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

Many-query problems, arising from uncertainty quantification, Bayesianinversion, Bayesian optimal experimental design, and optimization underuncertainty-require numerous evaluations of a parameter-to-output map. Theseevaluations become prohibitive if this parametric map is high-dimensional andinvolves expensive solution of partial differential equations (PDEs). To tacklethis challenge, we propose to construct surrogates for high-dimensionalPDE-governed parametric maps in the form of projected neural networks thatparsimoniously capture the geometry and intrinsic low-dimensionality of thesemaps. Specifically, we compute Jacobians of these PDE-based maps, and projectthe high-dimensional parameters onto a low-dimensional derivative-informedactive subspace; we also project the possibly high-dimensional outputs ontotheir principal subspace. This exploits the fact that many high-dimensionalPDE-governed parametric maps can be well-approximated in low-dimensionalparameter and output subspace. We use the projection basis vectors in theactive subspace as well as the principal output subspace to construct theweights for the first and last layers of the neural network, respectively. Thisfrees us to train the weights in only the low-dimensional layers of the neuralnetwork. The architecture of the resulting neural network captures to firstorder, the low-dimensional structure and geometry of the parametric map. Wedemonstrate that the proposed projected neural network achieves greatergeneralization accuracy than a full neural network, especially in the limitedtraining data regime afforded by expensive PDE-based parametric maps. Moreover,we show that the number of degrees of freedom of the inner layers of theprojected network is independent of the parameter and output dimensions, andhigh accuracy can be achieved with weight dimension independent of thediscretization dimension.

Derivative-Informed Projected Neural Networks for High-Dimensional Parametric Maps Governed by PDEs