we propose a new method to numerically solve elliptic and parabolic partial differential equations (PDEs) supplemented with boundary conditions. Since the construction of triangulations on unknown manifolds can be both difficult and expensive, both in terms of computational and data requirements, our goal is to solve these problems without a triangulation. Instead, we rely only on using the sample points to define quadrature formulas on the unknown manifold. Our main tool is the diffusion maps algorithm. We re-analyze this well-known method in a variational sense for manifolds with boundary. Our main result is that the variational diffusion maps graph Laplacian is a consistent estimator of the Dirichlet energy on the manifold. This improves upon previous results and provides a rigorous justification of the well-known relationship between diffusion maps and the Neumann eigenvalue problem. Moreover, using semigeodesic coordinates we derive the first uniform asymptotic expansion of the diffusion maps kernel integral operator for manifolds with boundary. This expansion relies on a novel lemma which relates the extrinsic Euclidean distance to the coordinate norm in a normal collar of the boundary. We then use a recently developed method of estimating the distance to boundary function (notice that the boundary location is assumed to be unknown) to construct a consistent estimator for boundary integrals. Finally, by combining these various estimators, we illustrate how to impose Dirichlet and Neumann conditions for some common PDEs based on the Laplacian. Several numerical examples illustrate our theoretical findings. 

The task of modeling and forecasting a dynamical system is one of the oldest problems, and itremains challenging. Broadly, this task has two subtasks: extracting the full dynamical informa-tion from a partial observation, and then explicitly learning the dynamics from this information.We present a mathematical framework in which the dynamical information is represented in theform of an embedding. The framework combines the two subtasks using the language of spaces,maps, and commutations. The framework also unifies two of the most common learning paradigms:delay-coordinates and reservoir computing. We use this framework as a platform for two otherinvestigations of the reconstructed system, its dynamical stability and the growth of error underiterations. We show that these questions are deeply tied to more fundamental properties of theunderlying system, i.e., the behavior of matrix cocycles over the base dynamics, its nonuniformhyperbolic behavior, and its decay of correlations. Thus, our framework bridges the gap betweenuniversally observed behavior of dynamics modeling and the spectral, differential, and ergodic prop-erties intrinsic to the dynamics.