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 {https://github.com/xuangu-fang/Gaussian-Process-Slover-for-High-Freq-PDE}.

Labeling sufficient data for supervised learning remains an open challenge in social network analysis. An alternative is to collect self-labeled data, i.e. the data labeled by their owners. Emmery et al show that standard models can be trained and perform well on self-labeled data, suggesting the effectiveness of this approach. In this paper, we argue self-labeled data may not be representative of the population. Taking Twitter demographic prediction as an example, we show the popular FastText model standardly trained on self-labeled data does not generalize well on random testing samples. We then present a new learner DeFastText that aims to correct data bias using the kernel means matching technique. In experiment, we show it achieves lower generalization errors than FastText. This research raises an attention of the data bias problem when learning from self-labeled data in social network analysis.