论文标题
线性偏微分方程系统的高斯工艺先验，具有恒定系数
Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficients
论文作者
论文摘要
部分微分方程（PDE）是建模物理系统的重要工具，将它们包括在机器学习模型中是合并物理知识的重要方法。鉴于任何具有恒定系数的线性PDE系统，我们提出了一个高斯工艺（GP）先验的家族，我们称之为EPGP，因此所有实现都是该系统的确切解决方案。我们将Ehrenpreis-Palamodov的基本原理（作为非线性傅立叶变换起）来构建GP内核镜像GP的标准光谱方法。我们的方法可以从任何数据（例如噪声测量）或定义定义的初始条件和边界条件等任何数据中推断出线性PDE系统的可能解决方案。构建EPGP-priors是算法，通常适用，并带有稀疏版本（S-EPGP），该版本可以学习相关的光谱频率，并且在大数据集中效果更好。我们展示了我们对三个PDE系统族的方法，热量方程，波动方程和麦克斯韦方程，在某些实验中，我们在计算时间和精度下以几个数量级来改善了计算时间和精度的最新状态。
Partial differential equations (PDEs) are important tools to model physical systems and including them into machine learning models is an important way of incorporating physical knowledge. Given any system of linear PDEs with constant coefficients, we propose a family of Gaussian process (GP) priors, which we call EPGP, such that all realizations are exact solutions of this system. We apply the Ehrenpreis-Palamodov fundamental principle, which works as a non-linear Fourier transform, to construct GP kernels mirroring standard spectral methods for GPs. Our approach can infer probable solutions of linear PDE systems from any data such as noisy measurements, or pointwise defined initial and boundary conditions. Constructing EPGP-priors is algorithmic, generally applicable, and comes with a sparse version (S-EPGP) that learns the relevant spectral frequencies and works better for big data sets. We demonstrate our approach on three families of systems of PDEs, the heat equation, wave equation, and Maxwell's equations, where we improve upon the state of the art in computation time and precision, in some experiments by several orders of magnitude.
