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

翻译：在本研究中，我们探讨了利用多保真度信息构建神经网络代理模型的一般性问题。受ReLU神经网络误差复杂度估计的启发，我们将廉价低保真度模型与昂贵高保真度模型之间的相关性表述为一个可能非线性的残差函数。该函数定义了两种映射关系：1）模型共享输入空间与低保真度模型输出之间的映射；2）两个模型输出间差异的映射。该计算框架通过协同训练两个神经网络实现：第一个网络在少量高、低保真度数据上学习残差函数；训练完成后，该网络用于生成额外的合成高保真度数据，进而训练第二个网络。最终训练完成的第二个网络将作为我们关注的高保真度量的代理模型。我们通过四个数值算例展示了所提框架的效能，证明当输出预测需要在较小容差范围内保持精确时，该方法能显著降低计算成本。