用多保真度信息提升非线性偏微分方程求解的高斯过程精度
Multifidelity Gaussian process regression for solving nonlinear partial differential equations
- 基于多保真度模拟构建可微分非平稳核，融合低保真数据
- 通过协克里金框架推导高保真核与均值，提升预测准确性
- 适用于物理信息建模场景，尤其适合数据稀疏的工程仿真
使用核方法求解非线性偏微分方程（PDEs）为传统数值求解器提供了有前景的替代方案。然而，其性能高度依赖核函数选择。本文针对信息本身具有多保真度的特点，提出一种基于协克里金的核学习方法，利用多保真度模拟的实证数据。首先，将可微分的非平稳核拟合至由低保真度模拟获得的经验核；其次，推导出带有估计超参数的高保真核，并在多保真度框架下构建相应的高保真均值。这些组件可直接用于高斯过程框架求解PDE。最后，我们在Burgers方程上验证了该物理信息方法的有效性。
原文摘要 · Abstract (English)
Solving nonlinear partial differential equations (PDEs) using kernel methods offers a compelling alternative to traditional numerical solvers. However, the performance of these methods strongly depends on the choice of kernel. In this work, as the available information is inherently multifidelity, we propose a kernel learning approach based on cokriging, leveraging empirical information from multifidelity simulations. In the first step, we fit a differentiable non-stationary kernel to an empirical kernel obtained from low-fidelity simulations. In the second step, we derive a high-fidelity kernel with estimated hyperparameters, and construct a corresponding high-fidelity mean using the multifidelity framework. These components can then be used within a Gaussian process framework for solving PDEs. Finally, we demonstrate the performance of the proposed physics-informed method on the Burgers' equation.
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