Shukai Du

Thomas Brown, Shukai Du, Hasan Eruslu et al.

We consider the problem of waves propagating in a viscoelastic solid. For the material properties of the solid we consider both classical and fractional differentiation in time versions of the Zener, Maxwell, and Voigt models, where the coupling of different models within the same solid are covered as well. Stability of each model is investigated in the Laplace domain, and these are then translated to time-domain estimates. With the use of semigroup theory, some time-domain results are also given which avoid using the Laplace transform and give sharper estimates. We take the time to develop and explain the theory necessary to understand the relation between the equations we solve in the Laplace domain and those in the time-domain which are written using the language of causal tempered distributions. Finally we offer some numerical experiments that highlight some of the differences between the models and how different parameters effect the results.

Shukai Du, Nailin Du

This paper investigates the least-squares projection method for bounded linear operators, which provides a natural regularization scheme by projection for many ill-posed problems. Yet, without additional assumptions, the convergence of this approximation scheme cannot be guaranteed. We reveal that the convergence of least-squares projection method is determined by two independent factors -- the kernel approximability and the offset angle. The kernel approximability is a necessary condition of convergence described with kernel $N(T)$ and its subspaces $N(T){\cap}X_n$, and we give several equivalent characterizations for it (Theorem 1). The offset angle of $X_n$ is defined as the largest canonical angle between space $T^*T(X_n)$ and $T^{\dagger}T(X_n)$ (which are subspaces of $N(T)^\bot$), and it geometrically reflects the rate of convergence (Theorem 2). The paper also presents new observations for the unconvergence examples of Seidman [10, Example 3.1] and Du [2, Example 2.10] under the notions of kernel approximability and offset angle.

Shukai Du, Junzhe Zhang, Yiming Li

Flow matching trains a neural velocity field by regression against a target velocity associated with a prescribed probability path connecting a simple initial distribution to the data distribution. A central design choice is the path itself. Existing constructions, including rectified and optimal-transport-based paths, transport samples along straight lines between coupled endpoints and thus cover only a narrow class of dynamics. We observe that this corresponds to the simplest case of the least-action principle in classical mechanics, in which the kinetic Lagrangian yields free-particle straight-line trajectories. Building on this observation, we propose Lagrangian flow matching, a physics-based framework in which the probability path and velocity field are determined by minimizing the action of a general Lagrangian subject to the continuity equation and the prescribed endpoints. We show that this dynamic problem admits an equivalent static optimal transport (OT) formulation, yielding a family of simulation-free training objectives that recover OT-based flow matching as the kinetic special case and the trigonometric variance-preserving diffusion path as the harmonic-oscillator case. More general Lagrangians give rise to new probability paths and velocity fields, and numerical experiments show that they induce meaningful changes in the learned dynamics while remaining competitive with existing conditional flow matching models.