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1. Basis-to-Basis Operator Learning Using Function Encoders

   https://doi.org/10.1016/j.cma.2024.117646  

   Ingebrand, Tyler; Thorpe, Adam J; Goswami, Somdatta; Kumar, Krishna; Topcu, Ufuk (February 2025, Computer Methods in Applied Mechanics and Engineering)

   We present Basis-to-Basis (B2B) operator learning, a novel approach for learning operators on Hilbert spaces of functions based on the foundational ideas of function encoders. We decompose the task of learning operators into two parts: learning sets of basis functions for both the input and output spaces and learning a potentially nonlinear mapping between the coefficients of the basis functions. B2B operator learning circumvents many challenges of prior works, such as requiring data to be at fixed locations, by leveraging classic techniques such as least squares to compute the coefficients. It is especially potent for linear operators, where we compute a mapping between bases as a single matrix transformation with a closed-form solution. Furthermore, with minimal modifications and using the deep theoretical connections between function encoders and functional analysis, we derive operator learning algorithms that are directly analogous to eigen-decomposition and singular value decomposition. We empirically validate B2B operator learning on seven benchmark operator learning tasks and show that it demonstrates a two-orders-of-magnitude improvement in accuracy over existing approaches on several benchmark tasks. 

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2. Nested gausslet basis sets

   https://doi.org/10.1063/5.0180092  

   White, Steven R; Lindsey, Michael J (December 2023, The Journal of Chemical Physics)

   We introduce nested gausslet bases, an improvement on previous gausslet bases that can treat systems containing atoms with much larger atomic numbers. We also introduce pure Gaussian distorted gausslet bases, which allow the Hamiltonian integrals to be performed analytically, as well as hybrid bases in which the gausslets are combined with standard Gaussian-type bases. All these bases feature the diagonal approximation for the electron–electron interactions so that the Hamiltonian is completely defined by two Nb × Nb matrices, where Nb ≈ 104 is small enough to permit fast calculations at the Hartree–Fock level. In constructing these bases, we have gained new mathematical insight into the construction of one-dimensional diagonal bases. In particular, we have proved an important theorem relating four key basis set properties: completeness, orthogonality, zero-moment conditions, and diagonalization of the coordinate operator matrix. We test our basis sets on small systems with a focus on high accuracy, obtaining, for example, an accuracy of 2 × 10−5 Ha for the total Hartree–Fock energy of the neon atom in the complete basis set limit. 

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3. Equivariant Hilbert Basis Theorem

   Erman, Daniel; Sam, Steven; Snowden, Andrew (January 2020, Mathematical research letters)

   null (Ed.)
4. Basis expansions for functional snippets

   https://doi.org/10.1093/biomet/asaa088  

   Lin, Zhenhua; Wang, Jane-Ling; Zhong, Qixian (October 2020, Biometrika)

   null (Ed.)

   Summary Estimation of mean and covariance functions is fundamental for functional data analysis. While this topic has been studied extensively in the literature, a key assumption is that there are enough data in the domain of interest to estimate both the mean and covariance functions. We investigate mean and covariance estimation for functional snippets in which observations from a subject are available only in an interval of length strictly, and often much, shorter than the length of the whole interval of interest. For such a sampling plan, no data is available for direct estimation of the off-diagonal region of the covariance function. We tackle this challenge via a basis representation of the covariance function. The proposed estimator enjoys a convergence rate that is adaptive to the smoothness of the underlying covariance function, and has superior finite-sample performance in simulation studies. 

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5. An equivariant Hilbert basis theorem

   Erman, Daniel; Sam, Steven V; Snowden, Andrew (January 2020, Mathematical research letters)