arXiv subjects

Jinqiao Duan

Publications and source records attributed to Jinqiao Duan.

At least 19 recordsLinked to original sources

arXiv2026

The Numerical Onsager-Machlup action functional for Euler discretized SDEs driven by fractional Brownian motion

In this work, different from previous results, the explicit expression of numerical Onsager-Machlup action functional for Euler discretized SDEs driven by fractional Brownian motion is derived provided the drift coefficient and numerical reference path satisfy some suitable conditions. Then numerical fractional Euler-Lagrange equations for numerical Onsager-Machlup action functional are also obtained. Finally, numerical experiments to illustrate and support the theoretical findings.

math.NA↗

arXiv2026

Noise-intensity bifurcations of transition paths by Morse index formula

We study most probable transition paths (MPTPs) of a gradient diffusion system in $\mathbb{R}^n$ with fixed endpoints, in the framework of Onsager--Machlup (OM) theory. The existence theory for such paths is developed within Lagrangian variational theory: above the Mañé critical value $c_u(L)$ the energy-penalized action admits global minimizers, while below the endpoint-dependent critical value $k_0(L;x_\pm)$ only time-truncated minimizers exist; at regular levels between them, free-time extremals exist. The main result is a $\{0,1\}$-index theorem: at a nondegenerate free-time extremal, the Morse index of the full Hessian equals the fixed-time Morse index plus a correction $ν\in\{0,1\}$, equal to $1$ exactly when the reduced (minimal fixed-time) action satisfies $S''(T)<0$. The loss of local minimality along a branch is thereby split into two mechanisms: a conjugate-point mechanism and a duration mechanism. Combined with a Hamiltonian spectral-flow formula, this yields a bifurcation criterion in the noise intensity $σ$ and a sharp local stability criterion expressed in terms of the crossing instants. As an example, the one-dimensional quartic double well is analyzed in detail.

math.DS↗

arXiv2026

Relaxation and Steady-State Entropy Production for Langevin SPDEs: A Dirichlet-Form Approach

We develop a Dirichlet-form framework for relaxation and steady-state entropy production in preconditioned Langevin stochastic partial differential equations. In infinite dimensions, the usual Fokker--Planck calculations based on ambient Lebesgue densities and the corresponding probability-current densities are generally unavailable. For the detailed-balance class, a quasi-regular symmetric Dirichlet form on the Gibbs state space yields an exact de Bruijn entropy-dissipation formula for regular densities and an integrated inequality for arbitrary finite-entropy initial laws. Self-adjointness gives detailed balance and stationary path reversal, while a coordinate-martingale criterion identifies the associated process with the prescribed SPDE. For the one-dimensional $Φ^4_1$ and convex Allen--Cahn-type Gibbs dynamics, we combine the established strong well-posedness theory with direct verification of the logarithmic derivatives, form closure, quasi-regularity and form--SPDE correspondence, and obtain relative-entropy decay bounds with exponents $2$ and $2(1-λ/m)$, respectively, with $m>λ$ in the latter case. Away from detailed balance, bounded skew-adjoint linear forcing preserves a Gaussian invariant law without requiring commutation between the forcing and covariance. We identify the antisymmetric action on cylinder observables and its Cameron--Martin current and prove that the squared current energy equals the steady-state entropy-production rate defined by forward--reverse path-space relative entropy per unit time, as well as the monotone limit of the Galerkin rates. Under commutation, we additionally obtain an explicit transient Onsager decomposition after a mass quench. Exclusion processes and Gaussian rotors provide finite-state and Gaussian benchmarks.

math.PR↗

arXiv2026

Linear Response Theory for Jump-Driven Stochastic Systems: Transient Statistics and Escape Dynamics

In this paper, we develop a linear response theory for a class of stochastic differential equations driven by jump processes. We investigate the response of the system to small time-dependent perturbations from three complementary perspectives: probability density functions, mean exit times, and escape probabilities. By performing perturbation analyses of the corresponding forward and backward Kolmogorov equations, we derive the first-order response equations governing these statistical quantities and establish explicit linear response formulas, which characterize the sensitivity of the evolution of the probability distribution and the escape behavior of the system to variations in its coefficients.

math.PR↗

arXiv2026

Entropy Production and Reversibility Criteria for Stochastic Evolution Equations

This paper develops a path-space theory of entropy production for a class of stochastic evolution equations on infinite-dimensional Hilbert spaces. Since such spaces have no canonical Lebesgue reference measure, the usual finite-dimensional density formulas do not extend directly. We instead work relative to the invariant Gaussian measure of a reversible Ornstein--Uhlenbeck reference process. Combining an infinite-dimensional Girsanov transform, time reversal of the reference process, and the stationary Fokker--Planck equation relative to the Gaussian measure, we derive an explicit entropy-production formula in terms of an irreversibility field. On the natural test class, this field represents the difference between the forward and reversed nonlinear drifts. Under the standing assumptions, vanishing entropy production is equivalent to vanishing stationary probability current, self-adjointness of the generator in the invariant Hilbert space, detailed balance, and invariance of the stationary path law under time reversal. The reversible case is therefore characterized by a Gaussian-reference gradient structure for the nonlinear drift.

math.AP↗

arXiv2026

Geometric Methods for Stochastic Dynamical Systems

Geometric methods are indispensable for analyzing, predicting, and mitigating the complex behaviors inherent in nonlinear systems. In this regime, the most probable transition path minimizes the Onsager-Machlup action functional, marking the likeliest route across an energy barrier. Lifting the analysis from individual sample paths to the infinite-dimensional space of probability densities, the book recasts these transitions as Schrödinger bridges - optimal paths between boundary distributions defined by minimizing relative entropy - and shows that the Onsager-Machlup path emerges as a special case when metastable states are idealized as Dirac masses. Generalizing further through α-divergences, which connect to entropies and the thermodynamic cost of nonequilibrium transitions, it introduces information geodesics as the resulting optimal density paths, offering a unified geometric account of how complex systems move between metastable regimes under uncertainty.

math.DS↗

arXiv2026

Joint Discovery of Graph Structure and Dynamics in Stochastic Interacting Particle Systems

We study the joint identification of network structure and governing dynamics in stochastic interacting particle systems, which consist of an unknown directed weighted interaction graph with unknown local and non-local interaction components. We formulate the problem as a coupled inverse problem for the graph and the associated basis coefficients, and develop two alternating least-squares-type estimators: a three-block scheme (TALS) and an integrated diagonal-augmented scheme (IALS). The IALS formulation combines the updates of the local and interaction coefficients into a single least-squares subproblem, and is particularly well suited to settings in which the nodewise local dynamics share a common functional template up to node-dependent scaling. We further establish an identifiability result under a rank-2 joint coercivity condition together with an appropriate normalization convention. Synthetic experiments show that the proposed estimators accurately recover both the interaction graph and the dynamical components, and remain robust under stochastic forcing, observation noise, and basis mismatch. We also provide an illustrative real-data application on ictal SEEG recordings, where the learned models produce stable and interpretable dynamical summaries across multiple basis configurations. This work advances a theoretically guaranteed scalable framework for learning stochastic interacting particle systems, with broad potential for data-driven identification in computational biology, neuroscience, and beyond.

cs.SI↗

arXiv2026

Critical Transitions in Interacting Particle Systems: An Onsager-Machlup Action Functional Framework

This paper establishes an indirect approximation theorem for the most probable transition pathway of a stochastic interacting particle system in the mean-field framework. This paper studied the problem of indirect approximation of the most probable transition pathway of an interacting particle system (i.e., a high-dimensional stochastic dynamical system) and its mean field limit equation (McKean-Vlasov stochastic differential equation). This study is based on the Onsager-Machlup action functional, reformulated the problem as an optimal control problem. This paper completes the derivation using the stochastic Pontryagin's Maximum Principle. This paper proves the existence and uniqueness theorem for the solution to the mean-field optimal control problem of McKean-Vlasov stochastic differential equations and establishes a system of equations that determine the control parameters $θ^{*}$ and $θ^{N}$ respectively. There are few studies on the most probable transition pathways of stochastic interacting particle systems, it is still a great challenge to solve the most probable transition pathways directly or to approximate it with the mean field limit system. Therefore, this paper first gave the proof of correspondence between the core equation of Pontryagin's Maximum Principle, that is, Hamiltonian extreme condition equation. In other words, this relationship indirectly illustrates the correspondence between the most probable transition pathways of stochastic interacting particle systems and those of mean-field systems.

math.DS↗

arXiv2026

Efficient Encrypted Computation in Convolutional Spiking Neural Networks with TFHE

With the rapid advancement of AI technology, we have seen more and more concerns on data privacy, leading to some cutting-edge research on machine learning with encrypted computation. Fully Homomorphic Encryption (FHE) is a crucial technology for privacy-preserving computation, while it struggles with continuous non-polynomial functions, as it operates on discrete integers and supports only addition and multiplication. Spiking Neural Networks (SNNs), which use discrete spike signals, naturally complement FHE's characteristics. In this paper, we introduce FHE-DiCSNN, a framework built on the TFHE scheme, utilizing the discrete nature of SNNs for secure and efficient computations. By leveraging bootstrapping techniques, we successfully implement Leaky Integrate-and-Fire (LIF) neuron models on ciphertexts, allowing SNNs of arbitrary depth. Our framework is adaptable to other spiking neuron models, offering a novel approach to homomorphic evaluation of SNNs. Additionally, we integrate convolutional methods inspired by CNNs to enhance accuracy and reduce the simulation time associated with random encoding. Parallel computation techniques further accelerate bootstrapping operations. Experimental results on the MNIST and FashionMNIST datasets validate the effectiveness of FHE-DiCSNN, with a loss of less than 3\% compared to plaintext, respectively, and computation times of under 1 second per prediction. We also apply the model into real medical image classification problems and analyze the parameter optimization and selection.

cs.CR↗

arXiv2026

Homogenization and corrector results for the stochastic non-homogeneous incompressible Navier-Stokes equations with rapid oscillation

In this paper we are concerned with the homogenization property of stochastic non-homogeneous incompressible Navier-Stokes equations with rapid oscillation in a smooth bounded domain of $\mathbb{R}^d$, $d=2,3$, and driven by multiplicative infinite-dimensional Wiener noise. Using two-scale convergence, stochastic compactness and the martingale representation theory, we first show the solutions of original equations converge to the solution of a stochastic non-homogeneous incompressible homogenized system. Also, the energy equation of the homogenized system is established. Furthermore, a corrector result is proved which strengthens the two-scale convergence from weak to strong in the regularity space $H^1(\mathcal{O})$. Since the continuity equation which is of transport type cannot confer any regularization effect, there are some issues for proving the two results, including the difficulties for establishing the stochastic compactness and passing to the limit. We develop new regularity estimates, a stochastic version of lower semicontinuity as well as energy equation to overcome these difficulties.

math.PR↗

arXiv2026

A Jacobi Field Approach to Splitting Detection in Schrödinger Bridge

We study the problem of detecting the onset of path splitting in stochastic interpolation between probability distributions. This question is especially subtle when the target distribution is nonconvex or supported on disconnected components, where interpolating trajectories may separate into distinct branches. Motivated by the stochastic control and Schrödinger bridge viewpoint, we propose a Jacobi field based indicator for identifying candidate splitting times and locations. Our approach is based on the Jacobi field associated with the linearization of an induced interpolating flow. Starting from a stochastic interpolation ansatz, we construct an Eulerian velocity field by conditional averaging and derive its spatial Jacobian in terms of the local posterior geometry of the target sample cloud. This allows us to interpret the symmetric part of the Jacobian as a local strain tensor and to use its spectral structure to quantify the amplification of infinitesimal perturbations along reference trajectories. Numerical experiments on non-convex and disconnected target distributions show that the proposed indicator consistently localizes the emergence of branching regions and captures the temporal development of splitting. These results suggest that Jacobi field analysis provides a natural mathematical framework for studying local instability and splitting phenomena in stochastic interpolation.

math.DS↗

arXiv2026

Beyond Distance: Quantifying Point Cloud Dynamics with Persistent Homology and Dynamic Optimal Transport

We introduce a framework for analyzing topological tipping in time-evolutionary point clouds by extending the recently proposed Topological Optimal Transport (TpOT) distance. While TpOT unifies geometric, homological, and higher-order relations into one metric, its global scalar distance can obscure transient, localized structural reorganizations during dynamic phase transitions. To overcome this limitation, we present a hierarchical dynamic evaluation framework driven by a novel topological and hypergraph reconstruction strategy. Instead of directly interpolating abstract network parameters, our method interpolates the underlying spatial geometry and rigorously recomputes the valid topological structures, ensuring physical fidelity. Along this geodesic, we introduce a set of multi-scale indicators: macroscopic metrics (Topological Distortion and Persistence Entropy) to capture global shifts, and a novel mesoscopic dual-perspective Hypergraph Entropy (node-perspective and edge-perspective) to detect highly sensitive, asynchronous local rewirings. We further propagate the cycle-level entropy change onto individual vertices to form a point-level topological field. Extensive evaluations on physical dynamical systems (Rayleigh-Van der Pol limit cycles, Double-Well cluster fusion), high-dimensional biological aggregation (D'Orsogna model), and longitudinal stroke fMRI data demonstrate the utility of combining transport-based alignment with multi-scale entropy diagnostics for dynamic topological analysis.

stat.ML↗

arXiv2026

Predicting the onset of period-doubling bifurcations via dominant eigenvalue extracted from autocorrelation

Predicting the occurrence of transitions in the qualitative dynamics of many natural systems is crucial, yet it remains a challenging task. Generic early warning signals like variance and lag-1 autocorrelation identify critical slowing down near tipping points but lack practical thresholds for predicting imminent transitions. More recent studies found that the dynamical eigenvalue is rooted in the framework of empirical dynamical modeling and then estimates the dominant eigenvalue of a system from time series, providing a threshold ($|$DEV$|$ = 1) to predict bifurcations and classify their types. However, its application requires careful calibration of the hyperparameters and focuses on reconstructing system dynamics directly from data. Here, we employ Ornstein-Uhlenbeck process to derive analytic approximations for the lag-$τ$ autocorrelation function prior to period-doubling bifurcation thereby estimating the dominant eigenvalue of dynamical systems, named dominant eigenvalue extracted from autocorrelation (DE-AC), and revealing its dynamic behaviour when approaching a period-doubling bifurcation. Theoretically, dominant eigenvalue tends to $-1$ when the system approaches a period-doubling bifurcation. In particular, we evaluated DE-AC on simulation data from cardiac alternans model and on experimental data from chick heart aggregates undergoing a period-doubling bifurcation. DE-AC reliably detected the beginning of the cardiac arrhythmia (period-doubling bifurcation) in most cases. Moreover, it demonstrated superior sensitivity and specificity as an early warning signal compared to the three widely used indicators -- variance, lag-1 autocorrelation, and dynamical eigenvalue. Our theoretical and empirical results suggest that DE-AC represents a quantitative measure for predicting the onset of potentially dangerous alternating rhythms in the heart.

nlin.CD↗

arXiv2026

Nonlocal Kramers-Moyal formulas and data-driven discovery of stochastic dynamical systems with multiplicative Lévy noise

Traditional data-driven methods, effective for deterministic systems or stochastic differential equations (SDEs) with Gaussian noise, fail to handle the discontinuous sample paths and heavy-tailed fluctuations characteristic of Lévy processes, particularly when the noise is state-dependent. To bridge this gap, we establish nonlocal Kramers-Moyal formulas, rigorously generalizing the classical Kramers-Moyal relations to SDEs with multiplicative Lévy noise. These formulas provide a direct link between short-time transition probability densities (or sample path statistics) and the underlying SDE coefficients: the drift vector, diffusion matrix, Lévy jump measure kernel, and Lévy noise intensity functions. Leveraging these theoretical foundations, we develop novel data-driven algorithms capable of simultaneously identifying all governing components from data and establish convergence results and error analysis for the algorithms. We validate the framework through extensive numerical experiments on prototypical systems. This work provides a principled and practical toolbox for discovering interpretable SDE models governing complex systems influenced by discontinuous, heavy-tailed, state-dependent fluctuations, with broad applicability in climate science, neuroscience, epidemiology, finance, and biological physics.

math.DS↗

arXiv2025

Learning Lévy density via adaptive RKHS regression with bi-level optimization

We propose a nonparametric method to learn the Lévy density from probability density data governed by a nonlocal Fokker-Planck equation. We recast the problem as identifying the kernel in a nonlocal integral operator from discrete data, which leads to an ill-posed inverse problem. To regularize it, we construct an adaptive reproducing kernel Hilbert space (RKHS) whose kernel is built directly from the data. Under standard source and spectral decay conditions, we show that the reconstruction error decays in the mesh size at a near optimal rate. Importantly, we develop a generalized singular value decomposition (GSVD)-based bilevel optimization algorithm to choose the regularization parameter, leading to efficient and robust computation of the regularized estimator. Numerical experiments for several Lévy densities, drift fields and data types (PDE-based densities and sample ensemble-based KDE reconstructions) demonstrate that our bilevel RKHS method outperforms classical L-curve and generalized cross-validation strategies and that the adaptive RKHS norm is more accurate and robust than $L^2_ρ$- and $\ell^2$-based regularization.

math.NA↗

arXiv2025

Artificial intelligence as a surrogate brain: Bridging neural dynamical models and data

Recent breakthroughs in artificial intelligence (AI) are reshaping the way we construct computational counterparts of the brain, giving rise to a new class of ``surrogate brains''. In contrast to conventional hypothesis-driven biophysical models, the AI-based surrogate brain encompasses a broad spectrum of data-driven approaches to solve the inverse problem, with the primary objective of accurately predicting future whole-brain dynamics with historical data. Here, we introduce a unified framework of constructing an AI-based surrogate brain that integrates forward modeling, inverse problem solving, and model evaluation. Leveraging the expressive power of AI models and large-scale brain data, surrogate brains open a new window for decoding neural systems and forecasting complex dynamics with high dimensionality, nonlinearity, and adaptability. We highlight that the learned surrogate brain serves as a simulation platform for dynamical systems analysis, virtual perturbation, and model-guided neurostimulation. We envision that the AI-based surrogate brain will provide a functional bridge between theoretical neuroscience and translational neuroengineering.

q-bio.NC↗

arXiv2025

Fokker-Planck equation for stochastic heat equations

This work is devoted to the study of the Fokker--Planck equation for a stochastic heat equation with an additive $Q$-Wiener noise and non-homogeneous boundary conditions. We explicitly construct the probability density function and establish the associated Fokker--Planck equation by applying the eigenfunction expansion technique. Moreover, the Feynman--Kac formula is used to obtain the probabilistic representation of the solution. The analysis is further extended to cases with multiplicative noise involving nonlocal diffusion operators under homogeneous boundary conditions, as well as the corresponding Kardar--Parisi--Zhang (KPZ) equation. Notably, the evolution of the probability density function for the stochastic heat equation depends critically on the spatial location.

math.PR↗

arXiv2025

Stochastic Perturbations in the Fractional Nonlinear Schrödinger Equation: Well-posedness and Blow-up

This work investigates radial solutions for nonlinear fractional Schrödinger equations driven by multiplicative noise. Leveraging radial deterministic and stochastic Strichartz estimates, we establish local well-posedness in the energy-subcritical regime for the stochastic fractional nonlinear Schrödinger equation. Global existence is subsequently demonstrated through stochastic evolution of mass and energy. In focusing supercritical settings, we derive blow-up criteria via localized virial inequality, revealing how multiplicative noise measurably suppresses blow-up formation compared to deterministic dynamics.

math.AP↗