In this work, we consider the general problem of constructing a neural network surrogate model using multi-fidelity information. Motivated by rigorous error and complexity estimates for ReLU neural networks, given an inexpensive low-fidelity and an expensive high-fidelity computational model, we present a residual multi-fidelity computational framework that formulates the correlation between models as a residual function, a possibly non-linear mapping between 1) the shared input space of the models together with the low-fidelity model output and 2) the discrepancy between the two model outputs. To accomplish this, we train two neural networks to work in concert. The first network learns the residual function on a small set of high-fidelity and low-fidelity data. Once trained, this network is used to generate additional synthetic high-fidelity data, which is used in the training of a second network. This second network, once trained, acts as our surrogate for the high-fidelity quantity of interest. We present three numerical examples to demonstrate the power of the proposed framework. In particular, we show that dramatic savings in computational cost may be achieved when the output predictions are desired to be accurate within small tolerances.

翻译：本文考虑利用多保真信息构建神经网络代理模型的一般性问题。受ReLU神经网络严格误差与复杂度估计的启发，针对计算成本低廉的低保真模型与计算成本高昂的高保真模型，我们提出一种残差多保真计算框架。该框架将模型间的关联性表述为残差函数——即模型共享输入空间与低保真模型输出之间的可能非线性映射，以及两模型输出间的差异。为此，我们训练两个神经网络协同工作：第一个网络通过少量高保真与低保真数据学习残差函数；训练完成后，该网络用于生成额外的合成高保真数据，以辅助第二个网络的训练。经训练后，第二个网络充当高保真目标量的代理模型。我们通过三个数值算例验证该框架的有效性，特别表明当输出预测需达到较小容差精度时，该方法可大幅节省计算成本。