Integer-order calculus fails to capture the long-range dependence (LRD) and memory effects found in many complex systems. Fractional calculus addresses these gaps through fractional-order integrals and derivatives, but fractional-order dynamical systems pose substantial challenges in system identification and optimal control tasks. In this paper, we theoretically derive the optimal control via linear quadratic regulator (LQR) for fractional-order linear time-invariant (FOLTI) systems and develop an end-to-end deep learning framework based on this theoretical foundation. Our approach establishes a rigorous mathematical model, derives analytical solutions, and incorporates deep learning to achieve data-driven optimal control of FOLTI systems. Our key contributions include: (i) proposing a novel method for system identification and optimal control strategy in FOLTI systems, (ii) developing the first end-to-end data-driven learning framework, Fractional-Order Learning for Optimal Control (FOLOC), that learns control policies from observed trajectories, and (iii) deriving theoretical bounds on the sample complexity for learning accurate control policies under fractional-order dynamics. Experimental results indicate that our method accurately approximates fractional-order system behaviors without relying on Gaussian noise assumptions, pointing to promising avenues for advanced optimal control.

Mixture-regularized bidirectional gated recurrent unit with attention (BiGAR) boosts the efficiency of decoding brain signals into hand movement trajectories. The novel neural decoder achieves an R-squared of over 0.8 in less than 0.2 ms of computation time on the MC_Maze dataset using fewer than 500 training trials. The expectation maximization (EM) algorithm used to extract neural hidden states improves R-squared and retains relatively low computation time for BiGAR. We further research on how different mixture regularizers impact the model performance. We generate mixture regularizers through pairwise weighted sum mixing of five individual regularizers associated with the Gaussian, Cauchy, Laplace, Sinc-squared, and Sin-fourth probability density functions. Experiments indicate that the improvement of model R-squared with mixture regularizers exceeds that of traditional individual regularizers and no regularizer.