Flow-HBM: A generative likelihood-free hierarchical Bayesian model updating framework with dual normalizing flow-based inference networks

Abstract

Hierarchical Bayesian modeling (HBM) has emerged as a powerful framework for quantifying uncertainties in structural dynamics by introducing hyperparameters that govern the distributions of model parameters. However, practical application of HBM is hindered by several challenges. Approximations such as Laplace and variational inference often impose restrictive assumptions, the sampling methods are computationally expensive. Most critically, the likelihood function in complex hierarchical models is typically intractable, limiting the feasibility of standard Bayesian inference. To address these challenges, this study proposes Flow-HBM, a novel data-driven, likelihood-free HBM framework based on normalizing flow generative model. First, synthetic datasets are generated by sampling hyperparameters from the prior and simulating responses using a finite element model. A normalizing flow model is then trained to learn the complex posterior distributions by minimizing the Kullback–Leibler divergence between the true and model-estimated posteriors via maximum likelihood training on the synthetic data. To efficiently estimate model parameters, hyperparameters, and prediction error, the joint posterior is factorized into two components: (1) the posterior of hyperparameters and prediction error given all data, and (2) the posterior of model parameters given the hyperparameters, prediction error, and individual dataset. This leads to two flow-based inference networks: a model inference network (MIN) for estimating the posterior distribution of model parameters conditioned on dataset-specific observations, and a hyper inference network (HIN) for inferring the posterior of hyperparameters and prediction error parameters conditioned on the aggregated data across all datasets. Both MIN and HIN are implemented using interleaved affine coupling and neural spline flow layers, and trained jointly in an offline phase. Once trained, the framework enables near-instant inference of all unknowns by sampling from a base Gaussian and applying the learned invertible mappings, bypassing the need for likelihood evaluation. The proposed method is validated on a four-story shear building and a reinforced concrete slab, demonstrating accurate parameter estimation and significant computational gains, paving the way for real-time hierarchical Bayesian model updating.