English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We propose a novel machine learning framework for solving optimizationproblems governed by large-scale partial differential equations (PDEs) withhigh-dimensional random parameters. Such optimization under uncertainty (OUU)problems may be computational prohibitive using classical methods, particularlywhen a large number of samples is needed to evaluate risk measures at everyiteration of an optimization algorithm, where each sample requires the solutionof an expensive-to-solve PDE. To address this challenge, we propose a newneural operator approximation of the PDE solution operator that has thecombined merits of (1) accurate approximation of not only the map from thejoint 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 architecturesthat are scalable to high-dimensional OUU problems, and (3) requiring only alimited number of training data to achieve high accuracy for both the PDEsolution and the OUU solution. We refer to such neural operators as multi-inputreduced basis derivative informed neural operators (MR-DINOs). We demonstratethe accuracy and efficiency our approach through several numerical experiments,i.e. the risk-averse control of a semilinear elliptic PDE and the steady stateNavier--Stokes equations in two and three spatial dimensions, each involvingrandom field inputs. Across the examples, MR-DINOs offer $10^{3}$--$10^{7}\times$ reductions in execution time, and are able to produce OUU solutions ofcomparable accuracies to those from standard PDE based solutions while beingover $10 \times$ more cost-efficient after factoring in the cost ofconstruction.

Efficient PDE-Constrained optimization under high-dimensional uncertainty using derivative-informed neural operators