J. Zech

activity

20142026

most citedDe Rham compatible Deep Neural Network FEM

12 citations · 20 across the 12 of their papers we have counts for

collaborators

14 papers

cs.LG2026

Operator Surrogates

Christoph Schwab, Josephine Westermann, Jakob Zech

This chapter provides a mathematical introduction to operator surrogates, which are finite-parametric maps approximating complex (typically nonlinear) operators between infinite-di…

math.NA2026

Neural and Spectral Operator Surrogates on Gaussian Spaces

Carlo Marcati, Mario Marić, Christoph Schwab +1

We prove expression rate bounds of finite-parametric, spectral and neural surrogates for holomorphic maps between separable Hilbert spaces. The surrogates have an encoder-approxima…

math.NA2025

Quantum Circuit Encodings of Polynomial Chaos Expansions

Junaid Aftab, Christoph Schwab, Haizhao Yang +1

This work investigates the expressive power of quantum circuits in approximating high-dimensional, real-valued functions. We focus on countably-parametric holomorphic maps $u:U\to…

math.NA2025

Sparsity for Infinite-Parametric Holomorphic Functions on Gaussian Spaces

Carlo Marcati, Christoph Schwab, Jakob Zech

We investigate the sparsity of Wiener polynomial chaos expansions of holomorphic maps $G$ on Gaussian Hilbert spaces, as arise in the coefficient-to-solution maps of line…

math.NA2023★ 2 cited

Deep Operator Network Approximation Rates for Lipschitz Operators

Christoph Schwab, Andreas Stein, Jakob Zech

We establish universality and expression rate bounds for a class of neural Deep Operator Networks (DON) emulating Lipschitz (or Hölder) continuous maps $\mathcal G:\mathcal X\to\ma…

math.NA2022

Multilevel Domain Uncertainty Quantification in Computational Electromagnetics

Rubén Aylwin, Carlos Jerez-Hanckes, Christoph Schwab +1

We continue our study [Domain Uncertainty Quantification in Computational Electromagnetics, JUQ (2020), 8:301--341] of the numerical approximation of time-harmonic electromagnetic…

math.NA2022★ 3 cited

Neural and spectral operator surrogates: unified construction and expression rate bounds

Lukas Herrmann, Christoph Schwab, Jakob Zech

Approximation rates are analyzed for deep surrogates of maps between infinite-dimensional function spaces, arising e.g. as data-to-solution maps of linear and nonlinear partial dif…

math.OC2022

Multilevel Optimization for Inverse Problems

Simon Weissmann, Ashia Wilson, Jakob Zech

Inverse problems occur in a variety of parameter identification tasks in engineering. Such problems are challenging in practice, as they require repeated evaluation of computationa…

math.NA2022★ 12 cited

De Rham compatible Deep Neural Network FEM

Marcello Longo, Joost A. A. Opschoor, Nico Disch +2

On general regular simplicial partitions $T$ of bounded polytopal domains $Ω \subset R^{d}$ , $d \in {2, 3}$ , we construct \emph{exact neural network (NN) emulations}…

math.NA2022★ 3 cited

Analyticity and sparsity in uncertainty quantification for PDEs with Gaussian random field inputs

Dinh Dũng, Van Kien Nguyen, Christoph Schwab +1

We establish sparsity and summability results for coefficient sequences of Wiener-Hermite polynomial chaos expansions of countably-parametric solutions of linear elliptic and parab…

math.NA2021

Deep Learning in High Dimension: Neural Network Approximation of Analytic Functions in $L^{2} (R^{d}, γ_{d})$

Christoph Schwab, Jakob Zech

For artificial deep neural networks, we prove expression rates for analytic functions $f : R^{d} \to R$ in the norm of $L^{2} (R^{d}, γ_{d})$ where $d\in {\mathbb{N}}\…

math.ST2021

Sparse approximation of triangular transports. Part II: the infinite dimensional case

Jakob Zech, Youssef Marzouk

For two probability measures $ρ$ and $π$ on $[- 1, 1]^{N}$ we investigate the approximation of the triangular Knothe-Rosenblatt transport $T:[-1,1]^{\mathbb{N}}\to [-1,1]^{\…

math.NA2020

Sparse approximation of triangular transports. Part I: the finite dimensional case