The development of efficient surrogates of partial differential equations (PDEs) is a critical step towards scalable modeling of complex, multiscale systems-of-systems. Convolutional neural networks (CNNs) have gained popularity as the basis for such surrogate models due to their success in capturing high-dimensional input-output mappings and the negligible cost of a forward pass. However, the high cost of generating training data -- typically via classical numerical solvers -- raises the question of whether these models are worth pursuing over more straightforward alternatives with well-established theoretical foundations, such as Monte Carlo methods. To reduce the cost of data generation, we propose training a CNN surrogate model on a mixture of numerical solutions to both the $d$-dimensional problem and its ($d-1$)-dimensional approximation, taking advantage of the efficiency savings guaranteed by the curse of dimensionality. We demonstrate our approach on a multiphase flow test problem, using transfer learning to train a dense fully-convolutional encoder-decoder CNN on the two classes of data. Numerical results from a sample uncertainty quantification task demonstrate that our surrogate model outperforms Monte Carlo with several times the data generation budget.

本研究解决了高成本训练数据生成的问题，提出通过混合数值解训练卷积神经网络（CNN）替代模型的创新方法。研究表明，该模型在多相流测试问题上超越传统的蒙特卡洛方法，在数据生成预算上实现了显著节省。此发现为复杂多尺度系统的建模提供了更高效的替代方案。

多维数据的迁移学习：基于神经网络的替代建模新方法