@Article{CSIAM-AM-1-715,
author = {Wang , Yating and Lin , Guang},
title = {MFPC-Net: Multi-Fidelity Physics-Constrained Neural Process},
journal = {CSIAM Transactions on Applied Mathematics},
year = {2020},
volume = {1},
number = {4},
pages = {715--739},
abstract = {<p style="text-align: justify;">Recently, there are numerous works on developing surrogate models under
the idea of deep learning. Many existing approaches use high fidelity input and solution labels for training. However, it is usually difficult to acquire sufficient high fidelity
data in practice. In this work, we propose a network which can utilize computational
cheap low-fidelity data together with limited high-fidelity data to train surrogate models, where the multi-fidelity data are generated from multiple underlying models. The
network takes a context set as input (physical observation points, low fidelity solution at observed points) and output (high fidelity solution at observed points) pairs. It
uses the neural process to learn a distribution over functions conditioned on context
sets and provide the mean and standard deviation at target sets. Moreover, the proposed framework also takes into account the available physical laws that govern the
data and imposes them as constraints in the loss function. The multi-fidelity physics-constrained network (MFPC-Net) (1) takes datasets obtained from multiple models
at the same time in the training, (2) takes advantage of the available physical information, (3) learns a stochastic process which can encode prior beliefs about the correlation
between two fidelity with a few observations, and (4) produces predictions with uncertainty. The ability of representing a class of functions is ensured by the property
of neural process and is achieved by the global latent variables in the neural network.
Physical constraints are added to the loss using Lagrange multipliers. An algorithm
to optimize the loss function is proposed to effectively train the parameters in the network on an ad hoc basis. Once trained, one can obtain fast evaluations of the entire
domain of interest given a few observation points from a new low- and high-fidelity
model pair. Particularly, one can further identify the unknown parameters such as
permeability fields in elliptic PDEs with a simple modification of the network. Several
numerical examples for both forward and inverse problems are presented to demonstrate the performance of the proposed method.</p>},
issn = {2708-0579},
doi = {https://doi.org/10.4208/csiam-am.2020-0039},
url = {http://global-sci.org/intro/article_detail/csiam-am/18543.html}
}