"Deep Neural Network Approximation of High-Dimensional Hilbert-Valued Functions From Limited Data"Dexter, NickThe approximation of solutions to parameterized partial differential equations (PDEs) is a key challenge in computational uncertainty quantification problems for engineering and science. In these problems, the solution is a function of both the physical and parametric variables, taking values in an infinite-dimensional Hilbert or Banach space. Compressed sensing (CS) has recently been extended to allow approximation of Hilbert-valued functions, and exponential rates of convergence for CS-based polynomial approximations have been established. However, the application of deep neural networks (DNNs) to such problems presents many potential benefits over the polynomial based approach. DNNs with ReLU activation functions have been shown to emulate virtually every known approximation scheme, and optimal approximation rates have been established for such networks on a wide array of function classes including holomorphic and piecewise smooth functions. In this work we present theoretical results on learning high-dimensional Hilbert-valued functions from limited datasets by DNNs. We also present numerical results demonstrating the efficacy of such approaches on function and parametric PDE approximation problems. |
| http://univie.ac.at/projektservice-mathematik/e/talks/Dexter_2021-06_ABDM_ICCHA_submission.pdf |
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