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Jinqiao Duan

Publications and source records attributed to Jinqiao Duan.

At least 55 records · Page 3Linked to original sources
arXiv2024

Fourier neural operator based fluid-structure interaction for predicting the vesicle dynamics

Solving complex fluid-structure interaction (FSI) problems, characterized by nonlinear partial differential equations, is crucial in various scientific and engineering applications. Traditional computational fluid dynamics (CFD) solvers are insufficient to meet the growing requirements for large-scale and long-period simulations. Fortunately, the rapid advancement in neural networks, especially neural operator learning mappings between function spaces, has introduced novel approaches to tackle these challenges via data-driven modeling. In this paper, we propose a Fourier neural operator-based fluid-structure interaction solver (FNO-based FSI solver) for efficient simulation of FSI problems, where the solid solver based on the finite difference method is seamlessly integrated with the Fourier neural operator to predict incompressible flow using the immersed boundary method. We analyze the performance of the FNO-based FSI solver in the following three situations: training data with or without the steady state, training method with one-step label or multi-step labels, and prediction in interpolation or extrapolation. We find that the best performance for interpolation is achieved by training the operator with multi-step labels using steady-state data. Finally, we train the FNO-based FSI solver using this optimal training method and apply it to vesicle dynamics. The results show that the FNO-based FSI solver is capable of capturing the variations in the fluid and the vesicle.

math.DS↗
arXiv2023

Learning effective dynamics from data-driven stochastic systems

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 period satisfying some unknown slow-fast stochastic systems, we propose a novel algorithm including a neural network called Auto-SDE to learn invariant slow manifold. Our approach captures the evolutionary nature of a series of time-dependent autoencoder neural networks with the loss constructed from a discretized stochastic differential equation. Our algorithm is also validated to be accurate, stable and effective through numerical experiments under various evaluation metrics.

stat.ML↗
arXiv2023

$C^{k}$ extension and invariant manifolds for the compactification of nonautonomous systems with autonomous limits

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 for the $C^{k}$ ($k\geq 2$) extension of the compactification. After compactifying nonautonomous systems, the compactified systems may gain an additional center direction. We prove the existence and uniqueness of center or center-stable manifolds for general compact invariant sets including normally hyperbolic invariant manifolds and admissible sets.

math.DS↗
arXiv2023

The Most Probable Transition Paths of Stochastic Dynamical Systems: A Sufficient and Necessary Characterization

The most probable transition paths of a stochastic dynamical system are the global minimizers of the Onsager-Machlup action functional and can be described by a necessary but not sufficient condition, the Euler-Lagrange equation (a second-order differential equation with initial-terminal conditions) from a variational principle. This work is devoted to showing a sufficient and necessary characterization for the most probable transition paths of stochastic dynamical systems with Brownian noise. We prove that, under appropriate conditions, the most probable transition paths are completely determined by a first-order ordinary differential equation. The equivalence is established by showing that the Onsager-Machlup action functional of the original system can be derived from the corresponding Markovian bridge process. For linear stochastic systems and the nonlinear Hongler's model, the first-order differential equations determining the most probable transition paths are shown analytically to imply the Euler-Lagrange equations of the Onsager-Machlup functional. For general nonlinear systems, the determining first-order differential equations can be approximated, in a short time or for the small noise case. Some numerical experiments are presented to illustrate our results.

math-ph↗
arXiv2023

Weak averaging principle for multiscale stochastic dynamical systems driven by stable processes

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 Lévy noises, whose stable indexes may be different. The homogenizing index $r_0$ of slow components has a relation with the stable index $α_1$ of the noise of fast components given by $0<r_0<2-2/{α_1}$. By first studying a nonlocal Poisson equation and then constructing suitable correctors, we obtain that the slow components weakly converge to a Lévy process as the scale parameter goes to zero.

math.DS↗
arXiv2023

Efficient Privacy-Preserving Convolutional Spiking Neural Networks with FHE

With the rapid development of AI technology, we have witnessed numerous innovations and conveniences. However, along with these advancements come privacy threats and risks. Fully Homomorphic Encryption (FHE) emerges as a key technology for privacy-preserving computation, enabling computations while maintaining data privacy. Nevertheless, FHE has limitations in processing continuous non-polynomial functions as it is restricted to discrete integers and supports only addition and multiplication. Spiking Neural Networks (SNNs) operate on discrete spike signals, naturally aligning with the properties of FHE. In this paper, we present a framework called FHE-DiCSNN. This framework is based on the efficient TFHE scheme and leverages the discrete properties of SNNs to achieve high prediction performance on ciphertexts. Firstly, by employing bootstrapping techniques, we successfully implement computations of the Leaky Integrate-and-Fire neuron model on ciphertexts. Through bootstrapping, we can facilitate computations for SNNs of arbitrary depth. This framework can be extended to other spiking neuron models, providing a novel framework for the homomorphic evaluation of SNNs. Secondly, inspired by CNNs, we adopt convolutional methods to replace Poisson encoding. This not only enhances accuracy but also mitigates the issue of prolonged simulation time caused by random encoding. Furthermore, we employ engineering techniques to parallelize the computation of bootstrapping, resulting in a significant improvement in computational efficiency. Finally, we evaluate our model on the MNIST dataset. Experimental results demonstrate that, with the optimal parameter configuration, FHE-DiCSNN achieves an accuracy of 97.94% on ciphertexts, with a loss of only 0.53% compared to the original network's accuracy of 98.47%. Moreover, each prediction requires only 0.75 seconds of computation time

cs.CR↗
arXiv2023

Large Deviations for SDE driven by Heavy-tailed Lévy Processes

We obtain sample-path large deviations for a class of one-dimensional stochastic differential equations with bounded drifts and heavy-tailed Lévy processes. These heavy-tailed Lévy processes do not satisfy the exponential integrability condition, which is a common restriction on the Lévy processes in existing large deviations contents. We further prove that the solution processes satisfy a weak large deviation principle with a discrete rate function and logarithmic speed. We also show that they do not satisfy the full large deviation principle.

math.PR↗
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↗
arXiv2023

An Optimal Control Method to Compute the Most Likely Transition Path for Stochastic Dynamical Systems with Jumps

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 certain scenarios. Based on the large deviation principle, the most likely transition path could be treated as the minimizer of the rate function upon paths that connect two points. One of the challenges to calculate the most likely transition path for stochastic dynamical systems under non-Gaussian Lévy noise is that the associated rate function can not be explicitly expressed by paths. For this reason, we formulate an optimal control problem to obtain the optimal state as the most likely transition path. We then develop a neural network method to solve this issue. Several experiments are investigated for both Gaussian and non-Gaussian cases.

math.NA↗
arXiv2023

Well-posedness and averaging principle for Lévy-type McKean-Vlasov stochastic differential equations under local Lipschitz conditions

In this paper, we investigate a class of McKean-Vlasov stochastic differential equations under Lévy-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 solutions of McKean-Vlasov stochastic differential equations can be approximated by the solutions of the associated averaged McKean-Vlasov stochastic differential equations in the sense of mean square convergence. In contrast to the existing work, a novel feature is the use of a much weaker condition -- local Lipschitzian in the state variables, allowing for possibly super-linearly growing drift, but linearly growing diffusion and jump coefficients. Therefore, our results are suitable for a wider class of McKean-Vlasov stochastic differential equations.

math.PR↗
arXiv2023

Privacy-Preserving Discretized Spiking Neural Networks

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 models could have really high computational costs. The third-generation neural network, SNN (Spiking Neural Network), mimics real neurons by using discrete spike signals, whose sequences exhibit strong sparsity, providing advantages such as low energy consumption and high efficiency. In this paper, we construct a framework to evaluate the homomorphic computation of SNN named FHE-DiSNN that enables SNN to achieve good prediction performance on encrypted data. First, benefitting from the discrete nature of spike signals, our proposed model avoids the errors introduced by discretizing activation functions. Second, by applying bootstrapping, we design new private preserving functions FHE-Fire and FHE-Reset, through which noise can be refreshed, allowing us to evaluate SNN for an arbitrary number of operations. Furthermore, We improve the computational efficiency of FHE-DiSNN while maintaining a high level of accuracy. Finally, we evaluate our model on the MNIST dataset. The experiments show that FHE-DiSNN with 30 neurons in the hidden layer achieves a minimum prediction accuracy of 94.4%. Under optimal parameters, it achieves a 95.1% accuracy, with only a 0.6% decrease compared to the original SNN (95.7%). These results demonstrate the superiority of SNN over second-generation neural networks for homomorphic evaluation.

cs.CR↗
arXiv2023

Reservoir Computing with Error Correction: Long-term Behaviors of Stochastic Dynamical Systems

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 traditional Reservoir Computing performance and integrates the virtues of both approaches. With few assumptions about the underlying stochastic dynamical systems, this model-free method successfully predicts the long-term evolution of stochastic dynamical systems and replicates dynamical behaviors. We verify the effectiveness of the proposed framework in several experiments, including the stochastic Van der Pal oscillator, El Niño-Southern Oscillation simplified model, and stochastic Lorenz system. These experiments consist of Markov/non-Markov and stationary/non-stationary stochastic processes which are defined by linear/nonlinear stochastic differential equations or stochastic delay differential equations. Additionally, we explore the noise-induced tipping phenomenon, relaxation oscillation, stochastic mixed-mode oscillation, and replication of the strange attractor.

math.DS↗
arXiv2023

Learning Stochastic Dynamical Systems as an Implicit Regularization with Graph Neural Networks

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. In particular, this novel framework enables us to investigate the implicit regularization effect of the noise terms in S-GGNs. We provide a theoretical guarantee for the proposed S-GGNs by deriving the difference between the two corresponding loss functions in a small neighborhood of weight. Then, we employ Kuramoto's model to generate data for comparing the spectral density from the Hessian Matrix of the two loss functions. Experimental results on real-world data, demonstrate that S-GGNs exhibit superior convergence, robustness, and generalization, compared with state-of-the-arts.

cs.LG↗
arXiv2023

Deep Reinforcement Learning in Finite-Horizon to Explore the Most Probable Transition Pathway

In many scientific and engineering problems, noise and nonlinearity are unavoidable, which could induce interesting mathematical problem such as transition phenomena. This paper focuses on efficiently discovering the most probable transition pathway for stochastic dynamical systems employing reinforcement learning. With the Onsager-Machlup action functional theory to quantify rare events in stochastic dynamical systems, finding the most probable pathway is equivalent to solving a variational problem on the action functional. When the action function cannot be explicitly expressed by paths near the reference orbit, the variational problem needs to be converted into an optimal control problem. First, by integrating terminal prediction into the reinforcement learning framework, we develop a Terminal Prediction Deep Deterministic Policy Gradient (TP-DDPG) algorithm to deal with the finite-horizon optimal control issue in a forward way. Next, we present the convergence analysis of our algorithm for the value function in terms of the neural network's approximation error and estimation error. Finally, we conduct various experiments in different dimensions for the transition problems in applications to illustrate the effectiveness of our algorithm.

math.OC↗
arXiv2023

Constraint Minimization Problem of the Nonlinear Schrödinger Equation with the Anderson Hamiltonian

We consider the two-dimensional nonlinear Schrödinger equation with a white noise potential, described by the Anderson hamiltonian. After define the corresponding energy space via the paracontrolled distribution framework from singular stochastic partial differential equations, we prove the existence of the minimizer as the least energy solution by studying a minimization problem of the corresponding energy functional subject to $L^2$ constraints. Subsequently, we study the regularity of the minimizer, which is a weak solution of the nonlinear Schrödinger equation. Finally, we derive a tail estimate for the distribution of the principal eigenvalue corresponding to the least energy solution by energy estimates.

math.PR↗
arXiv2023

Modulation analysis of the stochastic Camassa-Holm equation with pure jump noise

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 that the remainder converges to the solution of a stochastic linear equation as amplitude of the jump noise tends to zero.

math.PR↗
arXiv2023

Homogenization of Non-symmetric Jump Processes

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 scales. It turns out that the Feller process converges in distribution, as the scaling parameter goes to zero, to a Lévy process. As special cases of our result, some homogenization problems studied in previous works can be recovered. We also generalize the approach to the homogenization of symmetric stable-like processes with variable order. Moreover, we present some numerical experiments to demonstrate the usage of our homogenization results in the numerical approximation of first exit times.

math.PR↗
arXiv2023

Detecting the Most Probable High Dimensional Transition Pathway Based on Optimal Control Theory

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 variational principle and Onsager-Machlup action functional theory, the variational problem is transformed into deterministic optimal control problem. Different from traditional variational method, optimal control theory can handle high dimensional problems well. This paper describes the following three systems, double well system driven by additive noise, Maire-Stein system driven by multiplicative noise, and Nutrient-Phytoplankton-Nooplanktonr system.For numerical computation, based on Pontryagin's Maximum Principle, we adopt method of successive approximations to compute the optimal solution pathway, which is the most probable transition pathway in the sense of Onsager - Machlup.

math.OC↗