fractional calculus

To address the high memory consumption and computational complexity arising from forward-mode differentiation in training Neural Fractional Differential Equations (Neural FDEs), this work introduces, for the first time, adjoint-based backpropagation into the Neural FDE training framework. By formulating and solving an augmented fractional-order adjoint equation, our method enables efficient time-reversed gradient computation, overcoming the scalability limitations of conventional forward-mode differentiation in large-scale settings. The approach integrates fractional calculus, the adjoint state method, and neural differential equation theory, and is compatible with mainstream numerical FDE solvers. Experiments on tasks such as graph representation learning demonstrate performance on par with baseline models, while reducing memory usage by over 60% and accelerating training by 2–3×. This advancement significantly enhances the feasibility of Neural FDEs for large-scale dynamical system modeling.

This work addresses the high-precision numerical solution of abstract Cauchy problems involving Caputo-type time-fractional derivatives, unbounded operator coefficients, and nonzero initial data. A novel unified representation of the mild solution is proposed, valid for all α ∈ (0,2), thereby achieving the first formulation that simultaneously encompasses subparabolic, parabolic, and subhyperbolic regimes. Based on this representation, an adaptive Sinc quadrature method is developed—incorporating the operator’s spectral properties, the fractional order α, and the regularity of initial data. Theoretical analysis establishes exponential convergence over the entire time domain. The method preserves high-order accuracy at t = 0, requires no a priori extrapolation of the forcing term, and naturally enables multilevel parallelization. Numerical experiments rigorously validate the derived error estimates, demonstrating substantially improved accuracy and stability compared to classical finite difference and convolution quadrature methods.

This work addresses the challenge of reconstructing analytic ordinary differential equation (ODE) vector fields from limited discrete trajectory data. Methodologically, it introduces a novel approximation framework centered on the push-forward operator—employed here for the first time as the core modeling tool—combined with the Fourier–Borel transform and Fock space theory to construct finite-dimensional operator approximations within a local analytic functional space. Theoretically, it establishes rigorous convergence guarantees with explicit rates, proving that truncated least-squares polynomials achieve superior approximation both inside and outside their support domain. Experimentally, the method accurately recovers vector fields induced by analytic flow maps, exhibits strong extrapolation capability, and maintains numerical stability. Overall, it provides a new paradigm for analytic dynamical system modeling from sparse data.

Modeling real-world regression functions that are nonsmooth—specifically, those residing in the $L^2$ fractional Sobolev space of order $s in (0,1)$—poses a fundamental challenge for conventional nonparametric methods requiring higher-order differentiability. Method: This paper proposes a novel nonparametric regression framework based on the fractional Laplacian operator. It introduces, for the first time, fractional Laplacian feature mapping into nonparametric regression, circumventing restrictive smoothness assumptions and naturally accommodating canonical nonsmooth structures such as piecewise constants, sharp peaks, and fractional powers. The method integrates fractional differential operators, Sobolev space theory, and spectral regularization. Results: It achieves the minimax-optimal convergence rate $n^{-2s/(2s+d)}$, with a tight theoretical error bound. Numerical experiments demonstrate its substantial superiority over classical smooth kernel methods in fitting nonsmooth functions.

This work addresses the problem of computing rational function solutions to first-order algebraic ordinary differential equations (AODEs) with parameters. For equations whose coefficients are rational functions, we construct an equivalent algebraic system and establish, for the first time, necessary and sufficient conditions for the existence of rational solutions. Our method introduces a symbolic algorithmic framework based on parameter elimination and polynomial system solving, achieving full decidability for constant-coefficient cases and enabling the construction of general rational solutions—including those involving transcendental constants—for single-function coefficient cases. The main contributions are: (1) the first universal and exact criterion for deciding the existence of rational solutions to parametric first-order AODEs; (2) algorithmic resolution in two fundamental cases—constant coefficients and single-function coefficients; and (3) an open-source, executable prototype implementation for symbolic decision-making.

This work proposes a non-Markovian optimization framework to address the limitations of traditional methods that rely on the Markov assumption and are prone to noise in highly imbalanced data, often leading to the suppression of minority-class signals and overfitting. By embedding a weighted fractional-order Weyl integral as a memory operator into the optimization process, the approach replaces instantaneous gradients with dynamically weighted historical gradient sequences, thereby transcending local update constraints and effectively preserving weak signals. Empirical evaluations demonstrate a ~40% improvement in PR-AUC on financial fraud detection tasks and a significant reduction in overfitting in medical diagnosis scenarios, underscoring the method’s robustness and generalization capability.

This work addresses parametric numerical integration problems—including statistical functional evaluation, Chebyshev spectral approximation, and integrals arising from differential equations—by proposing a derivative-supervised differentiable machine learning framework. The method explicitly incorporates analytical derivative information into surrogate modeling of integrals for the first time, overcoming the limitations of conventional black-box regression while preserving physical consistency and substantially improving accuracy and generalization. Technically, it integrates deep neural networks with derivative-augmented training to enable efficient inference in high-dimensional parameter spaces. Evaluated on diverse benchmarks encompassing both smooth and ill-conditioned integrals, the approach achieves over 40% average reduction in mean squared error compared to standard architectures, while reducing sample requirements by a factor of 3–5. It thus delivers high accuracy, strong scalability, and exceptional sample efficiency.

This work addresses the challenge of solving partial differential equations involving the fractional Laplacian on bounded domains, particularly in regimes characterized by strong boundary singularities and long-time simulations. The authors propose a deterministic tensor neural network framework that integrates a geometry-adaptive near-field integral decomposition, trial functions informed by boundary singularity profiles, an automated strategy for selecting dominant singular exponents, and a separable spatiotemporal network architecture trained via alternating subspace optimization. Singular and regular components are efficiently handled through Gauss–Jacobi quadrature and deterministic angular integration, while residual low-rank decomposition enhances numerical stability and accuracy. Benchmark comparisons demonstrate that the proposed method significantly outperforms fractional physics-informed neural networks (fPINNs) and Monte Carlo approaches in both high-singularity and long-time settings.

This study addresses the limited flexibility of traditional fractionally differenced models in capturing long memory, which often fail to accommodate faster-decaying dependence structures observed in real-world data. To overcome this limitation, the authors propose a class of observation-driven models incorporating tempered fractional differencing, thereby preserving long-memory modeling capabilities while substantially enhancing flexibility and theoretical robustness. Parameter estimation is carried out via partial maximum likelihood, enabling hypothesis testing, confidence interval construction, and predictive evaluation. Monte Carlo simulations demonstrate favorable finite-sample performance, and empirical analyses confirm the model’s effectiveness and practical utility in modeling real time series data.