A dimension-reduced neural network-assisted approximate Bayesian computation for inverse heat conduction problems

Due to the flexibility and feasibility of addressing ill-posed problems, the Bayesian method has been widely used in inverse heat conduction problems (IHCPs). However, in the real science and engineering IHCPs, the likelihood function of the Bayesian method is commonly computationally expensive or analytically unavailable. In this study, in order to circumvent this intractable likelihood function, the approximate Bayesian computation (ABC) is expanded to the IHCPs. In ABC, the high dimensional observations in the intractable likelihood function are equalized by their low dimensional summary statistics. Thus, the performance of the ABC depends on the selection of summary statistics. In this study, a machine learning-based ABC (ML-ABC) is proposed to address the complicated selections of the summary statistics. The Auto-Encoder (AE) is a powerful Machine Learning (ML) framework which can compress the observations into very low dimensional summary statistics with little information loss. In addition, in order to accelerate the calculation of the proposed framework, another neural network (NN) is utilized to construct the mapping between the unknowns and the summary statistics. With this mapping, given arbitrary unknowns, the summary statistics can be obtained efficiently without solving the time-consuming forward problem with numerical method. Furthermore, an adaptive nested sampling method (ANSM) is developed to further improve the efficiency of sampling. The performance of the proposed method is demonstrated with two IHCP cases.