CFNN: Continued Fraction Neural Network

Fractional scientific machine learning requires numerical operators that can be differentiated, batched, accelerated, and composed with neural networks. When the dominant linear fractional evolution is known through a Mittag-Leffler propagator, repeatedly reconstructing that response with a history solver or relearning it from data is unnecessary. We present DFSC, a PyTorch environment organized around the Mittag-Leffler Spectral Layer (MLSL). The layer separates known fractional propagation from data-driven corrections, so neural modules learn only unresolved dynamics while fractional orders and residual-network parameters are optimized jointly. Its adaptive algorithm increases special-function truncation depth or Lanczos dimension until successive differentiable evaluations satisfy a requested tolerance. In the negative-real alternating-series regime, DFSC additionally returns a certified first-omitted-term bound; outside that regime it explicitly labels estimates as empirical. DFSC supports dense, sparse, matrix-free, self-adjoint, generalized, and controlled complex operator paths; trainable fractional orders; direct inverse problems; residual neural composition; and CPU/GPU execution. The certified series bound covers all 59 eligible reference cases, with median bound/error effectivity 1.246 for resolved errors. Reusing a prepared batched Lanczos basis gives identical fixed-path values and reduces repeated-query time by 4.61--7.11 times on CPU and 13.07--16.22 times on an RTX 5070, excluding one-time preparation. A 27-case inverse matrix finds full-rank local curvature throughout, while remaining explicitly model-conditional. External solver and mixed real-data results support DFSC as an error-aware optional primitive for matched fractional structure, rather than a general replacement for fractional solvers or neural models.

Kolmogorov-Arnold Networks (KANs) replace fixed scalar weights by learnable univariate maps, yet most spectral KANs still rely on Chebyshev polynomials that live on a bounded interval and therefore require a saturating squashing of the pre-activation. We introduce the Fractional Hermite Kolmogorov-Arnold Network (FHKAN), a neural architecture that equips KANs with an adaptive, trainable fractional-order generalized Hermite basis. With a nonnegative shape parameter, the underlying generalized Hermite functions are orthonormal on the entire real line and uniformly bounded under a Gaussian envelope, so they act directly on unbounded pre-activations without any saturating map. We derive closed-form derivatives of the generalized, fractional, and scaled basis away from the origin and determine its order of differentiability at the origin, where the fractional power map creates a singular point. On synthetic regression, FHKAN attains the lowest test error at every network depth among nine fixed activations and the spline-based KAN. On image classification, image denoising, and sentiment analysis, the best-performing fractional Hermite block yields the highest accuracy or reconstruction quality. On physics-informed problems, FHKAN reaches mean absolute errors of order 10^-6 to 10^-5 on the Lane-Emden equation, residuals of order 10^-4 on the Burgers equation, and absolute errors below 10^-4 on a fractional delay differential equation.

Kolmogorov-Arnold Networks (KANs) have recently emerged as a compelling alternative to multilayer perceptrons, offering enhanced interpretability via functional decomposition. However, existing KAN architectures, including spline-, wavelet-, radial-basis variants, etc., suffer from high computational cost and slow convergence, limiting scalability and practical applicability. Here, we introduce Feature-Enriched Kolmogorov-Arnold Networks (FEKAN), a simple yet effective extension that preserves all the advantages of KAN while improving computational efficiency and predictive accuracy through feature enrichment, without increasing the number of trainable parameters. By incorporating these additional features, FEKAN accelerates convergence, increases representation capacity, and substantially mitigates the computational overhead characteristic of state-of-the-art KAN architectures. We investigate FEKAN across a comprehensive set of benchmarks, including function-approximation tasks, physics-informed formulations for diverse partial differential equations (PDEs), and neural operator settings that map between input and output function spaces. For function approximation, we systematically compare FEKAN against a broad family of KAN variants, FastKAN, WavKAN, ReLUKAN, HRKAN, ChebyshevKAN, RBFKAN, and the original SplineKAN. Across all tasks, FEKAN demonstrates substantially faster convergence and consistently higher approximation accuracy than the underlying baseline architectures. We also establish the theoretical foundations for FEKAN, showing its superior representation capacity compared to KAN, which contributes to improved accuracy and efficiency.