Project Details
Projekt
Operator Learning for Optimal Control: Approximation and Statistical Theory
Subject Area
Mathematics
Term
since 2024
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 543965776
This proposal focuses on developing a comprehensive convergence analysis for operator learning, an emerging methodology used for the efficient approximation of data-to-solution maps for parameter dependent partial differential equations (PDEs). Our motivation stems from solving problems in optimal control, traditionally reliant on sequential, costly numerical PDE solves. To this end we will provide a complete error analysis aiming to dissect and understand the bias-variance trade-off and breaking the curse of dimensionality in operator learning. Additionally, the insights gained are expected to address computational challenges in parameter uncertainty quantification and estimation, prevalent across various applied mathematics domains. The project will delve into the development of suitable network architectures, focusing on their expressivity and error analysis, with a special focus on guaranteeing small covering numbers for the function class corresponding to all network realizations. Building upon this analysis, we aim to establish a robust statistical learning framework. This framework will extend the principles of empirical risk minimization to operator learning, and yield an analysis of regressing nonlinear mappings between infinite dimensional spaces. Key tools to achieve these goals will be the exploitation of low-dimensional structures stemming from known high regularity of parameter-to-solution maps, as well as the separation of the in- and output of the operator into important low-frequency, and less important high-frequency parts. This is achieved by a so-called encoder/decoder architecture which allows to represent in- and outputs in stable representation systems such as wavelets. The practical aspect of the project involves integrating operator learning models into optimization frameworks, in order to achieve a significant reduction of the computational effort for solving PDE-constrained optimal control problems.
DFG Programme
Priority Programmes
Subproject of
SPP 2298:
Theoretical Foundations of Deep Learning
