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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. 

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. 

The ability to sense propagating electromagnetic plane waves based on their directions of arrival (DOAs) is fundamental to a range of radio frequency (RF) sensing, communications, and imaging applications. This paper introduces an algorithm for the wideband true time delay digital delay Vandermonde matrix (DVM), utilizing Thiran fractional delays that are useful for realizing RF sensors having multiple look DOA support. The digital DVM algorithm leverages sparse matrix factorization to yield multiple simultaneous RF beams for low-complexity sensing applications. Consequently, the proposed algorithm offers a reduction in circuit complexity for multi-beam digital wideband beamforming systems employing Thiran fractional delays. Unlike finite impulse response filter-based approaches which are wideband but of a high filter order, the Thiran filters trade usable bandwidth in favor of low-complexity circuits. The phase and group delay responses of Thiran filters with delays of a fractional sampling period will be demonstrated. Thiran filters show favorable results for sample delay blocks with a temporal oversampling factor of three. Thiran fractional delays of orders three and four are mapped to Xilinx FPGA RF-SoC technologies for evaluation. The preliminary results of the APF-based Thiran fractional delays on FPGA can potentially be used to realize DVM factorizations using application-specific integrated circuit (ASIC) technology. 

The advancement of wireless networking has significantly enhanced beamforming capabilities in Autonomous Unmanned Aerial Systems (AUAS). This paper presents a simple and efficient classical algorithm to route a collection of AUAS or drone swarms extending our previous work on AUAS. The algorithm is based on the sparse factorization of frequency Vandermonde matrices that correspond to each drone, and its entries are determined through spatiotemporal data of drones in the AUAS. The algorithm relies on multibeam beamforming, making it suitable for large-scale AUAS networking in wireless communications. We show a reduction in the arithmetic and time complexities of the algorithm through theoretical and numerical results. Finally, we also present an ML-based AUAS routing algorithm using the classical AUAS algorithm and feed-forward neural networks. We compare the beamformed signals of the ML-based AUAS routing algorithm with the ground truth signals to minimize the error between them. The numerical error results show that the ML-based AUAS routing algorithm enhances the accuracy of the routing. This error, along with the numerical and theoretical results for over 100 drones, provides the basis for the scalability of the proposed ML-based AUAS algorithms for large-scale deployments. 

Computational limitations and Big Data analysis pose challenges in seeking efficient techniques to predict trajectories in three-body dynamics. Thus, a reduced-complexity classical algorithm is proposed utilizing predefined spacecraft's position and velocity data to achieve precise and accurate orbital trajectories of the spacecraft within three-body dynamics. The proposed algorithm seamlessly solves polynomial interpolation along with the boundary and interior conditions without the need for the spacecraft's acceleration data. Once the algorithm is derived, it will be tested across a diverse variety of periodic trajectories in the Earth-Moon system. Moreover, a comparative analysis is performed to evaluate the time complexity of the proposed algorithm compared with conventional orbit propagators. Finally, the proposed algorithm will be utilized and extended to learn and update distant retrograde orbits (DRO) while training a neural network with several initial conditions composing minimum predefined data. After the training is done, the neural network is used to accurately predict DRO trajectories for a given initial condition, demonstrating the exceptional accuracy and effectiveness of the proposed learning process. 

The emerging concept of drone swarms creates new opportunities with major societal implications. However, future drone swarm applications and services pose new communications and sensing challenges, particularly for collaborative tasks. To address these challenges, in this paper, we integrate sensor arrays and communication to propose a mathematical model to route a collection of autonomous unmanned aerial systems (AUAS), a so-called drone swarm or AUAS swarm, without having a base station of communication but communicating with each other using multiple spatio-temporal data. The theories of structured matrices, concepts in multi-beam beamforming, and sensor arrays are utilized to propose a swarm routing algorithm. We address the routing algorithm’s computational and arithmetic complexities, precision, and reliability. We measure bit-error-rate (BER) based on the number of elements in sensor arrays and beamformed output of the members of the swarm to authenticate and secure the routing for the decentralized AUAS networking. The proposed model has the potential to enable future drone swarm applications and services. Finally, we discuss future work on obtaining a machine-learning-based low-cost drone swarm routing algorithm.