We develop a general framework for data-driven approximation of input-output maps between infinite-dimensional spaces. The proposed approach is motivated by the recent successes of neural networks and deep learning, in combination with ideas from model reduction. This combination results in a neural network approximation which, in principle, is defined on infinite-dimensional spaces and, in practice, is robust to the dimension of finite-dimensional approximations of these spaces required for computation. For a class of input-output maps, and suitably chosen probability measures on the inputs, we prove convergence of the proposed approximation methodology. We also include numerical experiments which demonstrate the effectiveness of the method, showing convergence and robustness of the approximation scheme with respect to the size of the discretization, and compare it with existing algorithms from the literature; our examples include the mapping from coefficient to solution in a divergence form elliptic partial differential equation (PDE) problem, and the solution operator for viscous Burgers’ equation.

Keywords: approximation theory, deep learning, model reduction, neural networks, partial differential equations.

Kaushik Bhattacharya ^{1};
Bamdad Hosseini ^{2};
Nikola B. Kovachki ^{2};
Andrew M. Stuart ^{2}

@article{SMAI-JCM_2021__7__121_0, author = {Kaushik Bhattacharya and Bamdad Hosseini and Nikola B. Kovachki and Andrew M. Stuart}, title = {Model {Reduction} {And} {Neural} {Networks} {For} {Parametric} {PDEs}}, journal = {The SMAI Journal of computational mathematics}, pages = {121--157}, publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles}, volume = {7}, year = {2021}, doi = {10.5802/smai-jcm.74}, language = {en}, url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.74/} }

TY - JOUR AU - Kaushik Bhattacharya AU - Bamdad Hosseini AU - Nikola B. Kovachki AU - Andrew M. Stuart TI - Model Reduction And Neural Networks For Parametric PDEs JO - The SMAI Journal of computational mathematics PY - 2021 SP - 121 EP - 157 VL - 7 PB - Société de Mathématiques Appliquées et Industrielles UR - https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.74/ DO - 10.5802/smai-jcm.74 LA - en ID - SMAI-JCM_2021__7__121_0 ER -

%0 Journal Article %A Kaushik Bhattacharya %A Bamdad Hosseini %A Nikola B. Kovachki %A Andrew M. Stuart %T Model Reduction And Neural Networks For Parametric PDEs %J The SMAI Journal of computational mathematics %D 2021 %P 121-157 %V 7 %I Société de Mathématiques Appliquées et Industrielles %U https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.74/ %R 10.5802/smai-jcm.74 %G en %F SMAI-JCM_2021__7__121_0

Kaushik Bhattacharya; Bamdad Hosseini; Nikola B. Kovachki; Andrew M. Stuart. Model Reduction And Neural Networks For Parametric PDEs. The SMAI Journal of computational mathematics, Volume 7 (2021), pp. 121-157. doi : 10.5802/smai-jcm.74. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.74/

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