用深度乘积核构建可扩展的时空场高斯过程模型，支持任意坐标预测与高效不确定性量化。
Scalable Gaussian process modeling of parametrized spatio-temporal fields
- 采用深度乘积核与克罗内克矩阵代数，实现近线性复杂度训练
- 在1D非稳态Burgers方程上精度优于投影型降维模型
- 可对任意时空点做连续预测，且均值与方差计算成本接近
我们提出一种可扩展的高斯过程框架，结合深度乘积核，用于在固定或参数依赖域上对参数化时空场进行数据驱动建模。该框架学习连续表示，可在任意时空坐标处进行预测，不受训练数据分辨率限制。通过利用克罗内克矩阵代数，实现了近乎线性复杂度的高效训练，其计算开销随时空网格点总数近似线性增长。本方法的关键优势在于，后验方差的计算成本几乎与后验均值相当（在笛卡尔网格下精确相同，在非结构网格下通过严格界实现），从而支持可扩展的不确定性量化。在多个基准问题上的数值实验表明，该方法的精度与傅里叶神经算子、深度算子网络等算子学习方法相当。在一维非稳态Burgers方程上，其精度超过投影型降维模型。结果表明，该框架是数据驱动代理建模的有效工具，尤其适用于需要下游任务中不确定性估计的场景。
原文摘要 · Abstract (English)
We introduce a scalable Gaussian process (GP) framework with deep product kernels for data-driven learning of parametrized spatio-temporal fields over fixed or parameter-dependent domains. The proposed framework learns a continuous representation, enabling predictions at arbitrary spatio-temporal coordinates, independent of the training data resolution. We leverage Kronecker matrix algebra to formulate a computationally efficient training procedure with complexity that scales nearly linearly with the total number of spatio-temporal grid points. A key feature of our approach is the efficient computation of the posterior variance at essentially the same computational cost as the posterior mean (exactly for Cartesian grids and via rigorous bounds for unstructured grids), thereby enabling scalable uncertainty quantification. Numerical studies on a range of benchmark problems demonstrate that the proposed method achieves accuracy competitive with operator learning methods such as Fourier neural operators and deep operator networks. On the one-dimensional unsteady Burgers' equation, our method surpasses the accuracy of projection-based reduced-order models. These results establish the proposed framework as an effective tool for data-driven surrogate modeling, particularly when uncertainty estimates are required for downstream tasks.
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