Efficient PDE-Constrained optimization under high-dimensional
  uncertainty using derivative-informed neural operators
Dingcheng Luo Thomas O'Leary-Roseberry Peng Chen Omar Ghattas
We propose a novel machine learning framework for solving optimization
problems governed by large-scale partial differential equations (PDEs) with
high-dimensional random parameters. Such optimization under uncertainty (OUU)
problems may be computational prohibitive using classical methods, particularly
when a large number of samples is needed to evaluate risk measures at every
iteration of an optimization algorithm, where each sample requires the solution
of an expensive-to-solve PDE. To address this challenge, we propose a new
neural operator approximation of the PDE solution operator that has the
combined merits of (1) accurate approximation of not only the map from the
joint inputs of random parameters and optimization variables to the PDE state,
but also its derivative with respect to the optimization variables, (2)
efficient construction of the neural network using reduced basis architectures
that are scalable to high-dimensional OUU problems, and (3) requiring only a
limited number of training data to achieve high accuracy for both the PDE
solution and the OUU solution. We refer to such neural operators as multi-input
reduced basis derivative informed neural operators (MR-DINOs). We demonstrate
the accuracy and efficiency our approach through several numerical experiments,
i.e. the risk-averse control of a semilinear elliptic PDE and the steady state
Navier--Stokes equations in two and three spatial dimensions, each involving
random field inputs. Across the examples, MR-DINOs offer $10^{3}$--$10^{7}
\times$ reductions in execution time, and are able to produce OUU solutions of
comparable accuracies to those from standard PDE based solutions while being
over $10 \times$ more cost-efficient after factoring in the cost of
construction.