Learning elliptic partial differential equations with randomized linear algebra

Abstract

Given input-output pairs of an elliptic partial differential equation (PDE) in three dimensions, we derive the first theoretically-rigorous scheme for learning the associated Green's function $G$. By exploiting the hierarchical low-rank structure of $G$, we show that one can construct an approximant to $G$ that converges almost surely and achieves a relative error of $\mathcal{O}({\Gamma }_{\epsilon }^{-1/2}{\log }^3(1/\epsilon )\epsilon )$ using at most $\mathcal{O}({\epsilon }^{-6}{\log }^4(1/\epsilon ))$ input-output training pairs with high probability, for any $0<\epsilon <1$. The quantity $0<{\Gamma }_{\epsilon }\le 1$ characterizes the quality of the training dataset. Along the way, we extend the randomized singular value decomposition algorithm for learning matrices to Hilbert--Schmidt operators and characterize the quality of covariance kernels for PDE learning.

25 pages, 4 figures

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Cited by in corpus (9)

• Data-driven discovery of Green's functions with human-understandable deep learning
• Convergence Rates for Learning Linear Operators from Noisy Data
• A Mathematical Guide to Operator Learning
• Operator Learning Using Random Features: A Tool for Scientific Computing
• Elliptic PDE learning is provably data-efficient
• Principled interpolation of Green's functions learned from data
• Fast Macroscopic Forcing Method
• Learning Partial Differential Equations in Reproducing Kernel Hilbert Spaces
• A general error analysis for randomized low-rank approximation methods