用核方法融合物理规律，提升模型精度与计算效率。
Physics-informed kernel learning
- 将物理规律作为核函数约束，通过傅里叶近似实现可计算估计
- 在含噪声边界条件下，比传统求解器和PINN更准确且更快
- 提供收敛速度理论保证，适合需要物理一致性的科学建模
物理信息机器学习通常通过包含数据驱动项和偏微分方程（PDE）正则化的损失函数来整合物理先验。本文将问题建模为核回归任务，利用傅里叶方法近似相关核函数，提出一种可计算的物理信息风险最小化估计器，称为物理信息核学习（PIKL）。该框架提供理论保障，可量化物理先验对收敛速度的影响。数值实验表明，PIKL在混合建模和PDE求解中表现优异，其精度和计算时间均优于物理信息神经网络（PINN）。此外，在边界条件含噪声的情形下，PIKL超越传统PDE求解器。
原文摘要 · Abstract (English)
Physics-informed machine learning typically integrates physical priors into the learning process by minimizing a loss function that includes both a data-driven term and a partial differential equation (PDE) regularization. Building on the formulation of the problem as a kernel regression task, we use Fourier methods to approximate the associated kernel, and propose a tractable estimator that minimizes the physics-informed risk function. We refer to this approach as physics-informed kernel learning (PIKL). This framework provides theoretical guarantees, enabling the quantification of the physical prior's impact on convergence speed. We demonstrate the numerical performance of the PIKL estimator through simulations, both in the context of hybrid modeling and in solving PDEs. In particular, we show that PIKL can outperform physics-informed neural networks in terms of both accuracy and computation time. Additionally, we identify cases where PIKL surpasses traditional PDE solvers, particularly in scenarios with noisy boundary conditions.
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