Applied Mathematics Correction for “Elliptic PDE learning is provably data-efficient,” by Nicolas Boullé, Diana Halikias, and Alex Townsend, which published September 18, 2023; 10.1073/pnas.2303904120 (Proc. Natl. Acad. Sci. U.S.A. 120, e2303904120).
The authors note, “In the current article, we reference a preprint by Houman Owhadi and Florian Schäfer in the first section of the SI Appendix, along with several other works, as we wanted to have a nuanced discussion of prior works in the subject area of PDE learning. There are other highly relevant works that we discuss in the SI Appendix as well that are not cited in the main text due to space limitations.
The preprint of Owhadi and Schäfer addresses a related but fundamentally different problem, as they recover solution operators of elliptic PDEs using deterministic inputs, while our approach using hierarchical structure is considered more general by the community and explains the impact of the distribution of random inputs. We now agree that their work should have been discussed in the main text rather than solely in the SI Appendix, as is currently the case.”
The authors note that on page 2, left column, first full paragraph, line 2 “This work provides theoretical insights by constructing a provably data-efficient algorithm, showing that one can achieve exponential convergence when learning solution operators of elliptic PDEs.” should instead appear as “This work provides theoretical insights by constructing a provably data-efficient algorithm that can achieve exponential convergence for learning solution operators of elliptic PDEs.”
The authors note that on page 2, right column, second paragraph, line 6, “the Hilbert-Schmidt norm.” should instead appear as “the Hilbert–Schmidt norm, while (12) exploits sparse Gaussian elimination techniques and piecewise polynomial inputs to recover with pairs.”
The authors note that on page 2, right column, third paragraph, line 1, “Our main result dramatically improves the required amount of training data to construct an -approximation to by exploiting the hierarchical structure of G” should instead appear as “It is an open problem whether one can achieve exponential convergence while recovering the hierarchical structure of the solution operator (8, 13), as then the technique could potentially extend beyond elliptic solution operators. Our main result solves this open problem by exploiting the hierarchical structure of G”.
As a result of the changes above, authors have added a reference to the article. The complete reference appears below. A citation to the new reference should be included on page 2, right column, second paragraph, line 6, as described above.
F. Schäfer, H. Owhadi, Sparse recovery of elliptic solvers from matrix-vector products. arXiv [Preprint] (2021). https://doi.org/10.48550/arXiv.2110.05351.
The online version has been corrected.