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1. Beyond Regular Grids: Fourier-Based Neural Operators on Arbitrary Domains

   Lingsch, Levi E; Michelis, Mike Yan; Bezenac, Emmanuel De; Perera, Sirani M; Katzschmann, Robert K; Mishra, Siddhartha (July 2024, Proceedings of the 41st International Conference on Machine Learning, PMLR)

   The computational efficiency of many neural operators, widely used for learning solutions of PDEs, relies on the fast Fourier transform (FFT) for performing spectral computations. As the FFT is limited to equispaced (rectangular) grids, this limits the efficiency of such neural operators when applied to problems where the input and output functions need to be processed on general non-equispaced point distributions. Leveraging the observation that a limited set of Fourier (Spectral) modes suffice to provide the required expressivity of a neural operator, we propose a simple method, based on the efficient direct evaluation of the underlying spectral transformation, to extend neural operators to arbitrary domains. An efficient implementation of such direct spectral evaluations is coupled with existing neural operator models to allow the processing of data on arbitrary non-equispaced distributions of points. With extensive empirical evaluation, we demonstrate that the proposed method allows us to extend neural operators to arbitrary point distributions with significant gains in training speed over baselines, while retaining or improving the accuracy of Fourier neural operators (FNOs) and related neural operators. 

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2. Fast spectral separation method for kinetic equation with anisotropic non-stationary collision operator retaining micro-model fidelity

   Zhao, Yue; Lei, Huan (August 2025, arXivorg)

   In this paper, we present a generalized, data-driven collisional operator for one-component plasmas, learned from molecular dynamics simulations, to extend the collisional kinetic model beyond the weakly coupled regime. The proposed operator features an anisotropic, non-stationary collision kernel that accounts for particle correlations typically neglected in classical Landau formulations. To enable efficient numerical evaluation, we develop a fast spectral separation method that represents the kernel as a low-rank tensor product of univariate basis functions. This formulation admits an O(N log N) algorithm via fast Fourier transforms and preserves key physical properties, including discrete conservation laws and the H-theorem, through a structure-preserving central difference discretization. Numerical experiments demonstrate that the proposed model accurately captures plasma dynamics in the moderately coupled regime beyond the standard Landau model while maintaining high computational efficiency and structure-preserving properties. 

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3. Fourier Method for Approximating Eigenvalues of Indefinite Stekloff Operator

   https://doi.org/10.1007/978-3-319-97136-0_3  

   Wu, Yangqingxiang; Zikatanov, Ludmil T (January 2018, Lecture notes in computer science)

   Sistek, J.; Tichy, P; Kozubek, T.; Cermak, M.; Lukas, D.; Jaros, J.; Blaheta, R. (Ed.)

   We introduce an efficient method for computing the Stekloff eigenvalues associated with the indefinite Helmholtz equation. In general, this eigenvalue problem requires solving the Helmholtz equation with Dirichlet and/or Neumann boundary condition repeatedly. We propose solving the discretized problem with Fast Fourier Transform (FFT) based on carefully designed extensions and restrictions operators. The proposed Fourier method, combined with proper eigensolver, results in an efficient and easy approach for computing the Stekloff eigenvalues 

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4. Reflective prolate‐spheroidal operators and the adelic Grassmannian

   https://doi.org/10.1002/cpa.22118  

   Casper, W. Riley; Grünbaum, F. Alberto; Yakimov, Milen; Zurrián, Ignacio (July 2023, Communications on Pure and Applied Mathematics)

   Abstract Beginning with the work of Landau, Pollak and Slepian in the 1960s on time‐band limiting, commuting pairs of integral and differential operators have played a key role in signal processing, random matrix theory, and integrable systems. Previously, such pairs were constructed by ad hoc methods, which essentially worked because a commuting operator of low order could be found by a direct calculation. We describe a general approach to these problems that proves that every pointWof Wilson's infinite dimensional adelic Grassmannian gives rise to an integral operator , acting on for a contour , which reflects a differential operator with rational coefficients in the sense that on a dense subset of . By using analytic methods and methods from integrable systems, we show that the reflected differential operator can be constructed from the Fourier algebra of the associated bispectral function . The exact size of this algebra with respect to a bifiltration is in turn determined using algebro‐geometric methods. Intrinsic properties of four involutions of the adelic Grassmannian naturally lead us to consider the reflecting property above in place of plain commutativity. Furthermore, we prove that the time‐band limited operators of the generalized Laplace transforms with kernels given by the rank one bispectral functions always reflect a differential operator. A 90° rotation argument is used to prove that the time‐band limited operators of the generalized Fourier transforms with kernels admit a commuting differential operator. These methods produce vast collections of integral operators with prolate‐spheroidal properties, associated to the wave functions of all rational solutions of the KP hierarchy vanishing at infinity, introduced by Krichever in the late 1970s. 

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5. RNN Classification of Spectral EEG/EMG Data Associated with Facial Movements for Drone Control

   Alba Flores, Akintomide Adebile (April 2023, Proceedings from IEEE SouthEastcon)

   IEEE (Ed.)

   This research involves developing a drone control system that functions by relating EEG and EMG from the forehead to different facial movements using recurrent neural networks (RNN) such as long-short term memory (LSTM) and gated recurrent Unit (GRU). As current drone control methods are largely limited to handheld devices, regular operators are actively engaged while flying and cannot perform any passive control. Passive control of drones would prove advantageous in various applications as drone operators can focus on additional tasks. The advantages of the chosen methods and those of some alternative system designs are discussed. For this research, EEG signals were acquired at three frontal cortex locations (fp1, fpz , fp2 ) using electrodes from an OpenBCI headband and observed for patterns of Fast Fourier Transform (FFT) frequency-amplitude distributions. Five different facial expressions were repeated while recording EEG signals of 0-60Hz frequencies with two reference electrodes placed on both earlobes. EMG noise received during EEG measurements was not filtered away but was observed to be minimal. A dataset was first created for the actions done, and later categorized by a mean average error (MAE), a statistical error deviation analysis and then classified with both an LSTM and GRU neural network by relating FFT amplitudes to the actions. On average, the LSTM network had classification accuracy of 78.6%, and the GRU network had a classification accuracy of 81.8%. 

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