证明了深度梯度流方法解偏微分方程的泛化误差可趋于零。
Convergence of the generalization error for deep gradient flow methods for PDEs
- 将泛化误差分解为近似误差与训练误差，分别分析其极限行为。
- 在神经元数量和训练时间趋近无穷时，误差整体趋于零。
- 适用于高维偏微分方程求解，为深度学习方法提供理论支撑。
本文旨在为深度梯度流方法（DGFMs）求解（高维）偏微分方程（PDEs）提供坚实的数学基础。我们将DGFMs的泛化误差分解为近似误差和训练误差。首先证明，在合理且可验证的假设下，满足条件的PDE解可被神经网络逼近，因此当神经元数量趋于无穷时，近似误差趋于零。随后，我们推导出在‘宽网络极限’下训练过程所遵循的梯度流，并分析该流在训练时间趋于无穷时的极限行为。上述结果共同表明，当神经元数量和训练时间均趋于无穷时，DGFMs的泛化误差趋于零。
原文摘要 · Abstract (English)
The aim of this article is to provide a firm mathematical foundation for the application of deep gradient flow methods (DGFMs) for the solution of (high-dimensional) partial differential equations (PDEs). We decompose the generalization error of DGFMs into an approximation and a training error. We first show that the solution of PDEs that satisfy reasonable and verifiable assumptions can be approximated by neural networks, thus the approximation error tends to zero as the number of neurons tends to infinity. Then, we derive the gradient flow that the training process follows in the ``wide network limit'' and analyze the limit of this flow as the training time tends to infinity. These results combined show that the generalization error of DGFMs tends to zero as the number of neurons and the training time tend to infinity.
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