In this study, we develop a novel multi-fidelity deep learning approach that transforms low-fidelity solution maps into high-fidelity ones by incorporating parametric space information into a standard autoencoder architecture. It is shown that, due to the integration of parametric space data, this method requires significantly less training data to achieve effective performance in predicting high-fidelity solution from the low-fidelity one. In this study, our focus is on a 2D steady-state heat transfer analysis in highly heterogeneous materials microstructure, where the spatial distribution of heat conductivity coefficients for two distinct materials is condensed. Subsequently, the boundary value problem is solved on the coarsest grid using a pre-trained physics-informed neural operator network. Afterward, the calculated low-fidelity result is upscaled using the newly designed enhanced autoencoder. The novelty of the developed enhanced autoencoder lies in the concatenation of heat conductivity maps of different resolutions to the decoder segment in distinct steps. We then compare the outcomes of developed algorithm with the corresponding finite element results, standard U-Net architecture as well as other upscaling approaches such as interpolation functions of varying orders and feedforward neural networks (FFNN). The analysis of the results based on the new approach demonstrates superior performance compared to other approaches in terms of computational cost and error on the test cases. Therefore, as a potential supplement to neural operators networks, our architecture upscales low-fidelity solutions to high-fidelity ones while preserving critical details that are often lost in conventional upscaling methods, especially at sharp interfaces, such as those encountered with interpolation methods.

翻译：本研究开发了一种新颖的多保真深度学习方法，通过将参数空间信息融入标准自编码器架构，将低保真解映射转换为高保真解。研究表明，由于参数空间数据的整合，该方法在从低保真解预测高保真解时所需训练数据显著减少。本研究聚焦于高度非均匀材料微结构中的二维稳态传热分析，其中两种不同材料的热导系数空间分布被压缩。随后，利用预训练的物理信息神经算子网络在最粗网格上求解边值问题，并通过新设计的增强型自编码器对计算所得低保真结果进行升尺度处理。该增强型自编码器的创新之处在于，在不同解码器阶段逐步拼接不同分辨率的热导率图。我们将所提算法结果与有限元结果、标准U-Net架构以及其他升尺度方法（如不同阶次插值函数和前馈神经网络）进行对比。基于新方法的结果分析表明，在测试案例的计算成本与误差方面，该方法优于其他方法。因此，作为神经算子网络的潜在补充，本架构在将低保真解升尺度为高保真解的同时，保留了传统升尺度方法（尤其是插值法在尖锐界面处）常丢失的关键细节。