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 then apply the inverse Fourier transform to obtain the covariance function (according to the Wiener-Khinchin theorem). The covariance derived from the Gaussian mixture spectrum corresponds to the known spectral mixture kernel. We are the first to discover its rationale and effectiveness for PDE solving. 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 greatly promote computational efficiency and scalability, without any low-rank approximations. We show the advantage of our method in systematic experiments.

翻译：基于机器学习的求解器在物理模拟与科学计算领域备受关注，其中物理信息神经网络（PINNs）是典型代表。然而，PINNs在求解高频与多尺度偏微分方程时往往存在困难，这归因于神经网络训练中的谱偏差问题。针对该问题，我们转向高斯过程（GP）框架。为灵活捕捉主导频率，我们采用学生t混合模型或高斯混合模型对偏微分方程解的功率谱进行建模。随后依据维纳-辛钦定理，通过逆傅里叶变换获得协方差函数。高斯混合谱推导出的协方差函数对应已知的谱混合核函数——我们首次发现其在偏微分方程求解中的理论依据与有效性。接着，我们在对数域估计混合权重，并证明其等价于施加杰弗里斯先验。该机制可自动诱导稀疏性、剔除冗余频率，并将剩余频率调整至真实值方向。第三，为在海量配置点上实现高效可扩展计算（这对捕获高频至关重要），我们将配置点规则网格化，并在各输入维度上对协方差函数进行乘积操作。通过GP条件均值预测解及其导数，使其满足边界条件与方程约束。由此可推导出协方差矩阵的克罗内克积结构，我们利用克罗内克积特性与多重线性代数大幅提升计算效率与可扩展性，且无需任何低秩近似。系统性实验验证了本方法的优势。