Solving and Learning Nonlinear PDEs with Gaussian Processes
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.
Code (2)
Tasks
Gaussian ProcessesMethods 이 논문이 사용한 방법론
Similar Papers 제목 키워드 기반
Sparse Cholesky Factorization for Solving Nonlinear PDEs via Gaussian Processes
In recent years, there has been widespread adoption of machine learning-based approaches to automate the solving of partial differential equations (PDEs). Among these approaches, Gaussian processes (GPs) and kernel metho…
Gaussian ProcessesTensor Gaussian Processes: Efficient Solvers for Nonlinear PDEs
Machine learning solvers for partial differential equations (PDEs) have attracted growing interest. However, most existing approaches, such as neural network solvers, rely on stochastic training, which is inefficient and…
Gaussian ProcessesSolving Roughly Forced Nonlinear PDEs via Misspecified Kernel Methods and Neural Networks
We consider the use of Gaussian Processes (GPs) or Neural Networks (NNs) to numerically approximate the solutions to nonlinear partial differential equations (PDEs) with rough forcing or source terms, which commonly aris…
Gaussian ProcessesMultifidelity Gaussian process regression for solving nonlinear partial differential equations
Solving nonlinear partial differential equations (PDEs) using kernel methods offers a compelling alternative to traditional numerical solvers. However, the performance of these methods strongly depends on the choice of k…
Parameter Inference based on Gaussian Processes Informed by Nonlinear Partial Differential Equations
Partial differential equations (PDEs) are widely used for the description of physical and engineering phenomena. Some key parameters involved in PDEs, which represent certain physical properties with important scientific…
Gaussian ProcessesUncertainty Quantification