Parameter Estimation Method Of Differential Equations Based On Model And Data

As one of the classical inverse problems in the field of differential equations,the parameter estimation of differential equations has a wide range of applications.In this thesis,with the Gaussian process as the main research tool,the parameter estimation of partial differential equation and ordinary differential equation dynamical system is studied deeply by considering the observed data and the corresponding differential equation model.For partial differential equation,the thesis draws samples from the posterior distribution of unknown parameters based on the Markov chain Monte Carlo method under the framework of Bayesian inference.We estimate the state variables and their derivatives in the equation using Gaussian Process Regression with Constraints(GPRC),and propose a new "potential" function that incorporates data and model constraint.The experimental results show that,compared with the traditional two-stage method,the method has better adaptability to small samples and nonlinear problems,and the obtained estimation accuracy is higher.For dynamical system composed of ordinary differential equations,the thesis directly learns hyperparameters and optimal model parameters from the data by maximizing the marginal likelihood function of the parameter estimation model.In this method,the state variables of the dynamic system are regarded as multiple correlated Gaussian processes,and the differential equations are used as constraints to modify the Gaussian process priors according to the specific form of the differential operator.For nonlinear ordinary differential equation systems,the thesis proposes a piecewise first-order linearization strategy based on Taylor expansion;for sparse and non-uniform observed data,the thesis proposes to use the rejection sampling method to sample the time index constrained by the model.Finally,the numerical realization shows that the parameter estimation method has high precision.13 pictures,9 tables,60 references...