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Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equations
(2021)SAM Research ReportReport -
NGN2 induces diverse neuron types from human pluripotency
(2021)Stem Cell ReportsHuman neurons engineered from induced pluripotent stem cells (iPSCs) through neurogenin 2 (NGN2) overexpression are widely used to study neuronal differentiation mechanisms and to model neurological diseases. However, the differentiation paths and heterogeneity of emerged neurons have not been fully explored. Here, we used single-cell transcriptomics to dissect the cell states that emerge during NGN2 overexpression across a time course ...Report -
Multi-scale classification for electro-sensing
(2020)SAM Research ReportThis paper introduces premier and innovative (real-time) multi-scale method for target classification in electro-sensing. The intent is that of mimicking the behavior of the weakly electric fish, which is able to retrieve much more information about the target by approaching it. The method is based on a family of transform-invariant shape descriptors computed from generalized polarization tensors (GPTs) reconstructed at multiple scales. ...Report -
Deep learning in high dimension: ReLU network Expression Rates for Bayesian PDE inversion
(2020)SAM Research ReportWe establish dimension independent expression rates by deep ReLU networks for so-called (b,ε,X)-holomorphic functions. These are mappings from [−1,1]N→X, with X being a Banach space, that admit analytic extensions to certain polyellipses in each of the input variables. The significance of this function class has been established in previous works, where it was shown that functions of this type occur widely in uncertainty quantification ...Report -
Acoustic Scattering Problems with Convolution Quadrature and the Method of Fundamental Solutions
(2020)SAM Research ReportTime domain acoustic scattering problems in two and three dimensions are studied. The numerical scheme consists in the use of Convolution Quadrature method to reduce the time domain problem to solve frequency domain Helmholtz equations with complex wavenumbers. These equations are solved with the method of fundamental solutions (MFS), which approximates the solution by a linear combination of fundamental solutions defined at source points ...Report -
Analyticity and sparsity in uncertainty quantification for PDEs with Gaussian random field inputs
(2022)SAM Research ReportWe establish summability results for coefficient sequences of Wiener-Hermite polynomial chaos expansions for countably-parametric solutions of linear elliptic and parabolic divergence-form partial differential equations with Gaussian random field inputs. The novel proof technique developed here is based on analytic continuation of parametric solutions into the complex domain. It differs from previous works that used bootstrap arguments ...Report -
Graph-Coupled Oscillator Networks
(2022)SAM Research ReportWe propose Graph-Coupled Oscillator Networks (GraphCON), a novel framework for deep learning on graphs. It is based on discretizations of a second-order system of ordinary differential equations (ODEs), which model a network of nonlinear forced and damped oscillators, coupled via the adjacency structure of the underlying graph. The flexibility of our framework permits any basic GNN layer (e.g. convolutional or attentional) as the coupling ...Report -
On nonlinear Feynman-Kac formulas for viscosity solutions of semilinear parabolic partial differential equations
(2020)SAM Research ReportReport -
Boundary Integral Exterior Calculus
(2022)SAM Research ReportWe develop first-kind boundary integral equations for Hodge-Dirac and Hodge-Laplace operators associated with de Rham Hilbert complexes on compact Riemannian manifolds and Euclidean space. We show that the first-kind boundary integral operators associated with Hodge-Dirac and Hodge-Laplace boundary value problems posed on submanifolds with Lipschitz boundaries are Hodge-Dirac and Hodge-Laplace operators as well, but associated with a trace ...Report