Branched Signature Kernel Solvers for ODEs with rough Single-Trajectory signals

We develop a branched signature kernel solver for linear and nonlinear ordinary differential equations driven by a \emph{single observed trajectory} of a possibly rough forcing signal -- a setting that arises naturally in earthquake engineering, finance, biology, and structural health monitoring, where the forcing is observed exactly once and the solver must respect the underlying physical law without recourse to an ensemble of realizations. Two ingredients are new. First, a \emph{count-sampling} construction turns the single observation into a hierarchical family of $N+1$ nested training paths on which the branched signature kernel can be evaluated; this allows the signature kernel machinery, originally designed for multi-realization regression problems, to operate on a single-trajectory observation. Second, a kernel-collocation framework places the ansatz either on the highest-order derivative of the solution (with lower derivatives recovered by integrating the kernel) or on the solution itself (after $m$-fold integration of the ODE). We prove a universal approximation theorem for the branched signature kernel, leveraging the Hairer--Kelly morphism to express branched signature evaluations through geometric signatures of time-extended paths. The offline solver is extended to a streaming Test/Train/Retrain protocol with closed-form online updates in the linear case and scalar Newton steps in the nonlinear case. Numerical experiments on six benchmarks (El-Centro earthquake displacement, the Solow capital-stock model, an fBM-driven second-order ODE, a forced Duffing oscillator, a path-dependent Arias-intensity-degraded oscillator with variable coefficients, and a noisy Kuramoto phase-oscillator system) show that the branched signature-kernel solver delivers accurate, stable predictions across all regimes.

Advanced Efficient Strategy for Detection of Dark Objects Based on Spiking Network with Multi-Box Detection

Several deep learning algorithms have shown amazing performance for existing object detection tasks, but recognizing darker objects is the largest challenge. Moreover, those techniques struggled to detect or had a slow recognition rate, resulting in significant performance losses. As a result, an improved and accurate detection approach is required to address the above difficulty. The whole study proposes a combination of spiked and normal convolution layers as an energy-efficient and reliable object detector model. The proposed model is split into two sections. The first section is developed as a feature extractor, which utilizes pre-trained VGG16, and the second section of the proposal structure is the combination of spiked and normal Convolutional layers to detect the bounding boxes of images. We drew a pre-trained model for classifying detected objects. With state of the art Python libraries, spike layers can be trained efficiently. The proposed spike convolutional object detector (SCOD) has been evaluated on VOC and Ex-Dark datasets. SCOD reached 66.01% and 41.25% mAP for detecting 20 different objects in the VOC-12 and 12 objects in the Ex-Dark dataset. SCOD uses 14 Giga FLOPS for its forward path calculations. Experimental results indicated superior performance compared to Tiny YOLO, Spike YOLO, YOLO-LITE, Tinier YOLO and Center of loc+Xception based on mAP for the VOC dataset.