Christoph Schwab

We investigate the sparsity of Wiener polynomial chaos expansions of holomorphic maps [Formula: see text] on Gaussian Hilbert spaces. Representing the Gaussian random field input as an affine-parametric expansion, the nonlinear map becomes a countably parametric, deterministic holomorphic map of the coordinate sequence [Formula: see text]. We establish weighted summability results for the Wiener–Hermite coefficient sequences of images of affine-parametric expansions of the log-Gaussian input under [Formula: see text]. These results give rise to [Formula: see text]-term approximation rate bounds for the full range of input summability exponents [Formula: see text]. We show that these approximation rate bounds apply to parameter-to-solution maps for elliptic diffusion PDEs with lognormal coefficients.

We establish a framework for universality and expression rate bounds for a class of neural Deep Operator Networks (DONs) emulating Lipschitz (or Hölder) continuous maps [ ] between (subsets of) separable Hilbert spaces [ ], [ ]. The DON architecture considered uses linear encoders [ ] and decoders [ ] via (biorthogonal) Riesz bases of [ ], [ ], and an approximator network of an infinite-dimensional, parametric coordinate map that is Lipschitz continuous on the sequence space [ ]. Unlike previous works [L. Herrmann, C. Schwab and J. Zech, Neural and spectral operator surrogates: Construction and expression rate bounds, Adv. Comput. Math. 50(4) (2024) 72; C. Marcati and C. Schwab, Exponential convergence of deep operator networks for elliptic partial differential equations, SIAM J. Numer. Anal. 61(3) (2023) 1513–1545] which required for example [ ] to be holomorphic, the present expression rate results require mere Lipschitz (or Hölder) continuity of [ ]. Key in the proof of the present expression rate bounds is the use of either superexpressive activations (e.g., [Z. Shen, H. Yang and S. Zhang, Neural network approximation: Three hidden layers are enough, Neural Netw. 141 (2021) 160–173; Z. Shen, H. Yang and S. Zhang, Deep network approximation: Achieving arbitrary accuracy with fixed number of neurons, J. Mach. Learn. Res. 23(276) (2022) 1–60; D. Yarotsky, Elementary superexpressive activations, in Proc. 38th Int. Conf. Machine Learning (PMLR, 2021), pp. 11932–11940] and the references there) which are inspired by the Kolmogorov superposition theorem ([A. N. Kolmogorov, On the representation of continuous functions of many variables by superposition of continuous functions of one variable and addition, Dokl. Akad. Nauk SSSR 114 (1957) 953–956] or [G. G. Lorentz, Approximation of Functions (Holt, Rinehart and Winston, New York–Chicago, IL–Toronto, ON, 1966), Chap. 11] for a comprehensive exposition), or of nonstandard NN architectures with standard (ReLU) activations as recently proposed in [Z. Shen, H. Yang and S. Zhang, Deep network approximation: Achieving arbitrary accuracy with fixed number of neurons, J. Mach. Learn. Res. 23(276) (2022) 1–60; S. Zhang, Z. Shen and H. Yang, Neural network architecture beyond width and depth, in Advances in Neural Information Processing Systems, Vol. 35 (Curran Associates, 2022), pp. 5669–5681]. We illustrate the abstract results by approximation rate bounds for emulation of (a) solution operators for parametric elliptic variational inequalities and (b) Lipschitz maps of Hilbert–Schmidt operators.

On polytopal domains in [ ], we prove weighted analytic regularity of solutions to the Dirichlet problem for the shifted integral fractional Laplacian with analytic right-hand side. Employing the Caffarelli–Silvestre extension allows one to localize the problem and to decompose the regularity estimates into results on vertex, edge, face, vertex-edge, vertex-face, edge-face and vertex-edge-face neighborhoods of the boundary. Using tangential differentiability of the extended solutions, a bootstrapping argument based on Caccioppoli inequalities on dyadic decompositions of the neighborhoods provides weighted, analytic control of higher-order solution derivatives.

We introduce a conceptual framework for numerically solving linear elliptic, parabolic, and hyperbolic PDEs on bounded, polytopal domains in euclidean spaces by deep neural networks. The PDEs are recast as minimization of a least-squares (LSQ for short) residual of an equivalent, well-posed first-order system, over parametric families of deep neural networks. The associated LSQ residual is a) equal or proportional to a weak residual of the PDE, b) additive in terms of contributions from localized subnetworks, indicating locally ``out-of-equilibrium'' of neural networks with respect to the PDE residual, c) serves as numerical loss function for neural network training, and d) constitutes, even with incomplete training, a computable, (quasi-)optimal numerical error estimator in the context of adaptive LSQ finite element methods. In addition, an adaptive neural network growth strategy is proposed which, assuming exact numerical minimization of the LSQ loss functional, yields sequences of neural networks with realizations that converge rate-optimally to the exact solution of the first order system LSQ formulation.

The distribution of centered Gaussian random fields (GRFs) indexed by compacta such as smooth, bounded Euclidean domains or smooth, compact and orientable manifolds is determined by their covariance operators. We consider centered GRFs given as variational solutions to coloring operator equations driven by spatial white noise, with an elliptic self-adjoint pseudodifferential coloring operator from the Hörmander class. This includes the Matérn class of GRFs as a special case. Using biorthogonal multiresolution analyses on the manifold, we prove that the precision and covariance operators, respectively, may be identified with bi-infinite matrices and finite sections may be diagonally preconditioned rendering the condition number independent of the dimension p of this section. We prove that a tapering strategy by thresholding applied on finite sections of the bi-infinite precision and covariance matrices results in optimally numerically sparse approximations. That is, asymptotically only linearly many nonzero matrix entries are sufficient to approximate the original section of the bi-infinite covariance or precision matrix using this tapering strategy to arbitrary precision. The locations of these nonzero matrix entries can be determined a priori. The tapered covariance or precision matrices may also be optimally diagonally preconditioned. Analysis of the relative size of the entries of the tapered covariance matrices motivates novel, multilevel Monte Carlo (MLMC) oracles for covariance estimation, in sample complexity that scales log-linearly with respect to the number p of parameters. In addition, we propose and analyze novel compressive algorithms for simulating and kriging of GRFs. The complexity (work and memory vs. accuracy) of these three algorithms scales near-optimally in terms of the number of parameters p of the sample-wise approximation of the GRF in Sobolev scales.