Neural and spectral operator surrogates

Jakob Zech (Universität Heidelberg)

Abstract: In this talk we discuss the use of operator surrogates to 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. In this case, 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 output of the operator is the corresponding PDE solution. We will also present an alternative approach using interpolation, which allows for deterministic construction and eliminates the need for training the network weights. In both cases, algebraic and dimension-independent convergence rates are obtained.