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# Deep Neural Networks Are Effective At Learning High-Dimensional Hilbert-Valued Functions From Limited Data

*Proceedings of the 2nd Mathematical and Scientific Machine Learning Conference*, PMLR 145:1-36, 2022.

#### Abstract

The accurate approximation of scalar-valued functions from sample points is a key task in mathematical modelling and computational science. Recently, machine learning techniques based on Deep Neural Networks (DNNs) have begun to emerge as promising tools for function approximation in scientific computing problems, with some impressive results achieved on problems where the dimension of the underlying data or problem domain is large. In this work, we broaden this perspective by focusing on the approximation of functions that are \textit{Hilbert-valued}, i.e. they take values in a separable, but typically infinite-dimensional, Hilbert space. This problem arises in many science and engineering problems, in particular those involving the solution of parametric Partial Differential Equations (PDEs). Such problems are challenging for three reasons. First, pointwise samples are expensive to acquire. Second, the domain of the function is usually high dimensional, and third, the range lies in a Hilbert space. Our contributions are twofold. First, we present a novel result on DNN training for holomorphic functions with so-called \textit{hidden anisotropy}. This result introduces a DNN training procedure and a full theoretical analysis with explicit guarantees on the error and sample complexity. This error bound is explicit in the three key errors occurred in the approximation procedure: the best approximation error, the measurement error and the physical discretization error. Our result shows that there exists a procedure (albeit a non-standard one) for learning Hilbert-valued functions via DNNs that performs as well as, but no better than current best-in-class schemes. It therefore gives a benchmark lower bound for how well methods DNN training can perform on such problems. Second, we examine whether better performance can be achieved in practice through different types of architectures and training. We provide preliminary numerical results illustrating the practical performance of DNNs on Hilbert-valued functions arising as solutions to parametric PDEs. We consider different parameters, modify the DNN architecture to achieve better and competitive results and compare these to current best-in-class schemes.