Sparse Cholesky factorization for solving nonlinear PDEs via Gaussian processes (in EN)
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 methods have garnered considerable interest due to their flexibility, robust theoretical guarantees, and close ties to traditional methods. They can transform the solving of general nonlinear PDEs into solving quadratic optimization problems with nonlinear, PDE-induced constraints. However, the complexity bottleneck lies in computing with dense kernel matrices obtained from pointwise evaluations of the covariance kernel, and itspartial derivatives, a result of the PDE constraint and for which fast algorithms are scarce.
The primary goal of this paper is to provide a near-linear complexity algorithm for working with such kernel matrices. We present a sparse Cholesky factorization algorithm for these matrices based on the near-sparsity of the Cholesky factor under a novel ordering of pointwise and derivative measurements. The near-sparsity is rigorously justified by directly connecting the factor to GP regression and exponential decay of basis functions in numerical homogenization. We then employ the Vecchia approximation of GPs, which is optimal in the Kullback-Leibler divergence, to compute the approximate factor. This enables us to compute -approximate inverse Cholesky factors of the kernel matrices with complexity in space and in time. We integrate sparse Cholesky factorizations into optimization algorithms to obtain fast solvers of the nonlinear PDE. 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-Ampère equations. In summary, we provide a fast, scalable, and accurate method for solving general PDEs with GPs and kernel methods.
- Research Organization:
- Stanford Univ., CA (United States)
- Sponsoring Organization:
- USDOE Office of Science (SC)
- DOE Contract Number:
- SC0023163
- OSTI ID:
- 2578874
- Journal Information:
- Mathematics of Computation, Journal Name: Mathematics of Computation; ISSN 0025-5718
- Publisher:
- American Mathematical Society
- Country of Publication:
- United States
- Language:
- EN
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