Neural and spectral operator surrogates

Video record

In this talk we discuss expression rates for neural network-based operator surrogates, which are employed to approximate smooth maps between infinite-dimensional 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. Our analysis is based on representing the operator in- and outputs in stable bases and exploiting the resulting sparsity created by a separation in high- and low-frequency features. We show that algebraic convergence rates that are free from the curse of dimension can be achieved.

Jakob Zech is a Junior Professor at the Interdisciplinary Center for Scientific Computing at Heidelberg University, a position he holds since April 2020. Before this, he served as a postdoctoral fellow at the Massachusetts Institute of Technology (MIT) in the group of Youssef Marzouk, supported by an SNSF fellowship. He earned his PhD in Mathematics from ETH Zurich in December 2018, where his doctoral research, supervised by Christoph Schwab, focused on the Sparse-Grid Approximation of High-Dimensional Parametric Partial Differential Equations (PDEs).
Dr. Zech specializes in the methodological foundations and theoretical aspects of forward and inverse problems in Uncertainty Quantification. His research aims to develop and analyze algorithms for high-dimensional approximation and sampling for scientific applications, integrating numerics with machine learning tools such as neural networks and measure transport methods. Before his PhD, Jakob Zech completed a Master of Science in Applied Mathematics and a Bachelor of Science in Technical Mathematics at ETH Zurich and TU Wien, respectively.