Published January 1, 2025 | Published
Journal Article

Error analysis of kernel/GP methods for nonlinear and parametric PDEs

  • 1. ROR icon California Institute of Technology
  • 2. ROR icon University of Washington

Abstract

We introduce a priori Sobolev-space error estimates for the solution of arbitrary nonlinear, and possibly parametric, PDEs that are defined in the strong sense, using Gaussian process and kernel based methods. The primary assumptions are: (1) a continuous embedding of the reproducing kernel Hilbert space of the kernel into a Sobolev space of sufficient regularity; and (2) the stability of the differential operator and the solution map of the PDE between corresponding Sobolev spaces. The proof is articulated around Sobolev norm error estimates for kernel interpolants and relies on the minimizing norm property of the solution. The error estimates demonstrate dimension-benign convergence rates if the solution space of the PDE is smooth enough. We illustrate these points with applications to high-dimensional nonlinear elliptic PDEs and parametric PDEs. Although some recent machine learning methods have been presented as breaking the curse of dimensionality in solving high-dimensional PDEs, our analysis suggests a more nuanced picture: there is a trade-off between the regularity of the solution and the presence of the curse of dimensionality. Therefore, our results are in line with the understanding that the curse is absent when the solution is regular enough.

Copyright and License

© 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Acknowledgement

The authors gratefully acknowledge support by the Air Force Office of Scientific Research under MURI award number FA9550-20-1-0358 (Machine Learning and Physics-Based Modeling and Simulation). BH acknowledges support by the National Science Foundation grant number NSF-DMS-2208535 (Machine Learning for Bayesian Inverse Problems). HO also acknowledges support by the Department of Energy under award number DE-SC0023163 (SEA-CROGS: Scalable, Efficient and Accelerated Causal Reasoning Operators, Graphs and Spikes for Earth and Embedded Systems) and a Department of Defense Vannevar Bush Faculty Fellowship.

Contributions

Pau Batlle: Writing – original draft, Visualization, Validation, Software, Investigation, Formal analysis. Yifan Chen: Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Conceptualization. Bamdad Hosseini: Writing – original draft, Methodology, Investigation, Formal analysis, Conceptualization. Houman Owhadi: Writing – original draft, Supervision, Methodology, Investigation, Funding acquisition, Formal analysis, Conceptualization. Andrew M. Stuart: Writing – original draft, Supervision, Methodology, Investigation, Funding acquisition, Formal analysis, Conceptualization.

Data Availability

Data will be made available on request.

Additional details

Created:
November 11, 2024
Modified:
November 11, 2024