Applications Research Of Adaptive Physical-informed Neural Networks In Bayesian Inverse Problems

The Bayesian inverse problem has a wide range of important applications in practical engineering fields such as medical imaging,defense military,and industrial design.However,due to the complexity of the problem,solving such inverse problems often requires repeatedly solving the partial differential equation model to obtain the relevant information of the posterior solution.This causes a lot of computational time to be consumed when solving practical problems.In this paper,a fast Bayesian implementation method based on surrogate model will be designed in combination with Physics-informed neural networks(PINNs)to improve computational efficiency.In the process of specific numerical realization,we design two adaptive algorithms according to the characteristic that the posterior distribution of the Bayesian inverse problem has local concentration.The PINNs are first trained to have a rough but comprehensive approximation of the prior distribution of the parameters.And according to different equations,the PINNs model adopts adaptive point selection,adaptive weight,small batch training and other means to improve the accuracy.Then two different network architectures are adopted online and the network parameters are updated by adaptively selecting the training dataset.Specifically,the first is to fine-tune the last few layers of the PINNs,and the second is to use hybrid multi-fidelity network to learn the mapping from the predicted value of the PINNs at the observation location to the true value at the observation location.The two algorithms use PINNs as a surrogate model,reduce the need of high-precision solutions to the forward model,and reduce computing costs.Moreover,PINNs have learned location information.When the observation location changes,the knowledge saved by the previous model can still be used.In numerical experiments,we compared the proposed algorithm with the traditional method for three different models,and verified the accuracy and efficiency of the method.Numerical results show that the new format can greatly improve the computational efficiency while ensuring numerical accuracy.