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Deep ReLU network expression rates for option prices in high-dimensional, exponential Lévy models
(2020)SAM Research ReportWe study the expression rates of deep neural networks (DNNs for short) for option prices written on baskets of d risky assets, whose log-returns are modelled by a multivariate Lévy process with general correlation structure of jumps. We establish sufficient conditions on the characteristic triplet of the Lévy process X that ensure ε error of DNN expressed option prices with DNNs of size that grows polynomially with respect to O(ε−1), and ...Report -
Exponential ReLU Neural Network Approximation Rates for Point and Edge Singularities
(2020)SAM Research ReportWe prove exponential expressivity with stable ReLU Neural Networks (ReLU NNs) in H1(Ω) for weighted analytic function classes in certain polytopal domains Ω, in space dimension d=2,3. Functions in these classes are locally analytic on open subdomains D⊂Ω, but may exhibit isolated point singularities in the interior of Ω or corner and edge singularities at the boundary ∂Ω. The exponential expression rate bounds proved here imply uniform ...Report -
Multilevel Monte Carlo finite volume methods for random conservation laws with discontinuous flux
(2020)SAM Research ReportWe consider conservation laws with discontinuous flux where the initial datum, the flux function, and the discontinuous spatial dependency coefficient are subject to randomness. We establish a notion of random adapted entropy solutions to these equations and prove well-posedness provided that the spatial dependency coefficient is piecewise constant with finitely many discontinuities. In particular, the setting under consideration allows ...Report -
Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs
(2020)SAM Research ReportPhysics informed neural networks (PINNs) have recently been widely used for robust and accurate approximation of PDEs. We provide rigorous upper bounds on the generalization error of PINNs approximating solutions of the forward problem for PDEs. An abstract formalism is introduced and stability properties of the underlying PDE are leveraged to derive an estimate for the generalization error in terms of the training error and number of ...Report -
Higher-order Quasi-Monte Carlo Training of Deep Neural Networks
(2020)SAM Research ReportWe present a novel algorithmic approach and an error analysis leveraging Quasi-Monte Carlo points for training deep neural network (DNN) surrogates of Data-to-Observable (DtO) maps in engineering design. Our analysis reveals higher-order consistent, deterministic choices of training points in the input data space for deep and shallow Neural Networks with holomorphic activation functions such as tanh. These novel training points are proved ...Report -
Flux-stability for conservation laws with discontinuous flux and convergence rates of the front tracking method
(2020)SAM Research ReportWe prove that adapted entropy solutions of scalar conservation laws with discontinuous flux are stable with respect to changes in the flux under the assumption that the flux is strictly monotone in u and the spatial dependency is piecewise constant with finitely many discontinuities. We use this stability result to prove a convergence rate for the front tracking method -- a numerical method which is widely used in the field of conservation ...Report -
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Iterative Surrogate Model Optimization (ISMO): An active learning algorithm for PDE constrained optimization with deep neural networks
(2020)SAM Research ReportWe present a novel active learning algorithm, termed a iiterative surrogate model optimization (ISMO), for robust and efficient numerical approximation of PDE constrained optimization problems. This algorithm is based on deep neural networks and its key feature is the iterative selection of training data through a feedback loop between deep neural networks and any underlying standard optimization algorithm. Under suitable hypotheses, we ...Report -
Weighted analyticity of Hartree-Fock eigenfunctions
(2020)SAM Research ReportWe prove analytic-type estimates in weighted Sobolev spaces on the eigenfunctions of a class of elliptic and nonlinear eigenvalue problems with singular potentials, which includes the Hartree-Fock equations. Going beyond classical results on the analyticity of the wavefunctions away from the nuclei, we prove weighted estimates locally at each singular point, with precise control of the derivatives of all orders. Our estimates have ...Report -