arXiv subjects

Xiaoli Chen

Publications and source records attributed to Xiaoli Chen.

13 recordsLinked to original sources

arXiv2023

Constructing Custom Thermodynamics Using Deep Learning

One of the most exciting applications of artificial intelligence (AI) is automated scientific discovery based on previously amassed data, coupled with restrictions provided by known physical principles, including symmetries and conservation laws. Such automated hypothesis creation and verification can assist scientists in studying complex phenomena, where traditional physical intuition may fail. Here we develop a platform based on a generalized Onsager principle to learn macroscopic dynamical descriptions of arbitrary stochastic dissipative systems directly from observations of their microscopic trajectories. Our method simultaneously constructs reduced thermodynamic coordinates and interprets the dynamics on these coordinates. We demonstrate its effectiveness by studying theoretically and validating experimentally the stretching of long polymer chains in an externally applied field. Specifically, we learn three interpretable thermodynamic coordinates and build a dynamical landscape of polymer stretching, including the identification of stable and transition states and the control of the stretching rate. Our general methodology can be used to address a wide range of scientific and technological applications.

cond-mat.soft↗

arXiv2023

Detecting Stochastic Governing Laws with Observation on Stationary Distributions

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 method to learn the drift and diffusion terms of the stochastic differential equation. We introduce a new loss function containing the Hellinger distance between the observation data and the learned stationary probability density function. We discover that the learnt stochastic differential equation provides a fair approximation of the data-driven dynamical system after minimizing this loss function during the training method. The effectiveness of our method is demonstrated in numerical experiments.

math.DS↗

arXiv2022

Personal Privacy Protection Problems in the Digital Age

With the development of Internet technology, the issue of privacy leakage has attracted more and more attention from the public. In our daily life, mobile phone applications and identity documents that we use may bring the risk of privacy leakage, which had increasingly aroused public concern. The path of privacy protection in the digital age remains to be explored. To explore the source of this risk and how it can be reduced, we conducted this study by using personal experience, collecting data and applying the theory.

econ.GN↗

arXiv2022

A Deep Learning Approach to Nonconvex Energy Minimization for Martensitic Phase Transitions

We propose a mesh-free method to solve nonconvex energy minimization problems for martensitic phase transitions and twinning in crystals, using the deep learning approach. These problems pose multiple challenges to both analysis and computation, as they involve multiwell gradient energies with large numbers of local minima, each involving a topologically complex microstructure of free boundaries with gradient jumps. We use the Deep Ritz method, whereby candidates for minimizers are represented by parameter-dependent deep neural networks, and the energy is minimized with respect to network parameters. The new essential ingredient is a novel activation function proposed here, which is a smoothened rectified linear unit we call SmReLU; this captures the structure of minimizers where usual activation functions fail. The method is mesh-free and thus can approximate free boundaries essential to this problem without any special treatment, and is extremely simple to implement. We show the results of many numerical computations demonstrating the success of our method.

cs.CE↗

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 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 L\'evy 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 L\'evy 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↗

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↗

arXiv2020

Solving Inverse Stochastic Problems from Discrete Particle Observations Using the Fokker-Planck Equation and Physics-informed Neural Networks

The Fokker-Planck (FP) equation governing the evolution of the probability density function (PDF) is applicable to many disciplines but it requires specification of the coefficients for each case, which can be functions of space-time and not just constants, hence requiring the development of a data-driven modeling approach. When the data available is directly on the PDF, then there exist methods for inverse problems that can be employed to infer the coefficients and thus determine the FP equation and subsequently obtain its solution. Herein, we address a more realistic scenario, where only sparse data are given on the particles' positions at a few time instants, which are not sufficient to accurately construct directly the PDF even at those times from existing methods, e.g., kernel estimation algorithms. To this end, we develop a general framework based on physics-informed neural networks (PINNs) that introduces a new loss function using the Kullback-Leibler divergence to connect the stochastic samples with the FP equation, to simultaneously learn the equation and infer the multi-dimensional PDF at all times. In particular, we consider two types of inverse problems, type I where the FP equation is known but the initial PDF is unknown, and type II in which, in addition to unknown initial PDF, the drift and diffusion terms are also unknown. In both cases, we investigate problems with either Brownian or Levy noise or a combination of both. We demonstrate the new PINN framework in detail in the one-dimensional case (1D) but we also provide results for up to 5D demonstrating that we can infer both the FP equation and} dynamics simultaneously at all times with high accuracy using only very few discrete observations of the particles.

physics.comp-ph↗

arXiv2019

Learning and Meta-Learning of Stochastic Advection-Diffusion-Reaction Systems from Sparse Measurements

Physics-informed neural networks (PINNs) were recently proposed in [1] as an alternative way to solve partial differential equations (PDEs). A neural network (NN) represents the solution while a PDE-induced NN is coupled to the solution NN, and all differential operators are treated using automatic differentiation. Here, we first employ the standard PINN and a stochastic version, sPINN, to solve forward and inverse problems governed by a nonlinear advection-diffusion-reaction (ADR) equation, assuming we have some sparse measurements of the concentration field at random or pre-selected locations. Subsequently, we attempt to optimize the hyper-parameters of sPINN by using the Bayesian optimization method (meta-learning), and compare the results with the empirically selected hyper-parameters of sPINN.

physics.comp-ph↗

arXiv2018

Most probable dynamics of a genetic regulatory network under stable Lévy noise

Numerous studies have demonstrated the important role of noise in the dynamical behaviour of a complex system. The most probable trajectories of nonlinear systems under the influence of Gaussian noise have recently been studied already. However, there has been only a few works that examine how most probable trajectories in the two-dimensional system (MeKS network) are influenced under non-Gaussian stable Lévy noise. Therefore, we discuss the most probable trajectories of a two-dimensional model depicting the competence behaviour in B. subtilis under the influence of stable Lévy noise. On the basis of the Fokker-Planck equation, we describe the noise-induced most probable trajectories of the MeKS network from the low ComK protein concentration (vegetative state) to the high ComK protein concentration (competence state) under stable Lévy noise. We demonstrate choices of the non-Gaussianity index $α$ and the noise intensity $ε$ which generate the ComK protein escape from the low concentration to the high concentration. We also reveal the optimal combination of both parameters $α$ and $ε$ making the tipping time shortest. Moreover, we find that different initial concentrations around the low ComK protein concentration evolve to a metastable state, and provide the optimal $α$ and $ε$ such that the distance between the deterministic competence state and the metastable state is smallest.

math.DS↗

arXiv2018

A Stochastic Pitchfork Bifurcation in Most Probable Phase Portraits

We study stochastic bifurcation for a system under multiplicative stable Levy noise (an important class of non-Gaussian noise), by examining the qualitative changes of equilibrium states in its most probable phase portraits. We have found some peculiar bifurcation phenomena in contrast to the deterministic counterpart: (i) When the non-Gaussianity parameter in Levy noise varies, there is either one, two or none backward pitchfork type bifurcations; (ii) When a parameter in the vector field varies, there are two or three forward pitchfork bifurcations; (iii) The non-Gaussian Levy noise clearly leads to fundamentally more complex bifurcation scenarios, since in the special case of Gaussian noise, there is only one pitchfork bifurcation which is reminiscent of the deterministic situation.

math.DS↗

arXiv2018

Data assimilation and parameter estimation for a multiscale stochastic system with alpha-stable Levy noise

This work is about low dimensional reduction for a slow-fast data assimilation system with non-Gaussian $α-$stable Lévy noise via stochastic averaging. When the observations are only available for slow components, we show that the averaged, low dimensional filter approximates the original filter, by examining the corresponding Zakai stochastic partial differential equations. Furthermore, we demonstrate that the low dimensional slow system approximates the slow dynamics of the original system, by examining parameter estimation and most probable paths.

math.DS↗

arXiv2017

Lévy noise-induced transitions in gene regulatory networks

Important effects of noise on a one-dimensional gene expression model involving a single gene have recently been discussed. However, few works have been devoted to the transition in two-dimensional models which include the interaction of genes. Therefore, we investigate here, a quantitative two-dimensional model (MeKS network) of gene expression dynamics describing the competence development in the B. subtilis under the influence of Lévy as well as Brownian motions, where noises can do the B. subtilis a favor in nutrient depletion. To analyze the transitions between the vegetative and the competence regions therein, two deterministic quantities, the mean first exit time (MFET) and the first escape probability (FEP) from a microscopic perspective, as well as their averaged versions from a macroscopic perspective, are applied. The relative contribution factor (RCF), the ratio of non-Gaussian and Gaussian noise strengths, is adopted to implement optimal control in these transitions. Schematic representations indicate that there exists an optimum choice that makes the transition occurring at the highest probability. Additionally, we use a geometric concept, the stochastic basin of attraction, to exhibit a pictorial comprehension about the influence of the Lévy motion on the basin stability of the competence state.

q-bio.MN↗