(untitled) (CSIAM Transactions on Applied Mathematics, 2020)
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.