Computational Methods For Stochastic And Nonlocal Dynamical Systems

Noise-driven stochastic dynamical systems are widely used in physics,biology,economics,chemical and other fields.It is of great significance to describe the dynamical behaviors of stochastic dynamical systems.It is difficult to get the exact solution of the dynamical systems.So the numerical method provides an effective way to describe the dynamical behaviors.Dynamical systems driven by Brownian motion are characterized by local partial differential equation.But when fluctuations are present in certain events,such as burst-like events,Gaussian noise is not appropriate.In this case,it is more appropriate to model the random fluctuations by a non-Gaussian Lévy motion with heavy tails and bursting sample paths.Non-Gaussian dynamical systems are described by nonlocal partial differential equations,such as nonlocal related mean exit time,escape probability and Fokker-Planck equation.Another nonlocal operator is time fractional problem which has memory and describes the abnormal diffusion.In many practical applications,the parameters and structure of the model are unknown.In this case,it is necessary to combine the data and equation information to characterize the actual problem.We use machine learning methods to learn the parameters and solutions in the model based on observations.This thesis is organized as follows.(1)Firstly,we introduce some preparatory work before the thesis.We consider the effect of one-dimensional Lévy noise on random bifurcation,and numerical method of the nonlinear Schr ¨odinger-Poisson equation.We also consider the numerical method to solve the nonlinear Sobolev equation.And the convergence and stability of the scheme are proved by the energy method.We present the numerical experiments to illustrate the theoretical results.Comparing with the existing methods,we verify the effectiveness of the proposed method.(2)Secondly,we consider the nonlocal dynamics of non-Gaussian systems arising in a gene regulatory network.Numerous studies demonstrate the important role of noise in the dynamical behaviours of the complex systems.We use the mean exit time,escape probability,nonlocal Fokker-Planck equation and most probable trajectory to quantify dynamical behaviors of the stochastic dynamical system driven by non-Gaussian ?-stable Lévy motions.We analysis how the non-Gaussian index ? and noise intensity ? affect the gene regulation model.(3)Thirdly,we construct the linearized numerical methods for solving the twodimensional nonlinear time fractional Schr ¨odinger equations.We propose four linearized compact alternating direction implicit(ADI)methods by adding different correction terms.We prove the convergence and the stability of the proposed methods.We also give numerical examples to verify the accuracy and efficiency of the proposed schemes.(4)Lastly,we explore the deep learning method to solve the forward and inverse problems of the deterministic and stochastic advection-diffusion-reaction equation.First,we introduce the physics-informed neural network(PINN)to solve the deterministic differential equation,and the stochastic physics-informed neural network(s PINN)to solve the stochastic partial differential equation.Then we consider the inverse problem to the deterministic advection-diffusion-reaction equation.Given some observations of the solution,we learn the parameters and the solution.In the real problem,the observations are difficult to get.To better learn the unknown parameters and solution,we use the multi-fidelity neural network(m PINN)method to calculate the problem.Then,we consider the forward and inverse problems to solve the stochastic Advection-Diffusion-Reaction equation.We use Meta-learning to train the model.We know that the structure of neural networks,learning rates,etc.can have a big impact on the learning results.We use the Bayesian optimization method to obtain the optimal structure of the neural network.And by comparing with the previous results,we verify the benefits of Bayesian optimization.