Jinqiao Duan

• PhD
• Chair Professor at Great Bay University

Jacobi structures are known to generalize Poisson structures, encompassing symplectic, cosymplectic, and Lie-Poisson manifolds. Notably, other intriguing geometric structures -- such as contact and locally conformal symplectic manifolds -- also admit Jacobi structures but do not belong to the Poisson category. In this paper, we employ global stocha...

The Onsager--Machlup action functional is an important concept in statistical mechanics and thermodynamics to describe the probability of fluctuations in nonequilibrium systems. It provides a powerful tool for analyzing and predicting the behavior of complex stochastic systems. For diffusion process, the path integral method and the Girsanov trans...

Early warning for epilepsy patients is crucial for their safety and well being, in particular, to prevent or minimize the severity of seizures. Through the patients’ electroencephalography (EEG) data, we propose a meta learning framework to improve the prediction of early ictal signals. The proposed bilevel optimization framework can help automatic...

We introduce the Lebesgue--H\"{o}lder--Dini and Lebesgue--H\"{o}lder spaces $L^p(\mathbb{R};{\mathcal C}_{\vartheta,\varsigma}^{\alpha,\rho}({\mathbb R}^n))$ ($\vartheta\in \{l,b\}, \, \varsigma\in \{d,s,c,w\}$, $p\in (1,+\infty]$ and $\alpha\in [0,1)$), and then use a vector-valued Calder\'{o}n--Zygmund theorem to establish the maximal Lebesgue--H...

Stochastic contact Hamiltonian systems are a class of important mathematical models, which can describe the dissipative properties with odd dimensions in the stochastic environment. In this article, we investigate the numerical dynamics of the stochastic contact Hamiltonian systems via structure-preserving methods. The contact structure-preserving...

This paper proposes a general symplectic Euler scheme for a class of Hamiltonian stochastic differential equations driven by L´evy noise in the sense of Marcus form. The convergence of the symplectic Euler scheme for this Hamiltonian stochastic differential equations is investigated. Realizable numerical implementation of this scheme is also provid...

A general structure-preserving method is proposed for a class of Marcus stochastic Hamiltonian systems driven by additive Lévy noise. The convergence of the symplectic Euler scheme for this systems is investigated by Generalized Milstein Theorem. Realizable numerical implementation of this scheme is also provided in details. Numerical experiments a...

Discovering explicit governing equations of stochastic dynamical systems with both (Gaussian) Brownian noise and (non-Gaussian) L\'evy noise from data is chanllenging due to possible intricate functional forms and the inherent complexity of L\'evy motion. This present research endeavors to develop an evolutionary symbol sparse regression (ESSR) app...

The Onsager--Machlup action functional is an important concept in statistical mechanics and thermodynamics to describe the probability of fluctuations in nonequilibrium systems. It provides a powerful tool for analyzing and predicting the behavior of complex stochastic systems. For diffusion process, the path integral method and the Girsanov transf...

In this paper, we establish some Strichartz estimates for orthonormal functions and probabilistic convergence of density functions related to compact operators on manifolds. Firstly, we present the suitable bound of $\int_{a\leq|s|\leq b}e^{isx}s^{-1+i\gamma}ds(\gamma \in \mathbb{R},a\geq0,b>0),$ $\int_{a\leq s\leq b}e^{isx}s^{-1+i\gamma}ds(\gamma...

We study a rolling model from the perspective of probability. More precisely, we consider a Riemannian manifold rolling against Euclidean space, where the rolling is coupled with random slipping and twisting. The system is modeled by a stochastic differential equation of Stratonovich-type driven by semimartingales on the orthonormal frame bundle. T...

We prove a large deviation principle for stochastic differential equations driven by semimartingales, with additive controls. Conditions are given in terms of characteristics of driven semimartingales, so that if the noise-control pairs satisfy a large deviation principle with some good rate function, so do the solution processes. There is no joint...

Due to the advantages of beamforming technology in multi-antenna wireless communication systems, it has been widely studied and applied. However, its effectiveness in high-mobility scenarios may significantly deteriorate due to severe channel aging. In this paper, we propose an improved Lévy induced stochastic differential equation network (LDE-Ne...

The Schrödinger–Lohe model is an infinite-dimensional non-Abelian generalization of the Kuramoto model, which is brought up as an effective model for the study of quantum synchronization. In the Schrödinger–Lohe model, different wave functions interact with each other, which may lead to all the wave functions becoming arbitrarily close to each othe...

В работе доказывается принцип больших уклонений для стохастических дифференциальных уравнений по семимартингалам с аддитивными управлениями. В терминах характеристик семимартингалов приводятся условия, гарантирующие, что если пары "шум-управление" удовлетворяют принципу больших уклонений с некоторой хорошей функцией скорости, то это же будет верно...

We study the stochastic Camassa–Holm equation with pure jump noise. We prove that if the initial condition of the solution is a solitary wave solution of the unperturbed equation, the solution decomposes into the sum of a randomly modulated solitary wave and a small remainder. Moreover, we derive the equations for the modulation parameters and show...

Detecting early warning indicators for abrupt dynamical transitions in complex systems or high-dimensional observation data are essential in many real-world applications, such as brain diseases, natural disasters, and engineering reliability. To this end, we develop a novel approach: the directed anisotropic diffusion map that captures the latent e...

We study the small mass limit in mean field theory for an interacting particle system with non-Gaussian Lévy noise. When the Lévy noise has a finite second moment, we obtain the limit equation with convergence rate ε + 1 / ε N, by taking first the mean field limit N → ∞ and then the small mass limit ε → 0. If the order of the two limits is exchange...

Exit events induced by noise from the attracting domain containing a stable fixed point are ubiquitous phenomena in physical systems, wherein mean exit time is an important quantity which has been widely used in engineering, physical, chemical and biological fields. In this work, we devise a deep learning method to compute the mean exit time for dy...

We study the averaging principle for a family of multiscale stochastic dynamical systems. The fast and slow components of the systems are driven by two independent stable L\'evy noises, whose stable indexes may be different. The homogenizing index $r_0$ of slow components has a relation with the stable index $\alpha_1$ of the noise of fast componen...

The most probable transition paths (MPTPs) of a stochastic dynamical system are the global minimisers of the Onsager–Machlup action functional and can be described by a necessary but not sufficient condition, the Euler–Lagrange (EL) equation (a second-order differential equation with initial-terminal condi- tions) from a variational principle. This...

Recent advances in data science are opening up new research fields and broadening the range of applications of stochastic dynamical systems. Considering the complexities in real-world systems (e.g., noisy data sets and high dimensionality) and challenges in mathematical foundation of machine learning, this review presents two perspectives in the in...

Starting from a deterministic model, we propose and study a stochastic model for human papillomavirus infection and cervical cancer progression. Our analysis shows that the chronic infection state as random variables which have the ergodic invariant probability measure is necessary for progression from infected cell population to cervical cancer ce...

Bistability is a ubiquitous phenomenon in life sciences. In this paper, two kinds of bistable structures in two-dimensional dynamical systems are studied: one is two one-point attractors, another is a one-point attractor accompanied by a cycle attractor. By the Conley index theory, we prove that there exist other isolated invariant sets besides the...

Early warnings for dynamical transitions in complex systems or high-dimensional observation data are essential in many real world applications, such as gene mutation, brain diseases, natural disasters, financial crises, and engineering reliability. To effectively extract early warning signals, we develop a novel approach: the directed anisotropic d...

In this paper, we investigate a class of McKean-Vlasov stochastic differential equations under L\'evy-type perturbations. We first establish the existence and uniqueness theorem for solutions of the McKean-Vlasov stochastic differential equations by utilizing the Euler-like approximation. Then under some suitable conditions, we show that the soluti...

The rapid development of artificial intelligence has brought considerable convenience, yet also introduces significant security risks. One of the research hotspots is to balance data privacy and utility in the real world of artificial intelligence. The present second-generation artificial neural networks have made tremendous advances, but some big...

We establish the stochastic Strichartz estimate for the fractional Schr\"odinger equation with multiplicative noise. With the help of the deterministic Strichartz estimates, we prove the existence and uniqueness of a global solution to the stochastic fractional nonlinear Schr\"odinger equation in $L_2(\mathbb{R}^n)$ and $H^{1}(\mathbb{R}^n)$, respe...

The rapid development of quantitative portfolio optimization in financial engineering has produced promising results in AI-based algorithmic trading strategies. However, the complexity of financial markets poses challenges for comprehensive simulation due to various factors, such as abrupt transitions, unpredictable hidden causal factors, and heavy...

We study the compactification of nonautonomous systems with autonomous limits and related dynamics. Although the $C^{1}$ extension of the compactification was well established, a great number of problems arising in bifurcation and stability analysis require the compactified systems with high-order smoothness. Inspired by this, we give a criterion f...

Stochastic Gumbel graph networks are proposed to learn high-dimensional time series, where the observed dimensions are often spatially correlated. To that end, the observed randomness and spatial-correlations are captured by learning the drift and diffusion terms of the stochastic differential equation with a Gumble matrix embedding, respectively....

We study homogenization for a class of non-symmetric pure jump Feller processes. The jump intensity involves periodic and aperiodic constituents, as well as oscillating and non-oscillating constituents. This means that the noise can come both from the underlying periodic medium and from external environments, and is allowed to have different scales...

The existing data-driven identification methods for hybrid dynamical systems such as sparse optimization are usually limited to parameter identification for coefficients of pre-defined candidate functions or composition of prescribed function forms, which depend on the prior knowledge of the dynamical models. In this work, we propose a novel data-d...

The prediction of stochastic dynamical systems and the capture of dynamical behaviors are profound problems. In this article, we propose a data-driven framework combining Reservoir Computing and Normalizing Flow to study this issue, which mimics error modeling to improve the traditional Reservoir Computing performance and takes advantage of both ap...

This investigation focuses on discovering the most probable transition pathway for stochastic dynamical systems employing reinforcement learning. We first utilize Onsager-Machlup theory to quantify rare events in stochastic dynamical systems, and then convert the most likely transition path issue into a finite-horizon optimal control problem, becau...

In this work, we establish a stochastic contact variational integrator and its discrete version via stochastic Herglotz variational principle for stochastic contact Hamiltonian systems. A general structure-preserving stochastic contact method is provided to seek the stochastic contact variational integrators. Numerical experiments are performed to...

Multiscale stochastic dynamical systems have been widely adopted to a variety of scientific and engineering problems due to their capability of depicting complex phenomena in many real-world applications. This work is devoted to investigating the effective dynamics for slow–fast stochastic dynamical systems. Given observation data on a short-term p...

Financial applications such as stock price forecasting, usually face an issue that under the predefined labeling rules, it is hard to accurately predict the directions of stock movement. This is because traditional ways of labeling, taking Triple Barrier Method, for example, usually gives us inaccurate or even corrupted labels. To address this issu...

We study the homogenization for a class of non-symmetric pure jump Feller processes. The jump intensity involves periodic and aperiodic constituents, as well as oscillating and non-oscillating constituents. This means that the noise can come both from the underlying periodic medium and from external environments, and is allowed to have different sc...

Many natural systems exhibit phase transition where external environmental conditions spark a shift to a new and sometimes quite different state. Therefore, detecting the behavior of a stochastic dynamic system such as the most probable transition pathway, has made sense.We consider stochastic dynamic systems driven by Brownian motion. Based on var...

We devise a method for predicting certain receptor-ligand binding behaviors, based on stochastic dynamical modelling. We consider the dynamics of a receptor binding to a ligand on the cell membrane, where the receptor and ligand perform different motions and are thus modeled by stochastic differential equations with Gaussian noise or non-Gaussian n...

Transition phenomena between metastable states play an important role in complex systems due to noisy fluctuations. We introduce the Onsager–Machlup theory and the Freidlin–Wentzell theory to quantify rare events in stochastic differential equations. By the variational principle, the most probable transition pathway is the minimizer of the action f...

We study the effective approximation for a nonlocal stochastic Schrödinger equation with a rapidly oscillating, periodically time-dependent potential. We use the natural diffusive scaling of heterogeneous system and study the limit behavior as the scaling parameter tends to 0. This is motivated by data assimilation with non-Gaussian uncertainties....

We study the small mass limit for a class of Hamiltonian systems with multiplicative non-Gaussian Lévy noise. Derivation of the limiting equation depends on the structure of the stochastic Hamiltonian systems, in which a discontinuous noise-induced drift term arises. Firstly, we show that the momentum in the stochastic Hamiltonian system converges...

We establish the large deviation principle for the slow variables in slow-fast dynamical system driven by both Brownian noises and L\'evy noises. The fast variables evolve at much faster time scale than the slow variables, but they are fully inter-dependent. We study the asymptotics of the logarithmic functionals of the slow variables in the three...

We study the existence and stability of small-amplitude periodic waves emerging from fold-Hopf equilibria in a system of one reaction-diffusion equation coupled with one ordinary differential equation. This coupled system includes the FitzHugh-Nagumo system, caricature calcium models and other models in the real-world applications. Based on the rec...

In this article, we employ a collection of stochastic differential equations with drift and diffusion coefficients approximated by neural networks to predict the trend of chaotic time series which has big jump properties. Our contributions are, first, we propose a model called Lévy induced stochastic differential equation network, which explores co...

We establish the large deviation principle for the slow variables in slow-fast dynamical system driven by both Brownian noises and Lévy noises. The fast variables evolve at much faster time scale than the slow variables, but they are fully inter-dependent. We study the asymptotics of the logarithmic functionals of the slow variables in the three re...

Many natural systems exhibit tipping points where changing environmental conditions spark a sudden shift to a new and sometimes quite different state. Global climate change is often associated with the stability of marine carbon stocks. We consider a stochastic carbonate system of the upper ocean to capture such transition phenomena. Based on the O...

We study the spectra and spectral curves for a class of differential operators with asymptotically constant coefficients. These operators widely arise as the linearizations of nonlinear partial differential equations about patterns or nonlinear waves. We present a unified framework to prove the perturbation results on the related spectra and spectr...

In this paper, we firstly analyze the strong convergence of projective integration method for multiscale stochastic dynamical systems driven by α-stable processes, which is used to estimate the effect that the fast components have on slow ones. Then we obtain the pth moment error bounds between the solution of slow component produced by projective...

The information detection of complex systems from data is currently undergoing a revolution, driven by the emergence of big data and machine learning methodology. Discovering governing equations and quantifying the dynamical properties of complex systems are among the central challenges. In this work, we devised a nonparametric approach to learning...

We study the ``periodic homogenization'' for a class of nonlocal partial differential equations of parabolic-type with rapidly oscillating coefficients, related to stochastic differential equations driven by multiplicative isotropic $\alpha$-stable L\'evy noise ($1<\alpha<2$) which is nonlinear in the noise component. Our homogenization method is p...

The concept of quasi-potential plays a central role in understanding the mechanisms of rare events and characterizing the statistics of transition behaviors in stochastic dynamics. Despite its significance, the computation of quasi-potential is a challenging problem with limited existing techniques. In this paper, we devise a machine learning metho...

In this work we construct a stochastic contact variational integrator and its discrete version via stochastic Herglotz variational principle for stochastic contact Hamiltonian systems. A general structure-preserving stochastic contact method is devised, and the stochastic contact variational integrators are established. The implementation of this a...

Many natural systems exhibit tipping points where changing environmental conditions spark a sudden shift to a new and sometimes quite different state. Global climate change is often associated with the stability of marine carbon stocks. We consider a stochastic carbonate system of the upper ocean to capture such transition phenomena. Based on the O...

With the rapid development of computational techniques and scientific tools, great progress of data-driven analysis has been made to extract governing laws of dynamical systems from data. Despite the wide occurrences of non-Gaussian fluctuations, the effective data-driven methods to identify stochastic differential equations with non-Gaussian Lévy...

Recently, extracting data-driven governing laws of dynamical systems through deep learning frameworks has gained much attention in various fields. Moreover, a growing amount of research work tends to transfer deterministic dynamical systems to stochastic dynamical systems, especially those driven by non-Gaussian multiplicative noise. However, many...

In this work, we consider the nonparametric estimation problem of the drift function of stochastic differential equations driven by the [Formula: see text]-stable Lévy process. We first optimize the Kullback–Leibler divergence between the path probabilities of two stochastic differential equations with different drift functions. We then construct t...

Multiscale stochastic dynamical systems have been widely adopted to scientific and engineering problems due to their capability of depicting complex phenomena in many real world applications. This work is devoted to investigating the effective reduced dynamics for a slow-fast stochastic dynamical system. Given observation data on a short-term perio...

Many complex real world phenomena exhibit abrupt, intermittent, or jumping behaviors, which are more suitable to be described by stochastic differential equations under non-Gaussian Lévy noise. Among these complex phenomena, the most likely transition paths between metastable states are important since these rare events may have a high impact in ce...

Mathematical models for complex systems are often accompanied with uncertainties. The goal of this paper is to extract a stochastic differential equation governing model with observation on stationary probability distributions. We develop a neural network framework to learn the drift and diffusion terms of the stochastic differential equation. We i...

We investigate a quantitative network of gene expression dynamics describing the competence development in Bacillus subtilis. First, we introduce an Onsager–Machlup approach to quantify the most probable transition pathway for both excitable and bistable dynamics. Then, we apply a machine learning method to calculate the most probable transition pa...

Many complex real world phenomena exhibit abrupt, intermittent or jumping behaviors, which are more suitable to be described by stochastic differential equations under non-Gaussian L\'evy noise. Among these complex phenomena, the most likely transition paths between metastable states are important since these rare events may have high impact in cer...

We establish the compactly generated shape (H-shape) index theory for local semiflows on complete metric spaces via more general shape index pairs, which allows the phase space to be not separable. The main advantages are that the quotient space $N/E$ is not necessarily metrizable for the shape index pair $(N,E)$ and $N\setminus E$ need not to be a...

We investigate a quantitative network of gene expression dynamics describing the competence development in Bacillus subtilis. First, we introduce an Onsager-Machlup approach to quantify the most probable transition pathway for both excitable and bistable dynamics. Then, we apply a machine learning method to calculate the most probable transition pa...

In this work, we propose a method to learn multivariate probability distributions using sample path data from stochastic differential equations. Specifically, we consider temporally evolving probability distributions (e.g., those produced by integrating local or nonlocal Fokker-Planck equations). We analyze this evolution through machine learning a...

With the rapid increase of valuable observational, experimental and simulated data for complex systems, much efforts have been devoted to identifying governing laws underlying the evolution of these systems. Despite the wide applications of non-Gaussian fluctuations in numerous physical phenomena , the data-driven approaches to extract stochastic d...

During the COVID-19 pandemic, many institutions have announced that their counterparties are struggling to fulfill contracts. Therefore, it is necessary to consider the counterparty default risk when pricing options. After the 2008 financial crisis, a variety of value adjustments have been emphasized in the financial industry. The total value adjus...

Recently, extracting data-driven governing laws of dynamical systems through deep learning frameworks has gained a lot of attention in various fields. Moreover, a growing amount of research work tends to transfer deterministic dynamical systems to stochastic dynamical systems, especially those driven by non-Gaussian multiplicative noise. However, l...

Advances in data science are leading to new progresses in the analysis and understanding of complex dynamics for systems with experimental and observational data. With numerous physical phenomena exhibiting bursting, flights, hopping, and intermittent features, stochastic differential equations with non-Gaussian Lévy noise are suitable to model the...