Learning elliptic partial differential equations with randomized linear algebra
This provides a rigorous method for PDE learning, addressing a foundational challenge in scientific computing and physics.
The paper tackles the problem of learning Green's functions for elliptic PDEs from input-output pairs, achieving a relative error of O(Γ_ε^{-1/2} log^3(1/ε) ε) with O(ε^{-6} log^4(1/ε)) training pairs, with theoretical guarantees of almost sure convergence.
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 $\mathcal{O}(Γ_ε^{-1/2}\log^3(1/ε)ε)$ using at most $\mathcal{O}(ε^{-6}\log^4(1/ε))$ input-output training pairs with high probability, for any $0<ε<1$. The quantity $0<Γ_ε\leq 1$ 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.