Physics-informed neural networks have emerged as an alternative method for solving partial differential equations. However, for complex problems, the training of such networks can still require high-fidelity data which can be expensive to generate. To reduce or even eliminate the dependency on high-fidelity data, we propose a novel multi-fidelity architecture which is based on a feature space shared by the low- and high-fidelity solutions. In the feature space, the projections of the low-fidelity and high-fidelity solutions are adjacent by constraining their relative distance. The feature space is represented with an encoder and its mapping to the original solution space is effected through a decoder. The proposed multi-fidelity approach is validated on forward and inverse problems for steady and unsteady problems described by partial differential equations.

翻译：物理信息神经网络已成为求解偏微分方程的一种替代方法。然而，对于复杂问题，此类网络的训练仍可能依赖高保真度数据，而生成这类数据的成本较高。为降低甚至消除对高保真度数据的依赖，我们提出了一种新型多保真度架构，该架构基于低保真度解与高保真度解共享的特征空间。在该特征空间中，通过约束低保真度解与高保真度解投影之间的相对距离，使两者相互邻近。特征空间由编码器表示，其到原始解空间的映射通过解码器实现。所提出的多保真度方法在由偏微分方程描述的稳态与非稳态问题的正问题与反问题中均得到了验证。