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（GP）框架求解一般非线性偏微分方程（PDE）的计算可扩展性。该框架将求解PDE转化为求解具有非线性约束条件的二次优化问题。其复杂度瓶颈在于使用在协方差核函数在采样点及其偏导数处的点值来获取密的核矩阵，从而进行计算。我们提出了一种稀疏Cholesky分解算法，基于重新排列的Diracs和导数测量的几乎稀疏的Cholesky因子。我们严格地确定了稀疏模式，并量化了对应的Vecchia近似的指数收敛精度，该近似在Kullback-Leibler距离下是最优的。这使我们能够在空间中以$O(N\log^d(N/\epsilon))$的复杂度和时间中以$O(N\log^{2d}(N/\epsilon))$的复杂度计算$\epsilon$-近似逆Cholesky因子。使用这些稀疏的因子，基于梯度的优化方法变得可扩展。此外，我们可以使用通常更有效的Gauss-Newton方法，其中我们使用降低的核矩阵的稀疏因子作为预处理器，然后使用共轭梯度算法来解决线性系统。我们数值地说明了我们算法在广泛类别的非线性PDE（如非线性椭圆、Burgers和Monge-Ampere方程）中具有近线性的空间/时间复杂度。总之，我们提供了一种快速、可扩展和准确的方法，用于使用GP求解一般PDE。