We introduce a simple, rigorous, and unified framework for solving nonlinear partial differential equations (PDEs), and for solving inverse problems (IPs) involving the identification of parameters in PDEs, using the framework of Gaussian processes. The proposed approach: (1) provides a natural generalization of collocation kernel methods to nonlinear PDEs and IPs; (2) has guaranteed convergence for a very general class of PDEs, and comes equipped with a path to compute error bounds for specific PDE approximations; (3) inherits the state-of-the-art computational complexity of linear solvers for dense kernel matrices. The main idea of our method is to approximate the solution of a given PDE as the maximum a posteriori (MAP) estimator of a Gaussian process conditioned on solving the PDE at a finite number of collocation points. Although this optimization problem is infinite-dimensional, it can be reduced to a finite-dimensional one by introducing additional variables corresponding to the values of the derivatives of the solution at collocation points; this generalizes the representer theorem arising in Gaussian process regression. The reduced optimization problem has the form of a quadratic objective function subject to nonlinear constraints; it is solved with a variant of the Gauss–Newton method. The resulting algorithm (a) can be interpreted as solving successive linearizations of the nonlinear PDE, and (b) in practice is found to converge in a small number of iterations (2 to 10), for a wide range of PDEs. Most traditional approaches to IPs interleave parameter updates with numerical solution of the PDE; our algorithm solves for both parameter and PDE solution simultaneously. Experiments on nonlinear elliptic PDEs, Burgers' equation, a regularized Eikonal equation, and an IP for permeability identification in Darcy flow illustrate the efficacy and scope of our framework.

我们引入了一个简单、严格且统一的框架，用于求解非线性偏微分方程 (PDE)，以及使用高斯过程框架解决涉及 PDE 中参数识别的逆问题 (IP)。所提出的方法：（1）提供了对非线性偏微分方程和 IP 的搭配核方法的自然推广；(2) 保证了非常通用的 PDE 类的收敛性，并配备了计算特定 PDE 近似误差界限的路径；(3) 继承了用于密集核矩阵的线性求解器的最新计算复杂性。我们方法的主要思想是将给定 PDE 的解近似为高斯过程的最大后验 (MAP) 估计量，条件是在有限数量的搭配点处求解 PDE。虽然这个优化问题是无限维的，但可以通过引入对应于搭配点解的导数值的附加变量将其简化为有限维问题；这概括了高斯过程回归中出现的代表定理。简化优化问题具有受非线性约束的二次目标函数形式；它是用高斯-牛顿方法的变体来解决的。由此产生的算法 (a) 可以解释为求解非线性 PDE 的连续线性化，并且 (b) 在实践中被发现在少量迭代（2 到 10）中收敛，适用于广泛的 PDE。大多数传统的 IP 方法将参数更新与 PDE 的数值解交织在一起；我们的算法同时求解参数和 PDE 解。非线性椭圆偏微分方程、伯格斯方程、正则化 Eikonal 方程和达西流渗透率识别 IP 的实验说明了我们框架的有效性和范围。