We estimate the expressive power of certain deep neural networks (DNNs for short) on a class of countably-parametric, holomorphic maps [Formula: see text] on the parameter domain [Formula: see text]. Dimension-independent rates of best [Formula: see text]-term truncations of generalized polynomial chaos (gpc for short) approximations depend only on the summability exponent of the sequence of their gpc expansion coefficients. So-called [Formula: see text]-holomorphic maps [Formula: see text], with [Formula: see text] for some [Formula: see text], are known to allow gpc expansions with coefficient sequences in [Formula: see text]. Such maps arise for example as response surfaces of parametric PDEs, with applications in PDE uncertainty quantification (UQ) for many mathematical models in engineering and the sciences. Up to logarithmic terms, we establish the dimension independent approximation rate [Formula: see text] for these functions in terms of the total number [Formula: see text] of units and weights in the DNN. It follows that certain DNN architectures can overcome the curse of dimensionality when expressing possibly countably-parametric, real-valued maps with a certain degree of sparsity in the sequences of their gpc expansion coefficients. We also obtain rates of expressive power of DNNs for countably-parametric maps [Formula: see text], where [Formula: see text] is the Hilbert space [Formula: see text].
For a parameter dimension $$d\in {\mathbb {N}}$$ d ∈ N , we consider the approximation of many-parametric maps $$u: [-\,1,1]^d\rightarrow {\mathbb R}$$ u : [ - 1 , 1 ] d → R by deep ReLU neural networks. The input dimension d may possibly be large, and we assume quantitative control of the domain of holomorphy of u: i.e., u admits a holomorphic extension to a Bernstein polyellipse $${{\mathcal {E}}}_{\rho _1}\times \cdots \times {{\mathcal {E}}}_{\rho _d} \subset {\mathbb {C}}^d$$ E ρ 1 × ⋯ × E ρ d ⊂ C d of semiaxis sums $$\rho _i>1$$ ρ i > 1 containing $$[-\,1,1]^{d}$$ [ - 1 , 1 ] d . We establish the exponential rate $$O(\exp (-\,bN^{1/(d+1)}))$$ O ( exp ( - b N 1 / ( d + 1 ) ) ) of expressive power in terms of the total NN size N and of the input dimension d of the ReLU NN in $$W^{1,\infty }([-\,1,1]^d)$$ W 1 , ∞ ( [ - 1 , 1 ] d ) . The constant $$b>0$$ b > 0 depends on $$(\rho _j)_{j=1}^d$$ ( ρ j ) j = 1 d which characterizes the coordinate-wise sizes of the Bernstein-ellipses for u. We also prove exponential convergence in stronger norms for the approximation by DNNs with more regular, so-called “rectified power unit” activations. Finally, we extend DNN expression rate bounds also to two classes of non-holomorphic functions, in particular to d-variate, Gevrey-regular functions, and, by composition, to certain multivariate probability distribution functions with Lipschitz marginals.
We consider the stationary Stokes and Navier-Stokes Equations for viscous, incompressible flow in parameter dependent bounded domains D T , subject to homogeneous Dirichlet ("noslip") boundary conditions on ∂D T . Here, D T is the image of a given fixed nominal LipschitzWe establish shape holomorphy of Leray solutions which is to say, holomorphy of the mapdenotes the pullback of the corresponding weak solutions and T varies in W k,∞ with k ∈ {1, 2}, depending on the type of pullback. We consider in particular parametrized families {T y : y ∈ U } ⊆ W 1,∞ of domain mappings, with parameter domain U = [−1, 1] N and with affine dependence of T y on y. The presently obtained shape holomorphy implies summability results and n-term approximation rate bounds for gpc ("generalized polynomial chaos") expansions for the corresponding parametric solution map y → (û y ,p
We analyse convergence rates of Smolyak integration for parametric maps u: U → X taking values in a Banach space X, defined on the parameter domain U = [−1,1]N. For parametric maps which are sparse, as quantified by summability of their Taylor polynomial chaos coefficients, dimension-independent convergence rates superior to N-term approximation rates under the same sparsity are achievable. We propose a concrete Smolyak algorithm to a priori identify integrand-adapted sets of active multiindices (and thereby unisolvent sparse grids of quadrature points) via upper bounds for the integrands’ Taylor gpc coefficients. For so-called “(b,ε)-holomorphic” integrands u with b∈lp(∕) for some p ∈ (0, 1), we prove the dimension-independent convergence rate 2/p − 1 in terms of the number of quadrature points. The proposed Smolyak algorithm is proved to yield (essentially) the same rate in terms of the total computational cost for both nested and non-nested univariate quadrature points. Numerical experiments and a mathematical sparsity analysis accounting for cancellations in quadratures and in the combination formula demonstrate that the asymptotic rate 2/p − 1 is realized computationally for a moderate number of quadrature points under certain circumstances. By a refined analysis of model integrand classes we show that a generally large preasymptotic range otherwise precludes reaching the asymptotic rate 2/p − 1 for practically relevant numbers of quadrature points.
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