The Cost-Accuracy Trade-Off In Operator Learning With Neural Networks

03/24/2022
by   Maarten V. de Hoop, et al.
0

The term `surrogate modeling' in computational science and engineering refers to the development of computationally efficient approximations for expensive simulations, such as those arising from numerical solution of partial differential equations (PDEs). Surrogate modeling is an enabling methodology for many-query computations in science and engineering, which include iterative methods in optimization and sampling methods in uncertainty quantification. Over the last few years, several approaches to surrogate modeling for PDEs using neural networks have emerged, motivated by successes in using neural networks to approximate nonlinear maps in other areas. In principle, the relative merits of these different approaches can be evaluated by understanding, for each one, the cost required to achieve a given level of accuracy. However, the absence of a complete theory of approximation error for these approaches makes it difficult to assess this cost-accuracy trade-off. The purpose of the paper is to provide a careful numerical study of this issue, comparing a variety of different neural network architectures for operator approximation across a range of problems arising from PDE models in continuum mechanics.

READ FULL TEXT

page 10

page 11

page 12

page 13

page 19

research
04/22/2021

Bayesian Numerical Methods for Nonlinear Partial Differential Equations

The numerical solution of differential equations can be formulated as an...
research
01/30/2023

Fast Resolution Agnostic Neural Techniques to Solve Partial Differential Equations

Numerical approximations of partial differential equations (PDEs) are ro...
research
01/17/2023

Operator Learning Framework for Digital Twin and Complex Engineering Systems

With modern computational advancements and statistical analysis methods,...
research
10/06/2022

Residual-based error correction for neural operator accelerated infinite-dimensional Bayesian inverse problems

We explore using neural operators, or neural network representations of ...
research
01/13/2021

Bayesian neural networks for weak solution of PDEs with uncertainty quantification

Solving partial differential equations (PDEs) is the canonical approach ...
research
04/07/2021

Rademacher Complexity and Numerical Quadrature Analysis of Stable Neural Networks with Applications to Numerical PDEs

Methods for solving PDEs using neural networks have recently become a ve...
research
02/20/2020

Comparing recurrent and convolutional neural networks for predicting wave propagation

Dynamical systems can be modelled by partial differential equations and ...

Please sign up or login with your details

Forgot password? Click here to reset