  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.
