用跨维度迁移学习降低微分方程代理模型的数据成本。
Transfer Learning on Multi-Dimensional Data: A Novel Approach to Neural Network-Based Surrogate Modeling
- 在高维与低维问题解上联合训练CNN，利用维度降低的效率优势。
- 相同数据量下，代理模型误差比蒙特卡洛方法低3倍以上。
- 适合需要高效不确定性量化的大规模复杂系统建模者。
构建偏微分方程（PDE）的高效代理模型是实现复杂多尺度系统可扩展建模的关键步骤。卷积神经网络（CNN）因其在捕捉高维输入输出映射方面的成功及前向传播开销极小，被广泛用于此类代理模型。然而，生成训练数据的成本极高——通常依赖经典数值求解器。这引发疑问：这类模型是否值得投入，相比具有成熟理论基础的蒙特卡洛方法。为降低数据生成成本，我们提出在$d$维问题及其$(d-1)$维近似解的混合数据上训练CNN代理模型，利用维度降低带来的效率优势。我们在多相流测试问题上验证该方法，采用迁移学习在密集全卷积编码器-解码器结构上训练模型。在一次不确定性量化任务中，结果表明，在数倍于蒙特卡洛的数据生成预算下，我们的代理模型表现更优。
原文摘要 · Abstract (English)
The development of efficient surrogates for 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.
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