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Zhe Jiao

Publications and source records attributed to Zhe Jiao.

10 recordsLinked to original sources
arXiv2025

Controllability and mixing for acoustic wave motions

This paper concerns the dynamical behaviors of acoustic wave motion driven by a force acting through the boundary. If the boundary force is a suitable control, we show that the dynamical system associated to the acoustic wave motion is exactly controllable. Furthermore, when it is random perturbation of white noise type, we prove that the corresponding stochastic system is strong mixing. The bridge between these two problems is the observability inequality, which will be established in this work.

math.AP↗
arXiv2025

Solving a class of stochastic optimal control problems by physics-informed neural networks

The aim of this work is to develop a deep learning method for solving high-dimensional stochastic control problems based on the Hamilton--Jacobi--Bellman (HJB) equation and physics-informed learning. Our approach is to parameterize the feedback control and the value function using a decoupled neural network with multiple outputs. We train this network by using a loss function with penalty terms that enforce the HJB equation along the sampled trajectories generated by the controlled system. More significantly, numerical results on various applications are carried out to demonstrate that the proposed approach is efficient and applicable.

math.OC↗
arXiv2025

Non-convergence and convergence of bounded solutions to semilinear wave equation with dissipative boundary condition

This paper is concerned with the long-time dynamics of semilinear wave equation subject to dissipative boundary condition. To do so, we first analyze the set of equilibria, and show it could contain infinitely many elements. Second, we show that, for some nonlinear interior sources, the wave equations have solutions that do not stabilize to any single function, while they approach a continuum of such functions. Finally, if the interior source is a Łojasiewicz-type function, the solution of the wave equation converges to an equilibrium at a rate that depends on the Łojasiewicz exponent, although the set of equilibria is infinite.

math.AP↗
arXiv2025

Mixing for acoustic wave motion with boundary random force

This paper concerns about the large time behavior of acoustic wave motion driven by a random force acting through the boundary. We begin with an abstract result showing the interconnection between the regularity of Markov semigroup generated by a stochastic evolution equation and the observability property of the corresponding adjoint system. This result is then applied to study the mixing for acoustic wave system with a boundary random perturbation of the white noise type. We shall show there exists a unique invariant measure for the stochastic wave system, and the law of the solution to the system converges to this invariant measure weakly.

math.AP↗
arXiv2024

Implicit-explicit time discretization schemes for a class of semilinear wave equations with nonautonomous dampings

This paper is concerned about the implicit-explicit (IMEX) methods for a class of dissipative wave systems with time-varying velocity feedbacks and nonlinear potential energies, equipped with different boundary conditions. Firstly, we approximate the problems by using a vanilla IMEX method, which is a second-order scheme for the problems when the damping terms are time-independent. However, rigors analysis shows that the error rate declines from second to first order due to the nonautonomous dampings. To recover the convergence order, we propose a revised IMEX scheme and apply it to the nonautonomous wave equations with a kinetic boundary condition. Our numerical experiments demonstrate that the revised scheme can not only achieve second-order accuracy but also improve the computational efficiency.

math.NA↗
arXiv2024

The central limit theorem for slow-fast systems with Lévy noise

We consider a slow-fast stochastic differential system with Lévy noise. We will employ the perturbed test function method to study the normal deviation of the slow-fast system. Our main result states that the deviation can be approximated by a Gaussian process and the central limit theorem is obtained for the system.

math.PR↗
arXiv2024

Emergence of heavy tails in homogenized stochastic gradient descent

It has repeatedly been observed that loss minimization by stochastic gradient descent (SGD) leads to heavy-tailed distributions of neural network parameters. Here, we analyze a continuous diffusion approximation of SGD, called homogenized stochastic gradient descent, show that it behaves asymptotically heavy-tailed, and give explicit upper and lower bounds on its tail-index. We validate these bounds in numerical experiments and show that they are typically close approximations to the empirical tail-index of SGD iterates. In addition, their explicit form enables us to quantify the interplay between optimization parameters and the tail-index. Doing so, we contribute to the ongoing discussion on links between heavy tails and the generalization performance of neural networks as well as the ability of SGD to avoid suboptimal local minima.

stat.ML↗
arXiv2022

Stability for nonlinear wave motions damped by time-dependent frictions

We are concerned with the dynamical behavior of solutions to semilinear wave systems with time-varying damping and nonconvex force potential. Our result shows that the dynamical behavior of solution is asymptotically stable without any bifurcation and chaos. And it is a sharp condition on the damping coefficient for the solution to converge to some equilibrium. To illustrate our theoretical results, we provide some numerical simulations for dissipative sine-Gordon equation and dissipative Klein-Gordon equation.

math.AP↗
arXiv2021

Averaging principle of stochastic Burgers equation driven by Lévy processes

We are concerned about the averaging principle for the stochastic Burgers equation with slow-fast time scale. This slow-fast system is driven by Lévy processes. Under some appropriate conditions, we show that the slow component of this system strongly converges to a limit, which is characterized by the solution of stochastic Burgers equation whose coefficients are averaged with respect to the stationary measure of the fast-varying jump-diffusion. To illustrate our theoretical result, we provide some numerical simulations.

math.PR↗