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需要高保真数据，而生成这些数据的代价高昂。为减少甚至消除对高保真数据的依赖，我们提出了一种新型多保真架构，该架构基于低保真与高清真解共享的特征空间。在特征空间中，通过约束低清真与高清真解投影之间的相对距离，使得两者的投影保持邻近。特征空间由编码器表示，并通过解码器实现到原始解空间的映射。所提出的多保真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得到了验证。