Minghua Chen, Weihua Deng
This paper focuses on providing the high order algorithms for the space-time tempered fractional diffusion-wave equation. The designed schemes are unconditionally stable and have the global truncation error $\mathcal{O}(τ^2+h^2)$, being theoretically proved and numerically verified.
Minghua Chen, Weihua Deng
We provide and analyze the high order algorithms for the model describing the functional distributions of particles performing anomalous motion with power-law jump length and tempered power-law waiting time. The model is derived in [Wu, Deng, and Barkai, Phys. Rev. E., 84 (2016), 032151], being called the time-tempered fractional Feynman-Kac equation. The key step of designing the algorithms is to discretize the time tempered fractional substantial derivative, being defined as $${^S\!}D_t^{γ,\widetildeλ} G(x,p,t)\!=\!D_t^{γ,\widetildeλ} G(x,p,t)\!-\!λ^γG(x,p,t) ~{\rm with}~\widetildeλ=λ+ pU(x),\, p=ρ+Jη,\, J=\sqrt{-1},$$ where $$D_t^{γ,\widetildeλ} G(x,p,t) =\frac{1}{Γ(1-γ)} \left[\frac{\partial}{\partial t}+\widetildeλ \right] \int_{0}^t{\left(t-z\right)^{-γ}}e^{-\widetildeλ\cdot(t-z)}{G(x,p,z)}dz,$$ and $λ\ge 0$, $0<γ<1$, $ρ>0$, and $η$ is a real number. The designed schemes are unconditionally stable and have the global truncation error $\mathcal{O}(τ^2+h^2)$, being theoretically proved and numerically verified in {\em complex} space. Moreover, some simulations for the distributions of the first passage time are performed, and the second order convergence is also obtained for solving the `physical' equation (without artificial source term).
Minghua Chen, Weihua Deng
Recently, nonlocal models attract the wide interests of scientist. They mainly come from two applied scientific fields: peridyanmics and anomalous diffusion. Even though the matrices of the algebraic equation corresponding the nonlocal models are usually Toeplitz (denote a0 as the principal diagonal element, a1 as the trailing diagonal element, etc). There are still some differences for the models in these two fields. For the model of anomalous diffusion, a0/a1 is uniformly bounded; most of the time, a0/a1 of the model for peridyanmics is unbounded as the stepsize h tends to zero. Based on the uniform boundedness of a0/a1, the convergence of the two-grid method is well established [Chan, Chang, and Sun, SIAM J. Sci. Comput., 19 (1998), pp. 516--529; Pang and Sun, J. Comput. Phys., 231 (2012), pp. 693--703; Chen, Wang, Cheng, and Deng, BIT, 54 (2014), pp. 623--647]. This paper provides the detailed proof of the convergence of the two-grid method for the nonlocal model of peridynamics. Some special cases of the full multigrid and the V-cycle multigrid are also discussed. The numerical experiments are performed to verify the convergence.
Yanyan Yu, Weihua Deng, Yujiang Wu
The continuous time random walk (CTRW) underlies many fundamental processes in non-equilibrium statistical physics. When the jump length of CTRW obeys a power-law distribution, its corresponding Fokker-Planck equation has space fractional derivative, which characterizes Lévy flights. Sometimes the infinite variance of Lévy flight discourages it as a physical approach; exponentially tempering the power-law jump length of CTRW makes it more `physical' and the tempered space fractional diffusion equation appears. This paper provides the basic strategy of deriving the high order quasi-compact discretizations for space fractional derivative and tempered space fractional derivative. The fourth order quasi-compact discretization for space fractional derivative is applied to solve space fractional diffusion equation and the unconditional stability and convergence of the scheme are theoretically proved and numerically verified. Furthermore, the tempered space fractional diffusion equation is effectively solved by its counterpart of the fourth order quasi-compact scheme; and the convergence orders are verified numerically.
Jing Sun, Daxin Nie, Weihua Deng
Applying proper orthogonal decomposition to a usual finite element (FE) formulation for space fractional partial differential equation, we get a reduced FE model, which greatly reduces the complexity of computation. Then, the stability analysis and error estimate for the reduced model are presented. Finally, we verify the effectiveness of the algorithm by numerical experiments.
Zhijiang Zhang, Weihua Deng
For describing the probability distribution of the positions and times of particles performing anomalous motion, fractional PDEs are derived from the continuous time random walk models with waiting time distribution having divergent first order moment and/or jump length distribution which has divergent second order moment. It can be noted that the fractional PDEs are essentially dealing with the multiscale issues. Generally the regularity of the solutions for fractional PDEs is weak at the areas close to boundary and initial time. This paper focuses on developing the applications of wavelet bases to numerically solving fractional PDEs and digging out the potential benefits of wavelet methods comparing with other numerical methods, especially in the aspects of realizing preconditioning, adaptivity, and keeping the Toeplitz structure. More specifically, the contributions of this paper are as follows: 1. the techniques of efficiently generating stiffness matrix with computational cost $\mathcal{O}(2^J)$ are provided for first, second, and any order bases; 2. theoretically and numerically discuss the effective preconditioner for time-independent equation and multigrid method for time-dependent equation, respectively; 3. the wavelet adaptivity is detailedly discussed and numerically applied to solving the time-dependent (independent) equations. In fact, having reliable, simple, and local regularity indicators is the striking benefit of the wavelet in adaptively solving fractional PDEs (it seems hard to give a local posteriori error estimate for the adaptive finite element method because of the global property of the operator).
Wenjun Cui, Qiyu Kang, Xuhao Li et al.
Neural differential equation models have garnered significant attention in recent years for their effectiveness in machine learning applications.Among these, fractional differential equations (FDEs) have emerged as a promising tool due to their ability to capture memory-dependent dynamics, which are often challenging to model with traditional integer-order approaches.While existing models have primarily focused on constant-order fractional derivatives, variable-order fractional operators offer a more flexible and expressive framework for modeling complex memory patterns. In this work, we introduce the Neural Variable-Order Fractional Differential Equation network (NvoFDE), a novel neural network framework that integrates variable-order fractional derivatives with learnable neural networks.Our framework allows for the modeling of adaptive derivative orders dependent on hidden features, capturing more complex feature-updating dynamics and providing enhanced flexibility. We conduct extensive experiments across multiple graph datasets to validate the effectiveness of our approach.Our results demonstrate that NvoFDE outperforms traditional constant-order fractional and integer models across a range of tasks, showcasing its superior adaptability and performance.
Qiyu Kang, Xuhao Li, Kai Zhao et al.
Fractional-order differential equations (FDEs) enhance traditional differential equations by extending the order of differential operators from integers to real numbers, offering greater flexibility in modeling complex dynamical systems with nonlocal characteristics. Recent progress at the intersection of FDEs and deep learning has catalyzed a new wave of innovative models, demonstrating the potential to address challenges such as graph representation learning. However, training neural FDEs has primarily relied on direct differentiation through forward-pass operations in FDE numerical solvers, leading to increased memory usage and computational complexity, particularly in large-scale applications. To address these challenges, we propose a scalable adjoint backpropagation method for training neural FDEs by solving an augmented FDE backward in time, which substantially reduces memory requirements. This approach provides a practical neural FDE toolbox and holds considerable promise for diverse applications. We demonstrate the effectiveness of our method in several tasks, achieving performance comparable to baseline models while significantly reducing computational overhead.
Chongcan Li, Yong Cong, Weihua Deng
We preprocess the raw NMR spectrum and extract key characteristic features by using two different methodologies, called equidistant sampling and peak sampling for subsequent substructure pattern recognition; meanwhile may provide the alternative strategy to address the imbalance issue of the NMR dataset frequently encountered in dataset collection of statistical modeling and establish two conventional SVM and KNN models to assess the capability of two feature selection, respectively. Our results in this study show that the models using the selected features of peak sampling outperform the ones using the other. Then we build the Recurrent Neural Network (RNN) model trained by Data B collected from peak sampling. Furthermore, we illustrate the easier optimization of hyper parameters and the better generalization ability of the RNN deep learning model by comparison with traditional machine learning SVM and KNN models in detail.
Minghua Chen, Weihua Deng, Stefano Serra-Capizzano
In this paper we derive new uniform convergence estimates for the V-cycle MGM applied to symmetric positive definite Toeplitz block tridiagonal matrices, by also discussing few connections with previous results. More concretely, the contributions of this paper are as follows: (1) It tackles the Toeplitz systems directly for the elliptic PDEs. (2) Simple (traditional) restriction operator and prolongation operator are employed in order to handle general Toeplitz systems at each level of the recursion. Such a technique is then applied to systems of algebraic equations generated by the difference scheme of the two-dimensional fractional Feynman-Kac equation, which describes the joint probability density function of non-Brownian motion. In particular, we consider the two coarsening strategies, i.e., doubling the mesh size (geometric MGM) and Galerkin approach (algebraic MGM), which lead to the distinct coarsening stiffness matrices in the general case: however, several numerical experiments show that the two algorithms produce almost the same error behaviour.
Zhijiang Zhang, Weihua Deng
The functional distributions of particle trajectories have wide applications, including the occupation time in half-space, the first passage time, and the maximal displacement, etc. The models discussed in this paper are for characterizing the distribution of the functionals of the paths of anomalous diffusion described by time-space fractional diffusion equation. This paper focuses on providing effective computation methods for the models. Two kinds of time stepping schemes are proposed for the fractional substantial derivative. The multiresolution Galerkin method with wavelet B-spline is used for space approximation. Compared with the finite element or spectral polynomial bases, the wavelet B-spline bases have the advantage of keeping the Toeplitz structure of the stiffness matrix, and being easy to generate the matrix elements and to perform preconditioning. The unconditional stability and convergence of the provided schemes are theoretically proved and numerically verified. Finally, we also discuss the efficient implementations and some extensions of the schemes, such as the wavelet preconditioning and the non-uniform time discretization.