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. This method's integration of parametric space information significantly reduces the need for training data to effectively predict high-fidelity solutions from low-fidelity ones. In this study, we examine a two-dimensional steady-state heat transfer analysis within a highly heterogeneous materials microstructure. The heat conductivity coefficients for two different materials are condensed from a 101 x 101 grid to smaller grids. We then solve the boundary value problem on the coarsest grid using a pre-trained physics-informed neural operator network known as Finite Operator Learning (FOL). The resulting low-fidelity solution is subsequently upscaled back to a 101 x 101 grid using a 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. Hence the developed algorithm is named microstructure-embedded autoencoder (MEA). We compare the MEA outcomes with those from finite element methods, the standard U-Net, and various other upscaling techniques, including interpolation functions and feedforward neural networks (FFNN). Our analysis shows that MEA outperforms these methods in terms of computational efficiency and error on test cases. As a result, the MEA serves as a potential supplement to neural operator networks, effectively upscaling low-fidelity solutions to high fidelity while preserving critical details often lost in traditional upscaling methods, particularly at sharp interfaces like those seen with interpolation.

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。该方法对参数空间信息的整合显著降低了从低保真解有效预测高保真解所需的训练数据量。我们针对高度异质材料微结构中的二维稳态传热问题进行了分析，将两种不同材料的热导率系数从101×101网格压缩至更小网格。通过预训练的物理信息神经算子网络——即有限算子学习（FOL），在最粗网格上求解边值问题，并利用新设计的增强型自编码器将所得低保真解上采样回101×101网格。该增强型自编码器的创新之处在于将不同分辨率的热导率图分步拼接至解码器部分，因此所提算法被命名为微结构嵌入自编码器（MEA）。我们将MEA的结果与有限元方法、标准U-Net以及包括插值函数和前馈神经网络（FFNN）在内的多种上采样技术进行对比。分析表明，在测试案例中，MEA在计算效率和误差方面均优于这些方法。因此，MEA可作为神经算子网络的有效补充，在将低保真解上采样至高保真解的同时，保留传统上采样方法（如插值法）在尖锐界面处常丢失的关键细节。