Dongling Wang

Dongling Wang, Aiguo Xiao, Jun Zou

In this work, we study the long time behaviors, including asymptotic contractivity and dissipativity, of the solutions to several numerical methods for fractional ordinary differential equations (F-ODEs). The existing algebraic contractivity and dissipativity rates of the solutions to the scalar F-ODEs are first improved. In order to study the long time behavior of numerical solutions to fractional backward differential formulas (F-BDFs), two crucial analytical techniques are developed, with the first one for the discrete version of the fractional generalization of the traditional Leibniz rule, and the other for the algebraic decay rate of the solution to a linear Volterra difference equation. By mens of these auxiliary tools and some natural conditions, the solutions to F-BDFs are shown to be contractive and dissipative, and also preserve the exact contractivity rate of the continuous solutions. Two typical F-BDFs, based on the Grunwald-Letnikov formula and L1 method respectively, are studied. For high order F-BDFs, including some second order F-BDFs and $3-α$ order method, their numerical contractivity and dissipativity are also developed under some slightly stronger conditions. Numerical experiments are presented to validate the long time qualitative characteristics of the solutions to F-BDFs, revealing very different decay rates of the numerical solutions in terms of the the initial values between F-ODEs and integer ODEs and demonstrating the superiority of the structure-preserving numerical methods.

Zihao Shi, Dongling Wang

Neural networks have demonstrated significant potential in solving partial differential equations (PDEs). While global approaches such as Physics-Informed Neural Networks (PINNs) offer promising capabilities, they often lack inherent temporal causality, which can limit their accuracy and stability for time-dependent problems. In contrast, local training frameworks that progressively update network parameters over time are naturally suited for evolving PDEs. However, a critical challenge remains: many physical systems possess intrinsic invariants -- such as energy or mass -- that must be preserved to ensure physically meaningful solutions. This paper addresses this challenge by enhancing the Time-Evolving Natural Gradient (TENG) method, a recently proposed local training framework. We introduce two complementary techniques: (i) a relaxation algorithm that ensures the target solution $u_{\text{target}}$ preserves both quadratic and general nonlinear invariants of the original system, providing a structure-preserving learning target; and (ii) a projection technique that maps the updated network parameters $θ(t)$ back onto the invariant manifold, ensuring the final neural network solution strictly adheres to the conservation laws. Numerical experiments on the inviscid Burgers equation, Korteweg-de Vries equation, and acoustic wave equation demonstrate that our proposed approach significantly improves conservation properties while maintaining high accuracy.

Renjie Ding, Dongling Wang, Jun Zou

The Kaczmarz method is widely recognized as an efficient iterative algorithm for solving large-scale linear systems, owing to its simplicity and low memory requirements. However, the development of its nonlinear extensions for solving large-scale nonlinear systems has seen limited progress. In this work, we introduce a new family of momentum-accelerated averaging block nonlinear Kaczmarz methods tailored for large-scale nonlinear systems and ill-posed problems. Our contributions are twofold: (1) We develop an adaptive strategy for selecting step sizes and momentum coefficients, leading to a new average block nonlinear Kaczmarz method with adaptive momentum (ABNKAm). This algorithm achieves high computational efficiency by requiring only minimal inner-product computations per iteration, which significantly reduces both arithmetic complexity and memory usage. (2) We establish rigorous convergence of the ABNKAm under mild assumptions, proving that the method converges exponentially to the unique solution nearest to the initial point. Moreover, under suitable conditions, we provide a theoretical justification of acceleration of the proposed ABNKAm with momentum. Extensive numerical experiments demonstrate that ABNKAm outperforms existing nonlinear Kaczmarz variants in terms of both iteration count and computational time, with particularly notable gains in large-scale problems.