Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.16663 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916972513460224 |
|---|---|
| author | Huang, Jianlei Härkönen, Marc Lange-Hegermann, Markus Raiţă, Bogdan |
| author_facet | Huang, Jianlei Härkönen, Marc Lange-Hegermann, Markus Raiţă, Bogdan |
| contents | Working with systems of partial differential equations (PDEs) is a fundamental task in computational science. Well-posed systems are addressed by numerical solvers or neural operators, whereas systems described by data are often addressed by PINNs or Gaussian processes. In this work, we propose Boundary Ehrenpreis--Palamodov Gaussian Processes (B-EPGPs), a novel probabilistic framework for constructing GP priors that satisfy both general systems of linear PDEs with constant coefficients and linear boundary conditions and can be conditioned on a finite data set. We explicitly construct GP priors for representative PDE systems with practical boundary conditions. Formal proofs of correctness are provided and empirical results demonstrating significant accuracy and computational resource improvements over state-of-the-art approaches. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_16663 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gaussian Process Priors for Boundary Value Problems of Linear Partial Differential Equations Huang, Jianlei Härkönen, Marc Lange-Hegermann, Markus Raiţă, Bogdan Machine Learning Numerical Analysis Commutative Algebra 60G15, 13N10, 13P25, 60-08, 35G35 Working with systems of partial differential equations (PDEs) is a fundamental task in computational science. Well-posed systems are addressed by numerical solvers or neural operators, whereas systems described by data are often addressed by PINNs or Gaussian processes. In this work, we propose Boundary Ehrenpreis--Palamodov Gaussian Processes (B-EPGPs), a novel probabilistic framework for constructing GP priors that satisfy both general systems of linear PDEs with constant coefficients and linear boundary conditions and can be conditioned on a finite data set. We explicitly construct GP priors for representative PDE systems with practical boundary conditions. Formal proofs of correctness are provided and empirical results demonstrating significant accuracy and computational resource improvements over state-of-the-art approaches. |
| title | Gaussian Process Priors for Boundary Value Problems of Linear Partial Differential Equations |
| topic | Machine Learning Numerical Analysis Commutative Algebra 60G15, 13N10, 13P25, 60-08, 35G35 |
| url | https://arxiv.org/abs/2411.16663 |