DMS Applied and Computational Mathematics Seminar

Speaker: Nick Dexter, Florida State University

Title: Learning High-Dimensional Banach-Valued Functions from Limited Data with Deep Neural Networks

Abstract: Reconstructing high-dimensional functions from few samples is important for uncertainty quantification in computational science. Deep learning has achieved impressive results in parameterized PDE problems with solutions in Hilbert or Banach spaces. This work proposes a novel algorithmic approach using DL, compressed sensing, orthogonal polynomials, and finite elements to approximate smooth functions in infinite-dimensional Banach spaces. Theoretical analysis provides explicit guarantees on error and sample complexity, and numerical experiments demonstrate accurate approximations on challenging benchmark problems.