高维偏微分方程求解新方法，兼顾精度与不确定性量化。
PDE-DKL: PDE-constrained deep kernel learning in high dimensionality
- 用神经网络降维，再用高斯过程在物理约束下回归。
- 数据少时仍保持高精度，且能给出可信的不确定性估计。
- 适合科学计算中数据稀缺的高维复杂系统建模。
针对高维偏微分方程（PDE）求解中数据稀缺与计算复杂度高的问题，本文提出一种受偏微分方程约束的深度核学习框架（PDE-DKL）。该方法结合深度学习（DL）与高斯过程（GP），利用神经网络学习高维输入的低维潜在表示，降低问题复杂度；随后在满足控制方程的条件下，由高斯过程进行核回归，实现精确求解与可解释的不确定性量化。实验表明，该方法在数据有限时仍能保持高精度，兼具高计算效率与鲁棒性，适用于科学与工程中的复杂高维PDE建模任务。
原文摘要 · Abstract (English)
Many physics-informed machine learning methods for PDE-based problems rely on Gaussian processes (GPs) or neural networks (NNs). However, both face limitations when data are scarce and the dimensionality is high. Although GPs are known for their robust uncertainty quantification in low-dimensional settings, their computational complexity becomes prohibitive as the dimensionality increases. In contrast, while conventional NNs can accommodate high-dimensional input, they often require extensive training data and do not offer uncertainty quantification. To address these challenges, we propose a PDE-constrained Deep Kernel Learning (PDE-DKL) framework that combines DL and GPs under explicit PDE constraints. Specifically, NNs learn a low-dimensional latent representation of the high-dimensional PDE problem, reducing the complexity of the problem. GPs then perform kernel regression subject to the governing PDEs, ensuring accurate solutions and principled uncertainty quantification, even when available data are limited. This synergy unifies the strengths of both NNs and GPs, yielding high accuracy, robust uncertainty estimates, and computational efficiency for high-dimensional PDEs. Numerical experiments demonstrate that PDE-DKL achieves high accuracy with reduced data requirements. They highlight its potential as a practical, reliable, and scalable solver for complex PDE-based applications in science and engineering.
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