Georgia Institute of Technology College of Sciences

In the past few decades the problem of reconstructing high-dimensional functions taking values in abstract spaces from limited samples has received increasing attention, largely due to its relevance to uncertainty quantification (UQ) for computational science and engineering. These UQ problems are often posed in terms of parameterized partial differential equations whose solutions take values in Hilbert or Banach spaces. Impressive results have been achieved on such problems with deep learning (DL), i.e. machine learning with deep neural networks (DNN). This work focuses on approximating high-dimensional smooth functions taking values in reflexive and typically infinite-dimensional Banach spaces. Our novel approach to this problem is fully algorithmic, combining DL, compressed sensing, orthogonal polynomials, and finite element discretization. We present a full theoretical analysis for DNN approximation with explicit guarantees on the error and sample complexity, and a clear accounting of all sources of error. We also provide numerical experiments demonstrating the efficiency of DL at approximating such high-dimensional functions from limited data in UQ applications.