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Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equations
(2021)SAM Research ReportReport -
Weighted analytic regularity for the integral fractional Laplacian in polygons
(2021)SAM Research ReportWe prove weighted analytic regularity of solutions to the Dirichlet problem for the integral fractional Laplacian in polygons with analytic right-hand side. We localize the problem through the Caffarelli-Silvestre extension and study the tangential differentiability of the extended solutions, followed by bootstrapping based on Caccioppoli inequalities on dyadic decompositions of vertex, edge, and edge-vertex neighborhoods.Report -
Deep Solution Operators for Variational Inequalities via Proximal Neural Networks
(2021)SAM Research ReportWe introduce ProxNet, a collection of deep neural networks with ReLU activation which emulate numerical solution operators of variational inequalities (VIs). We analyze the expression rates of ProxNets in emulating solution operators for variational inequality problems posed on closed, convex cones in separable Hilbert spaces, covering the classical contact problems in mechanics, and early exercise problems as arise, e.g. in valuation of ...Report -
Deep Learning in High Dimension: Neural Network Expression Rates for Analytic Functions in $L^2(ℝ^d,\gamma_d)$
(2021)SAM Research ReportReport -
DeepCME: A deep learning framework for computing solution statistics of the Chemical Master Equation
(2021)SAM Research ReportStochastic models of biomolecular reaction networks are commonly employed in systems and synthetic biology to study the effects of stochastic fluctuations emanating from reactions involving species with low copy-numbers. For such models, the Kolmogorov’s forward equation is called the chemical master equation (CME), and it is a fundamental system of linear ordinary differential equations (ODEs) that describes the evolution of the probability ...Report -
Analytic regularity for the Navier-Stokes equations in polygons with mixed boundary conditions
(2021)SAM Research ReportWe prove weighted analytic regularity of Leray-Hopf variational solutions for the stationary, incompressible Navier-Stokes Equations (NSE) in plane polygonal domains, subject to analytic body forces. We admit mixed boundary conditions which may change type at each vertex, under the assumption that homogeneous Dirichlet (''no-slip'') boundary conditions are prescribed on at least one side at each vertex of the domain. The weighted analytic ...Report -
Multilevel approximation of Gaussian random fields: Covariance compression, estimation and spatial prediction
(2021)SAM Research ReportCentered Gaussian random fields (GRFs) indexed by compacta such as smooth, bounded domains in Euclidean space or smooth, compact and orientable manifolds are determined by their covariance operators. We consider centered GRFs given sample-wise as variational solutions to coloring operator equations driven by spatial white noise, with pseudodifferential coloring operator being elliptic, self-adjoint and positive from the Hörmander class. ...Report -
Deep ReLU Neural Network Approximation for stochastic differential equations with jumps
(2021)SAM Research ReportDeep neural networks (DNNs) with ReLU activation function are proved to be able to express viscosity solutions of linear partial integrodifferental equations (PIDEs) on state spaces of possibly high dimension d.Admissible PIDEs comprise Kolmogorov equations for high-dimensional diffusion, advection, and for pure jump Lévy processes. We prove for such PIDEs arising from a class of jump-diffusions on Rd, that for any compact K⊂Rd, there ...Report -
Constructive Deep ReLU Neural Network Approximation
(2021)SAM Research ReportWe propose an efficient, deterministic algorithm for constructing exponentially convergent deep neural network (DNN) approximations of multivariate, analytic maps f:[−1,1]K→R. We address in particular networks with the rectified linear unit (ReLU) activation function. Similar results and proofs apply for many other popular activation functions. The algorithm is based on collocating f in deterministic families of grid points with small ...Report -
Exponential Convergence of hp FEM for Spectral Fractional Diffusion in Polygons
(2020)SAM Research ReportReport