We study the computational scalability of a Gaussian process (GP) framework for solving general nonlinear partial differential equations (PDEs). This framework transforms solving PDEs to solving quadratic optimization problem with nonlinear constraints. Its complexity bottleneck lies in computing with dense kernel matrices obtained from pointwise evaluations of the covariance kernel of the GP and its partial derivatives at collocation points. We present a sparse Cholesky factorization algorithm for such kernel matrices based on the near-sparsity of the Cholesky factor under a new ordering of Diracs and derivative measurements. We rigorously identify the sparsity pattern and quantify the exponentially convergent accuracy of the corresponding Vecchia approximation of the GP, which is optimal in the Kullback-Leibler divergence. This enables us to compute $\epsilon$-approximate inverse Cholesky factors of the kernel matrices with complexity $O(N\log^d(N/\epsilon))$ in space and $O(N\log^{2d}(N/\epsilon))$ in time. With the sparse factors, gradient-based optimization methods become scalable. Furthermore, we can use the oftentimes more efficient Gauss-Newton method, for which we apply the conjugate gradient algorithm with the sparse factor of a reduced kernel matrix as a preconditioner to solve the linear system. We numerically illustrate our algorithm's near-linear space/time complexity for a broad class of nonlinear PDEs such as the nonlinear elliptic, Burgers, and Monge-Amp\`ere equations. In summary, we provide a fast, scalable, and accurate method for solving general PDEs with GPs.

翻译：我们研究了高斯过程框架在求解一般非线性偏微分方程时的计算可扩展性。该框架将偏微分方程求解转化为带非线性约束的二次优化问题，其计算瓶颈在于处理由高斯过程协方差核及其偏导数在配置点处的逐点评估所生成的稠密核矩阵。针对此类核矩阵，我们提出一种基于狄拉克测度与导数测量新序的稀疏乔列斯基分解算法——该算法利用乔列斯基因子在特定排序下的近似稀疏性。我们严格刻画了稀疏模式，并量化了相应高斯过程Vecchia近似（在库尔贝克-莱布勒散度意义下最优）的指数收敛精度。据此可计算核矩阵的 $\epsilon$ 近似逆乔列斯基因子，空间复杂度为 $O(N\log^d(N/\epsilon))$，时间复杂度为 $O(N\log^{2d}(N/\epsilon))$。通过引入稀疏因子，基于梯度的优化方法变得可扩展。此外，我们可采用通常更高效的 Gauss-Newton 方法，该方法以约化核矩阵的稀疏因子作为预条件子，结合共轭梯度算法求解线性系统。我们通过数值实验验证了算法在非线性椭圆方程、Burgers方程及Monge-Ampère方程等广谱非线性偏微分方程上具有近线性时空复杂度。综上，我们提出了一种快速、可扩展且精确的高斯过程偏微分方程通用求解方法。