We propose a framework to perform Bayesian inference using conditional score-based diffusion models to solve a class of inverse problems in mechanics involving the inference of a specimen's spatially varying material properties from noisy measurements of its mechanical response to loading. Conditional score-based diffusion models are generative models that learn to approximate the score function of a conditional distribution using samples from the joint distribution. More specifically, the score functions corresponding to multiple realizations of the measurement are approximated using a single neural network, the so-called score network, which is subsequently used to sample the posterior distribution using an appropriate Markov chain Monte Carlo scheme based on Langevin dynamics. Training the score network only requires simulating the forward model. Hence, the proposed approach can accommodate black-box forward models and complex measurement noise. Moreover, once the score network has been trained, it can be re-used to solve the inverse problem for different realizations of the measurements. We demonstrate the efficacy of the proposed approach on a suite of high-dimensional inverse problems in mechanics that involve inferring heterogeneous material properties from noisy measurements. Some examples we consider involve synthetic data, while others include data collected from actual elastography experiments. Further, our applications demonstrate that the proposed approach can handle different measurement modalities, complex patterns in the inferred quantities, non-Gaussian and non-additive noise models, and nonlinear black-box forward models. The results show that the proposed framework can solve large-scale physics-based inverse problems efficiently.

In this work, we train conditional Wasserstein generative adversarial networks to effectively sample from the posterior of physics-based Bayesian inference problems. The generator is constructed using a U-Net architecture, with the latent information injected using conditional instance normalization. The former facilitates a multiscale inverse map, while the latter enables the decoupling of the latent space dimension from the dimension of the measurement, and introduces stochasticity at all scales of the U-Net. We solve PDE-based inverse problems to demonstrate the performance of our approach in quantifying the uncertainty in the inferred field. Further, we show the generator can learn inverse maps which are local in nature, which in turn promotes generalizability when testing with out-of-distribution samples.