A1052
Title: Statistical learning theory for neural operators
Authors: Jakob Zech - Heidelberg University (Germany) [presenting]
Niklas Reinhardt - Heidelberg University (Germany)
Sven Wang - HU Berlin (Germany)
Abstract: Convergence rates for neural network-based operator surrogates are discussed, which approximate smooth maps between infinite-dimensional Hilbert spaces. Such surrogates have a wide range of applications and can be used in uncertainty quantification and parameter estimation problems in fields such as classical mechanics, fluid mechanics, electrodynamics, earth sciences, etc. The operator input represents the problem configuration and models initial conditions, material properties, forcing terms, and/or the domain of a partial differential equation (PDE) describing the underlying physics. The operator output is the corresponding PDE solution. The analysis demonstrates that, under suitable smoothness assumptions, the empirical risk minimizer for specific neural network architectures can overcome the curse of dimensionality in terms of required network parameters and the input-output pairs needed for training.
