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1. Coupled Generation

   https://doi.org/10.1080/01621459.2020.1844719  

   Dai, Ben; Shen, Xiaotong; Wong, Wing (January 2021, Journal of the American Statistical Association)

   null (Ed.)
2. Coupled embeddability

   https://doi.org/10.1112/blms.12646  

   Frick, Florian; Harrison, Michael (April 2022, Bulletin of the London Mathematical Society)
3. Operator Relationship between Conventional Coupled Cluster and Unitary Coupled Cluster

   https://doi.org/10.3390/sym14030494  

   Freericks, James K. (March 2022, Symmetry)

   The chemistry community has long sought the exact relationship between the conventional and the unitary coupled cluster ansatz for a single-reference system, especially given the interest in performing quantum chemistry on quantum computers. In this work, we show how one can use the operator manipulations given by the exponential disentangling identity and the Hadamard lemma to relate the factorized form of the unitary coupled-cluster approximation to a factorized form of the conventional coupled cluster approximation (the factorized form is required, because some amplitudes are operator-valued and do not commute with other terms). By employing the Trotter product formula, one can then relate the factorized form to the standard form of the unitary coupled cluster ansatz. The operator dependence of the factorized form of the coupled cluster approximation can also be removed at the expense of requiring even more higher-rank operators, finally yielding the conventional coupled cluster. The algebraic manipulations of this approach are daunting to carry out by hand, but can be automated on a computer for small enough systems. 

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4. Coupled Multiwavelet Neural Operator Learning for Coupled Partial Differential Equations

   Xiao, X; Cao, D.; Yang, R.; Gupta, G.; Liu, G.; Yin, C.; Balan, R.; Bogdan, P. (April 2023, ICLR 2023)

   Coupled partial differential equations (PDEs) are key tasks in modeling the complex dynamics of many physical processes. Recently, neural operators have shown the ability to solve PDEs by learning the integral kernel directly in Fourier/Wavelet space, so the difficulty for solving the coupled PDEs depends on dealing with the coupled mappings between the functions. Towards this end, we propose a coupled multiwavelets neural operator (CMWNO) learning scheme by decoupling the coupled integral kernels during the multiwavelet decomposition and reconstruction procedures in the Wavelet space. The proposed model achieves significantly higher accuracy compared to previous learning-based solvers in solving the coupled PDEs including Gray-Scott (GS) equations and the non-local mean field game (MFG) problem. According to our experimental results, the proposed model exhibits a 2ˆ „ 4ˆ improvement relative L2 error compared to the best results from the state-of-the-art models. 

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5. Coupled Multiwavelet Neural Operator Learning for Coupled Partial Differential Equations

   Xiongye Xiao, Defu Cao (May 2023, Proceedings of the International Conference on Learning Representations (ICLR))

   Yan Liu and Been Kim (Ed.)

   Coupled partial differential equations (PDEs) are key tasks in modeling the complex dynamics of many physical processes. Recently, neural operators have shown the ability to solve PDEs by learning the integral kernel directly in Fourier/Wavelet space, so the difficulty for solving the coupled PDEs depends on dealing with the coupled mappings between the functions. Towards this end, we propose a coupled multiwavelets neural operator (CMWNO) learning scheme by decoupling the coupled integral kernels during the multiwavelet decomposition and reconstruction procedures in the Wavelet space. The proposed model achieves significantly higher accuracy compared to previous learning-based solvers in solving the coupled PDEs including Gray-Scott (GS) equations and the non-local mean field game (MFG) problem. According to our experimental results, the proposed model exhibits a 2ˆ „ 4ˆ improvement relative L2 error compared to the best results from the state-of-the-art models. 

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