arXiv subjects

Jinqiao Duan

Publications and source records attributed to Jinqiao Duan.

At least 37 records · Page 2Linked to original sources

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↗

arXiv2025

Adaptive control for multi-scale stochastic dynamical systems with stochastic next generation reservoir computing

The rapid advancement of neuroscience and machine learning has established data-driven stochastic dynamical system modeling as a powerful tool for understanding and controlling high-dimensional, spatio-temporal processes. We introduce the stochastic next-generation reservoir computing (NG-RC) controller, a framework that integrates the computational efficiency of NG-RC with stochastic analysis to enable robust event-triggered control in multiscale stochastic systems. The asymptotic stability of the controller is rigorously proven via an extended stochastic LaSalle theorem, providing theoretical guarantees for amplitude regulation in nonlinear stochastic dynamics. Numerical experiments on a stochastic Van-der-Pol system subject to both additive and multiplicative noise validate the algorithm, demonstrating its convergence rate across varying temporal scales and noise intensities. To bridge theoretical insights with real-world applications, we deploy the controller to modulate pathological dynamics reconstructed from epileptic EEG data. This work advances a theoretically guaranteed scalable framework for adaptive control of stochastic systems, with broad potential for data-driven decision making in engineering, neuroscience, and beyond.

math.DS↗

arXiv2025

The stochastic Hamilton-Jacobi-Bellman equation on Jacobi structures

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 stochastic analysis techniques, initially developed by Meyer and Schwartz, to rigorously introduce stochastic Hamiltonian systems on Jacobi manifolds. We then propose a stochastic Hamilton-Jacobi-Bellman (HJB) framework as an alternative perspective on the underlying dynamics. We emphasize that many of our results extend the work of Bismut [Bis80, Bis81] and Lázaro-Camí \& Ortega [LCO08, LCO09]. Furthermore, aspects of our geometric Hamilton-Jacobi theory in the stochastic setting draw inspiration from the deterministic contributions of Abraham \& Marsden [AM78], de León \& Sardón [dLS17], Esen et al. [EdLSZ21], and related literature.

math.DG↗

arXiv2025

Action Functional as an Early Warning Indicator in the Space of Probability Measures via Schrödinger Bridge

Critical transitions and tipping phenomena between two meta-stable states in stochastic dynamical systems are a significant scientific issue. In this work, we expand the methodology of identifying the most probable transition pathway between two meta-stable states with Onsager-Machlup action functional, to investigate the evolutionary transition dynamics between two meta-stable invariant sets with Schrödinger bridge. In contrast to existing methodologies such as statistical analysis, bifurcation theory, information theory, statistical physics, topology, and graph theory for early warning indicators, we introduce a novel framework on Early Warning Signals (EWS) within the realm of probability measures that align with the entropy production rate (EPR). To validate our framework, we apply it to the Morris-Lecar model and investigate the transition dynamics between a meta-stable state and a stable invariant set (the limit cycle or homoclinic orbit) under various conditions. Additionally, we analyze real Alzheimer's data from the Alzheimer's Disease Neuroimaging Initiative database to explore EWS indicating the transition from healthy to pre-AD states. This framework not only expands the transition pathway to encompass measures between two specified densities on invariant sets, but also demonstrates the potential of our early warning indicators for complex diseases.

math.DS↗

arXiv2024

Probability Flow Approach to the Onsager--Machlup Functional for Jump-Diffusion Processes

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 transformation are two main approaches to construct the Onsager--Machlup functional. However, it is a long-standing challenge to apply these two methods to the jump-diffusion process, because the complexity of jump noise presents intrinsic technical barriers to derive the Onsager--Machlup functional. In this work, we propose a new strategy to solve this problem by utilizing the equivalent probabilistic flow between the pure diffusion process and the jump-diffusion process. For the first time, we rigorously establish the closed-form expression of the Onsager--Machlup functional for jump-diffusion processes with finite jump activity, which include an important term of the Lévy intensity at the origin. The same probability flow approach is further applied to the Lévy process with infinite jump activity, and yields a time-discrete version of the Onsager--Machlup functional.

math.PR↗

arXiv2024

A new maximal regularity for parabolic equations and an application

We introduce the Lebesgue--Hölder--Dini and Lebesgue--Hölder spaces $L^p(\mathbb{R};{\mathcal C}_{\vartheta,ς}^{α,ρ}({\mathbb R}^n))$ ($\vartheta\in \{l,b\}, \, ς\in \{d,s,c,w\}$, $p\in (1,+\infty]$ and $α\in [0,1)$), and then use a vector-valued Calderón--Zygmund theorem to establish the maximal Lebesgue--Hölder--Dini and Lebesgue--Hölder regularity for a class of parabolic equations. As an application, we obtain the unique strong solvability of the following stochastic differential equation \begin{eqnarray*} X_{s,t}(x)=x+\int\limits_s^tb(r,X_{s,r}(x))dr+W_t-W_{s}, \ \ t\in [s,T], \ x\in \mathbb{R}^n, \ s\in [0,T], \end{eqnarray*} for the low regularity growing drift in critical Lebesgue--Hölder--Dini spaces $L^p([0,T];{\mathcal C}^{\frac{2}{p}-1,ρ}_{l,d}({\mathbb R}^n;{\mathbb R}^n))$ ($p\in (1,2]$), where $\{W_t\}_{0\leq t\leq T}$ is a $n$-dimensional standard Wiener process. In particular, when $p=2$ we give a partially affirmative answer to a longstanding open problem, which was proposed by Krylov and Röckner for $b\in L^2([0,T];L^\infty({\mathbb R}^n;{\mathbb R}^n))$ based upon their work ({\em Probab. Theory Relat. Fields 131(2): 154--196, 2005}).

math.PR↗

arXiv2024

Numerical integrations of stochastic contact Hamiltonian systems via stochastic contact Hamilton-Jacobi equation

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 schemes are constructed by the stochastic contact Hamilton-Jacobi equation. A general numerical approximation method of the stochastic contact Hamilton-Jacobi equation is devised, and the convergent order theorem is provided, too. Numerical tests are shown to confirm the theoretical results and the usability of proposed approach.

math.NA↗

arXiv2024

Strichartz estimates for orthonormal functions and probabilistic convergence of density functions of compact operators on manifolds

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 $\int_{a\leq|s|\leq b}e^{isx}s^{-1+iγ}ds$ for the cases $γ\in \mathbb{R},a\geq0,b>0,$ $γ\in \mathbb{R},γ\neq0,a,b\in \mathbb{R}$ and $γ\in \mathbb{R}$, which extends the result of Page 204 of Vega (199-211,IMA Vol. Math. Appl., 42, 1992.) Secondly, we prove that $\left|γ\int_{a}^{b}e^{isx}s^{-1+iγ}ds\right|\leq C(1+|γ|)^{2}(γ\in \mathbb{R},a,b\in \mathbb{R}),$ where $C$ is independent of $γ,a, b$, which extends Lemma 1 of Bez et al. (Forum of Mathematics, Sigma, 9(2021), 1-52). Thirdly, we extend the result of Theorems 8, 9 of R. Frank, J. Sabin (Amer. J. Math. 139(2017), 1649-1691.) with the aid of the suitable bound of the above complex integrals established in this paper. Fourthly, we establish the Strichartz estimates for orthonormal functions related to Boussinesq operator on the real line for both small time interval and large time interval and on the torus with small time interval; we also establish the convergence result of some compact operators in Schatten norm. Fifthly, we establish the convergence result related to nonlinear part of the solution to some operator equations in Schatten spaces. Finally, inspired by the work of Hadama and Yamamoto (Probabilistic Strichartz estimates in Schatten classes and their applications to Hartree equation, arxiv:2311.02713v1.), for $γ_{0}\in \mathfrak{S}^{2}$, we establish the probabilistic convergence of density functions of compact operator on manifolds with full randomization, which improves the result of Corollary 1.2 of Bez et al. (Selecta Math. 26(2020), 24 pp) in the probabilistic sense.

math.PR↗

arXiv2024

An evolutionary approach for discovering non-Gaussian stochastic dynamical systems based on nonlocal Kramers-Moyal formulas

Discovering explicit governing equations of stochastic dynamical systems with both (Gaussian) Brownian noise and (non-Gaussian) Lévy noise from data is chanllenging due to possible intricate functional forms and the inherent complexity of Lévy motion. This present research endeavors to develop an evolutionary symbol sparse regression (ESSR) approach to extract non-Gaussian stochastic dynamical systems from sample path data, based on nonlocal Kramers-Moyal formulas, genetic programming, and sparse regression. More specifically, the genetic programming is employed to generate a diverse array of candidate functions, the sparse regression technique aims at learning the coefficients associated with these candidates, and the nonlocal Kramers-Moyal formulas serve as the foundation for constructing the fitness measure in genetic programming and the loss function in sparse regression. The efficacy and capabilities of this approach are showcased through its application to several illustrative models. This approach stands out as a potent instrument for deciphering non-Gaussian stochastic dynamics from available datasets, indicating a wide range of applications across different fields.

stat.ML↗

arXiv2024

Large Deviations for Stochastic Differential Equations Driven by Semimartingales

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 exponential tightness assumption for noise-control-solution triplets and no uniform exponential tightness assumption for noise.

math.PR↗

arXiv2024

The stochastic fractional nonlinear Schrödinger equations in $H^α$ and structure-preserving algorithm

In this paper, we first investigate the global existence of a solution for the stochastic fractional nonlinear Schrödinger equation with radially symmetric initial data in a suitable energy space $H^α$. We then show that the stochastic fractional nonlinear Schrödinger equation in the Stratonovich sense forms an infinite-dimensional stochastic Hamiltonian system, with its phase flow preserving symplecticity. Finally, we develop a stochastic midpoint scheme for the stochastic fractional nonlinear Schrödinger equation from the perspective of symplectic geometry. It is proved that the stochastic midpoint scheme satisfies the corresponding symplectic law in the discrete sense. A numerical example is conducted to validate the efficiency of the theory.

math.NA↗

arXiv2024

Early warning indicators via latent stochastic dynamical systems

Detecting early warning indicators for abrupt dynamical transitions in complex systems or high-dimensional observation data is 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 evolutionary dynamics in the low-dimensional manifold. Then three effective warning signals (Onsager-Machlup Indicator, Sample Entropy Indicator, and Transition Probability Indicator) are derived through the latent coordinates and the latent stochastic dynamical systems. To validate our framework, we apply this methodology to authentic electroencephalogram (EEG) data. We find that our early warning indicators are capable of detecting the tipping point during state transition. This framework not only bridges the latent dynamics with real-world data but also shows the potential ability for automatic labeling on complex high-dimensional time series.

stat.ML↗

arXiv2024

Stochastic parameter reduced-order model based on hybrid machine learning approaches

Establishing appropriate mathematical models for complex systems in natural phenomena not only helps deepen our understanding of nature but can also be used for state estimation and prediction. However, the extreme complexity of natural phenomena makes it extremely challenging to develop full-order models (FOMs) and apply them to studying many quantities of interest. In contrast, appropriate reduced-order models (ROMs) are favored due to their high computational efficiency and ability to describe the key dynamics and statistical characteristics of natural phenomena. Taking the viscous Burgers equation as an example, this paper constructs a Convolutional Autoencoder-Reservoir Computing-Normalizing Flow algorithm framework, where the Convolutional Autoencoder is used to construct latent space representations, and the Reservoir Computing-Normalizing Flow framework is used to characterize the evolution of latent state variables. In this way, a data-driven stochastic parameter reduced-order model is constructed to describe the complex system and its dynamic behavior.

cs.LG↗

arXiv2024

Multi-task Meta Label Correction for Time Series Prediction

Time series classification faces two unavoidable problems. One is partial feature information and the other is poor label quality, which may affect model performance. To address the above issues, we create a label correction method to time series data with meta-learning under a multi-task framework. There are three main contributions. First, we train the label correction model with a two-branch neural network in the outer loop. While in the model-agnostic inner loop, we use pre-existing classification models in a multi-task way and jointly update the meta-knowledge so as to help us achieve adaptive labeling on complex time series. Second, we devise new data visualization methods for both image patterns of the historical data and data in the prediction horizon. Finally, we test our method with various financial datasets, including XOM, S\&P500, and SZ50. Results show that our method is more effective and accurate than some existing label correction techniques.

cs.LG↗

arXiv2024

Diffusion Model Conditioning on Gaussian Mixture Model and Negative Gaussian Mixture Gradient

Diffusion models (DMs) are a type of generative model that has a huge impact on image synthesis and beyond. They achieve state-of-the-art generation results in various generative tasks. A great diversity of conditioning inputs, such as text or bounding boxes, are accessible to control the generation. In this work, we propose a conditioning mechanism utilizing Gaussian mixture models (GMMs) as feature conditioning to guide the denoising process. Based on set theory, we provide a comprehensive theoretical analysis that shows that conditional latent distribution based on features and classes is significantly different, so that conditional latent distribution on features produces fewer defect generations than conditioning on classes. Two diffusion models conditioned on the Gaussian mixture model are trained separately for comparison. Experiments support our findings. A novel gradient function called the negative Gaussian mixture gradient (NGMG) is proposed and applied in diffusion model training with an additional classifier. Training stability has improved. We also theoretically prove that NGMG shares the same benefit as the Earth Mover distance (Wasserstein) as a more sensible cost function when learning distributions supported by low-dimensional manifolds.

cs.LG↗

arXiv2024

Early Warning Prediction with Automatic Labeling in Epilepsy Patients

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' EEG data, we propose a meta learning framework to improve the prediction of early ictal signals. The proposed bi-level optimization framework can help automatically label noisy data at the early ictal stage, as well as optimize the training accuracy of the backbone model. To validate our approach, we conduct a series of experiments to predict seizure onset in various long-term windows, with LSTM and ResNet implemented as the baseline models. Our study demonstrates that not only the ictal prediction accuracy obtained by meta learning is significantly improved, but also the resulting model captures some intrinsic patterns of the noisy data that a single backbone model could not learn. As a result, the predicted probability generated by the meta network serves as a highly effective early warning indicator.

cs.LG↗