Machine learning based solvers have garnered much attention in physical simulation and scientific computing, with a prominent example, physics-informed neural networks (PINNs). However, PINNs often struggle to solve high-frequency and multi-scale PDEs, which can be due to spectral bias during neural network training. To address this problem, we resort to the Gaussian process (GP) framework. To flexibly capture the dominant frequencies, we model the power spectrum of the PDE solution with a student $t$ mixture or Gaussian mixture. We apply the inverse Fourier transform to obtain the covariance function (by Wiener-Khinchin theorem). The covariance derived from the Gaussian mixture spectrum corresponds to the known spectral mixture kernel. Next, we estimate the mixture weights in the log domain, which we show is equivalent to placing a Jeffreys prior. It automatically induces sparsity, prunes excessive frequencies, and adjusts the remaining toward the ground truth. Third, to enable efficient and scalable computation on massive collocation points, which are critical to capture high frequencies, we place the collocation points on a grid, and multiply our covariance function at each input dimension. We use the GP conditional mean to predict the solution and its derivatives so as to fit the boundary condition and the equation itself. As a result, we can derive a Kronecker product structure in the covariance matrix. We use Kronecker product properties and multilinear algebra to promote computational efficiency and scalability, without low-rank approximations. We show the advantage of our method in systematic experiments. The code is released at \url{https://github.com/xuangu-fang/Gaussian-Process-Slover-for-High-Freq-PDE}.

翻译：基于机器学习的求解器在物理模拟和科学计算领域备受关注，其中物理信息神经网络（PINNs）是一个典型代表。然而，PINNs 在求解高频和多尺度偏微分方程时往往遇到困难，这可能是由于神经网络训练中的谱偏差问题。为了解决这一问题，本文转向高斯过程（GP）框架。为了灵活捕捉主要频率，我们采用学生 $t$ 混合或高斯混合模型对 PDE 解的功率谱进行建模，并应用傅里叶逆变换得到协方差函数（依据 Wiener-Khinchin 定理）。由高斯混合谱导出的协方差对应于已知的谱混合核。接着，我们在对数域中估计混合权重，并证明这等价于施加 Jeffreys 先验，能自动诱导稀疏性、剪除多余频率，并将剩余频率调整至真实解方向。第三，为了在海量配置点上实现高效可扩展的计算——这对捕捉高频至关重要——我们将配置点放置在网格上，并在每个输入维度上乘以协方差函数。利用 GP 条件均值预测解及其导数，以拟合边界条件和方程本身。由此，协方差矩阵可推导出 Kronecker 积结构，我们利用 Kronecker 积性质与多重线性代数提升计算效率与可扩展性，无需低秩近似。通过系统实验证明了本方法的优势。代码已发布于 \url{https://github.com/xuangu-fang/Gaussian-Process-Slover-for-High-Freq-PDE}。