Model Reduction and Neural Networks for Parametric PDEs

05/07/2020
by   Kaushik Bhattacharya, et al.
39

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. Numerically we demonstrate the effectiveness of the method on a class of parametric elliptic PDE problems, showing convergence and robustness of the approximation scheme with respect to the size of the discretization, and compare our method with existing algorithms from the literature.

READ FULL TEXT
research
05/20/2020

The Random Feature Model for Input-Output Maps between Banach Spaces

Well known to the machine learning community, the random feature model, ...
research
05/28/2022

Approximation of Functionals by Neural Network without Curse of Dimensionality

In this paper, we establish a neural network to approximate functionals,...
research
10/03/2019

On Universal Approximation by Neural Networks with Uniform Guarantees on Approximation of Infinite Dimensional Maps

The study of universal approximation of arbitrary functions f: X→Y by ne...
research
05/21/2018

Category coding with neural network application

In many applications of neural network, it is common to introduce huge a...
research
06/10/2023

Any-dimensional equivariant neural networks

Traditional supervised learning aims to learn an unknown mapping by fitt...
research
12/14/2021

Simplicial approximation to CW complexes in practice

We describe an algorithm that takes as an input a CW complex and returns...
research
08/28/2023

Solving parametric elliptic interface problems via interfaced operator network

Learning operator mapping between infinite-dimensional Banach spaces via...

Please sign up or login with your details

Forgot password? Click here to reset