A Neural Network Surrogate Model Algorithm And Its Application In Bayesian Inverse Problems

The parameter identification of the steady-state heat equation is a subject in the field of inverse problems,which has important theoretical and engineering significance.The thesis focuses on the Bayesian inversion method of this kind of problems.In recent years,Bayesian approach has become an important tool for dealing with inverse problems.Compared with traditional regularization algorithms,Bayesian approach has two major advantages.First,Bayesian approach not only gives a specific estimation of solution,but also gives its uncertainty information,which provides reliability analysis for decision makers in practical engineering applications.Second,many algorithms based on Bayesian approach don't have derivative,which provides convenience for practical application in engineering.Based on these advantages,the Bayesian approach has been widely used to solve various inverse problems.Inverse problems's Bayesian solution are the posterior distribution of the unknown.In general,it is difficult to determine the posterior distribution,especially in the case that the likelihood function contains complex forward problem operators.Therefore,finding the approximation of the posterior distribution is the subject of our research.At present,one of the ways to obtain the approximate posterior distribution is to estimate the relevant statistics of the posterior distribution according to lots of posterior samples.The thesis focuses on sampling algorithms based on Markov chain Monte Carlo(MCMC),such as Metropolis-Hastings(MH)algorithm and pCN algorithm.This kind of method draws samples from the posterior distribution by rejecting and receiving the proposal.In each sampling process,the likelihood function must be calculated.In general,a high-fidelity numerical approach to the forward problem is required in the process of numerical evaluation to the likelihood function,which means that the calculation of forward problem is expensive.Therefore,we hope to have a low-cost surrogate model of the forward problem in the process of generating a large number of samples.The main methods for constructing surrogate model include polynomial chaos expansion,Gauss regression,neural network and so on.With the development of big data science in various fields,machine learning has been widely used in various disciplines.As a key method of machine learning,neural network has the characteristics of easy to execute and high calculation accuracy.It has gradually developed into a major tool of big data and has been widely used in image recognition,model evaluation,risk prediction and so on.Therefore,we consider replacing the forward problem with a neural network model that is easy to execute and high-precision in the MCMC sampling process.So it can avoid using a high-fidelity numerical method to calculate the forward problem.There are many algorithms for training neural network models.The thesis uses the BackPropagation(BP)algorithm.Through the training of sample data,the weight and threshold values of the network are constantly modified to make the error function decrease along the negative gradient direction,so as to obtain the ideal precision of the neural network model.In order to verify that the neural network surrogate model algorithm can accelerate the Bayesian inversion process,We use the general Bayes-MCMC algorithm and the neural network surrogate model algorithm to calculate the inverse heat problem with two and six parameters.We compare the running time,acceptance rate,relative error and thermal conductivity of the two methods.Numerical results show that the calculation efficiency of the neural network surrogate model algorithm is about 20 times higher than the general Bayes-MCMC algorithm.