arXiv subjects

Jianyu Hu

Publications and source records attributed to Jianyu Hu.

16 recordsLinked to original sources

arXiv2026

A study of path measures based on second-order Hamilton--Jacobi equations and their applications in stochastic thermodynamics

This paper provides a systematic investigation of the mathematical structure of path measures and their profound connections to stochastic differential equations (SDEs) through the framework of second-order Hamilton--Jacobi (HJ) equations. This approach establishes a unified methodology for analyzing large deviation principles (LDPs), entropy minimization, and entropy production in stochastic systems. Second-order HJ equations are shown to play a central role in bridging stochastic dynamics and measure theory while forming the foundation of stochastic geometric mechanics and their applications in stochastic thermodynamics. The large deviation rate function is rigorously derived from the probabilistic structure of path measures and proved to be equivalent to the Onsager--Machlup functional of stochastic gradient systems coupled with second-order HJ equations. We revisit entropy minimization problems, including finite time horizon problems and Schrödinger's problem, demonstrating the connections with stochastic geometric mechanics. Furthermore, we present a novel decomposition of entropy production for stochastic systems, revealing that thermodynamic irreversibility can be interpreted as the difference of the corresponding forward and backward second-order HJ equations. Together, this work establishes a comprehensive mathematical study of the relations between path measures and stochastic dynamical systems, and their diverse applications in stochastic thermodynamics and beyond.

math-ph↗

arXiv2026

Kernel Learning of PDE Solution Operators

A kernel-based approach for the learning of the solution operator of general nonhomogeneous partial differential equations (PDEs) is proposed. The method incorporates physical priors, typically encoded through the PDE operator, into a kernel ridge regression framework, and employs a regularization-based formulation to construct an operator learner. This yields a closed-form estimator that is independent of the input functions that determine the underlying PDE. From the perspective of regularization theory, the resulting estimator induces a well-defined operator that links input and output spaces, which contain the functions that define a Dirichlet problem and its solution, respectively. Consequently, it effectively shifts from a PDE solver to an operator-based solver. In contrast to standard supervised learning methods, it does not rely on paired input--output training data and enables systematic extrapolation beyond observed regimes. A full error analysis is conducted, providing convergence rates for the operator-based solver under suitable choices of regularization parameters. Extensive numerical experiments, including Darcy flow and Helmholtz equations, demonstrate that the proposed method achieves high accuracy and efficiency across a range of problem settings, and compares favorably with operator learning approaches in both approximation quality and computational cost.

math.NA↗

arXiv2025

A kernel method for the learning of Wasserstein geometric flows

Wasserstein gradient and Hamiltonian flows have emerged as essential tools for modeling complex dynamics in the natural sciences, providing a unifying geometric formulation of many partial differential equations (PDEs) and finding applications in fields ranging from optimal transport to quantum mechanics and information geometry. Despite their significance, the inverse identification of potential functions and interaction kernels underlying these flows remains relatively unexplored. In this work, we tackle this challenge by addressing the inverse problem of simultaneously recovering the potential function and interaction kernel from discretized observations of the density flow. We formulate the problem as an optimization task that minimizes a loss function specifically designed to enforce the underlying variational structure of Wasserstein flows, ensuring consistency with the geometric properties of the density manifold. Our framework employs a kernel-based operator approach using the associated {reproducing kernel Hilbert space (RKHS)}, which provides a closed-form representation of the unknown components. Furthermore, we conduct a comprehensive error analysis, providing convergence rates under adaptive regularization parameters as the temporal and spatial discretization mesh sizes tend to zero. Moreover, a stability analysis is presented to bridge the gap between discrete trajectory data and continuous-time flow dynamics for the Wasserstein Hamiltonian flow. {Finally, numerical experiments on both Wasserstein gradient and Hamiltonian flows demonstrate accurate and robust recovery from discrete density observations, while comparisons with sparse-learning approaches illustrate the strong dependence of sparse recovery on the choice of dictionary.}

math.NA↗

arXiv2025

A global structure-preserving kernel method for the learning of Poisson systems

A structure-preserving kernel ridge regression method is presented that allows the recovery of globally defined, potentially high-dimensional, and nonlinear Hamiltonian functions on Poisson manifolds out of datasets made of noisy observations of Hamiltonian vector fields. The proposed method is based on finding the solution of a non-standard kernel ridge regression where the observed data is generated as the noisy image by a vector bundle map of the differential of the function that one is trying to estimate. Additionally, it is shown how a suitable regularization solves the intrinsic non-identifiability of the learning problem due to the degeneracy of the Poisson tensor and the presence of Casimir functions. A full error analysis is conducted that provides convergence rates using fixed and adaptive regularization parameters. The good performance of the proposed estimator is illustrated with several numerical experiments.

math.NA↗

arXiv2025

A Structure-Preserving Kernel Method for Learning Hamiltonian Systems

A structure-preserving kernel ridge regression method is presented that allows the recovery of nonlinear Hamiltonian functions out of datasets made of noisy observations of Hamiltonian vector fields. The method proposes a closed-form solution that yields excellent numerical performances that surpass other techniques proposed in the literature in this setup. From the methodological point of view, the paper extends kernel regression methods to problems in which loss functions involving linear functions of gradients are required and, in particular, a differential reproducing property and a Representer Theorem are proved in this context. The relation between the structure-preserving kernel estimator and the Gaussian posterior mean estimator is analyzed. A full error analysis is conducted that provides convergence rates using fixed and adaptive regularization parameters. The good performance of the proposed estimator together with the convergence rate is illustrated with various numerical experiments.

stat.ML↗

arXiv2024

Coupling Between Local and Global Oscillations in Palladium-Catalysed Methane Oxidation

The interplay between order and disorder is crucial across various fields, especially in understanding oscillatory phenomena. Periodic oscillations are frequently observed in heterogeneous catalysis, yet their underlying mechanisms need deeper exploration. Here, we investigate how periodic oscillations arise during methane oxidation catalysed by palladium nanoparticles (Pd NPs), utilizing a suite of complementary operando techniques across various spatial scales. We found that reaction intensity and collective dynamic modes can be tuned by the reactant gas-flow rate. At lower gas-flow rates, we observed periodic facet reconstruction of Pd NPs correlated with repeated bubbling behaviour at the Pd/PdO interface, without evident global oscillatory responses. Conversely, at higher gas-flow rates, Pd NPs undergo chaotic transformations between metallic and oxidized states, resulting in overall oscillation. Integrating our observations at different gas-flow rates, we attributed the emergence of global oscillation to thermal coupling regulated by gas flow and connected local and global dynamics through a weak synchronization mechanism. This work demonstrates the correlations between open surfaces and interfaces, chaos and regularity, and dissipative processes and coupling behaviour. Our findings offer critical insights into the complexity behind catalytic oscillations and provide guidance for modulating oscillatory behaviours in catalytic processes, with significant implications for both science and industry.

physics.chem-ph↗

arXiv2023

The Most Likely Transition Path for a Class of Distribution-Dependent Stochastic Systems

Distribution-dependent stochastic dynamical systems arise widely in engineering and science. We consider a class of such systems which model the limit behaviors of interacting particles moving in a vector field with random fluctuations. We aim to examine the most likely transition path between equilibrium stable states of the vector field. In the small noise regime, the action functional does not involve the solution of the skeleton equation which describes the unperturbed deterministic flow of the vector field shifted by the interaction at zero distance. As a result, we are led to study the most likely transition path for a stochastic differential equation without distribution dependency. This enables the computation of the most likely transition path for these distribution-dependent stochastic dynamical systems by the adaptive minimum action method and we illustrate our approach in two examples.

math.NA↗

arXiv2023

Most Probable Transitions from Metastable to Oscillatory Regimes in a Carbon Cycle System

Global climate changes are related to the ocean's store of carbon. We study a carbonate system of the upper ocean, which has metastable and oscillatory regimes, under small random fluctuations. We calculate the most probable transition path via a geometric minimum action method in the context of the large deviations theory. By examining the most probable transition paths from metastable to oscillatory regimes for various external carbon input rates, we find two different transition patterns, which gives us an early warning sign for the dramatic change in the carbonate state of the ocean.

math.NA↗

arXiv2022

A Data-Driven Approach for Discovering the Most Probable Transition Pathway for a Stochastic Carbon Cycle System

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 Onsager-Machlup action functional theory, we calculate the most probable transition pathway between the metastable and oscillatory states via a neural shooting method, and further explore the effects of external random carbon input rates on the most probable transition pathway, which provides a basis to recognize naturally occurring tipping points. Particularly, we investigate the effect of the transition time on the transition pathway and further compute the optimal transition time using physics informed neural network, towards the maximum carbonate concentration state in the oscillatory regimes. This work offers some insights on the effects of random carbon input on climate transition in a simple model. Key words: Onsager-Machlup action functional, the most probable transition pathway, neural shooting method, stochastic carbon cycle system.

math.DS↗

arXiv2022

Data-driven method to learn the most probable transition pathway and stochastic differential equations

Transition phenomena between metastable states play an important role in complex systems due to noisy fluctuations. In this paper, the physics informed neural networks (PINNs) are presented to compute the most probable transition pathway. It is shown that the expected loss is bounded by the empirical loss. And the convergence result for the empirical loss is obtained. Then, a sampling method of rare events is presented to simulate the transition path by the Markovian bridge process. And we investigate the inverse problem to extract the stochastic differential equation from the most probable transition pathway data and the Markovian bridge process data, respectively. Finally, several numerical experiments are presented to verify the effectiveness of our methods.

math.NA↗

arXiv2022

An Onsager-Machlup approach to the most probable transition pathway for a genetic regulatory network

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 pathway via the Euler-Lagrangian equation. Finally, we analyze how the noise intensity affects the transition phenomena.

q-bio.MN↗

arXiv2021

Nonparametric inference of stochastic differential equations based on the relative entropy rate

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 dynamical properties of complex systems are among central challenges. In this work, we devise a nonparametric approach to learn the relative entropy rate from observations of stochastic differential equations with different drift functions.The estimator corresponding to the relative entropy rate then is presented via the Gaussian process kernel theory. Meanwhile, this approach enables to extract the governing equations. We illustrate our approach in several examples. Numerical experiments show the proposed approach performs well for rational drift functions, not only polynomial drift functions.

math.DS↗

arXiv2021

Variational inference of the drift function for stochastic differential equations driven by Lévy processes

In this paper, we consider the nonparametric estimation problem of the drift function of stochastic differential equations driven by $α$-stable Lévy motion. First, the Kullback-Leibler divergence between the path probabilities of two stochastic differential equations with different drift functions is optimized. By using the Lagrangian multiplier, the variational formula based on the stationary Fokker-Planck equation is constructed. Then combined with the data information, the empirical distribution is used to replace the stationary density, and the drift function is estimated non-parametrically from the perspective of the process. In the numerical experiment, the different amounts of data and different $α$ values are studied. The experimental results show that the estimation result of the drift function is related to both. When the amount of data increases, the estimation result will be better, and when the $α$ value increases, the estimation result is also better.

math.ST↗

arXiv2021

Transition pathways for a class of high dimensional stochastic dynamical systems with Lévy noise

This work is devoted to deriving the Onsager-Machlup action functional for a class of stochastic differential equations with (non-Gaussian) Lévy process as well as Brownian motion in high dimensions. This is achieved by applying the Girsanov transformation for probability measures and then by a path representation. The Poincaré lemma is essential to handle such path representation problem in high dimensions. We provide a sufficient condition on the vector field such that this path representation holds in high dimensions. Moreover, this Onsager-Machlup action functional may be considered as the integral of a Lagrangian. Finally, by a variational principle, we investigate the most probable transition pathways analytically and numerically.

math.DS↗

arXiv2020

Onsager-Machlup action functional for stochastic partial differential equations with Levy noise

This work is devoted to deriving the Onsager-Machlup action functional for stochastic partial differential equations with (non-Gaussian) Levy process as well as Gaussian Brownian motion. This is achieved by applying the Girsanov transformation for probability measures and then by a path representation. This enables the investigation of the most probable transition path for infinite dimensional stochastic dynamical systems modeled by stochastic partial differential equations, by minimizing the Onsager-Machlup action functional.

math.PR↗

arXiv2017

Slow manifolds for stochastic systems with non-Gaussian stable Lévy noise

This work is concerned with the dynamics of a class of slow-fast stochastic dynamical systems with non-Gaussian stable Lévy noise with a scale parameter. Slow manifolds with exponentially tracking property are constructed, eliminating the fast variables to reduce the dimension of these coupled dynamical systems. It is shown that as the scale parameter tends to zero, the slow manifolds converge to critical manifolds in distribution, which helps understand long time dynamics. The approximation of slow manifolds with error estimate in distribution are also considered.

math.DS↗