TY - JOUR
T1 - The Cost-Accuracy Trade-Off in Operator Learning with Neural Networks
AU - Hoop , Maarten V. de
AU - Huang , Daniel Zhengyu
AU - Qian , Elizabeth
AU - Stuart , Andrew M.
JO - Journal of Machine Learning
VL - 3
SP - 299
EP - 341
PY - 2022
DA - 2022/09
SN - 1
DO - http://doi.org/10.4208/jml.220509
UR - https://global-sci.org/intro/article_detail/jml/21030.html
KW - Computational partial differential equations, Surrogate modeling, Operator approximation, Neural networks, Computational complexity.
AB - <p style="text-align: justify;">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.</p>