Co-Design of Neural Operators and Stochastic Optimization Algorithms for Learning Surrogates for PDE-Constrained Optimization Under Uncertainty

University Of Texas At Austin, Austin TX

Investigators

Abstract

One of the great promises of modeling & simulation is that the models can serve as a basis for optimal decision making for complex physical systems. In many cases, the models for these systems are not fully known, and as a result contain uncertain parameters. This gives rise to problems in optimization under uncertainty (OUU). In the common situation in which the models take the form of partial differential equations (PDEs), for example characterizing fluid flow, solid mechanics, heat transfer, acoustics, and electromagnetics, the problems are known as PDE-constrained optimization under uncertainty (PDE-OUU). The recent development of so-called neural operators (NOs) promises to overcome the intractability of PDE-OUU problems by replacing the PDE model with a rapid-to-evaluate machine-learned surrogate. This project is developing a new integrated framework for both construction and training of NOs so that they better capture the mathematical structure of parameter and decision space and their impact on model outputs that drive decision making under uncertainty. These NOs will enable scalable, efficient, and accurate solution of PDE-OUU problems across a broad range of model-predictive decision-making under uncertainty problems of great societal or technological importance. Examples of such problems include those in climate change and natural hazard mitigation, design of new materials, operation of critical infrastructure, patient-specific disease treatment planning, and environmental observing system design. To facilitate the adoption of these algorithms, all software developed in this project will be released in open source form, building on existing successful libraries such as hIPPYlib. Two PhD students are being trained at the interdisciplinary interfaces of scientific machine learning, stochastic optimization, and PDE-constrained optimization. Despite their great importance in many technological, scientific, engineering, and medical fields, PDE-OUU problems are typically intractable when the uncertain parameter or decision variable dimensions are large, or when the models are large-scale and complex. However, many current methods for constructing NOs, as well as stochastic optimization methods to train them, do not exploit mathematical properties of the underlying models and as such are not sufficiently accurate to serve as proxies for the PDEs in OUU, particularly when the training data are limited due to the expense of obtaining them. To exploit mathematical properties of the PDE-governed maps from joint uncertain parameter and decision variable input space to model outputs that inform the optimization objective, this project seeks to extract knowledge of the geometry, smoothness, and intrinsic low dimensionality of the maps to synergistically co-design (1) training loss formulations, (2) neural architectures, and (3) stochastic optimization algorithms for training. The resulting NOs will exhibit greater accuracy with fewer PDE solves needed for training data, with accuracy measured over joint parameter–decision space. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.