Conditional score-based diffusion models for solving inverse problems in mechanics

Abstract

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.

Figures (18)

• Figure 1: Schematic diagram of the proposed framework for solving inverse problems in mechanics using cSDMs. Step 1 involves creating the dataset by sampling realizations of $({\bm{X}}, {\bm{Y}})$ from the joint distribution $\mathrm{p}_{{\bm{X}}{\bm{Y}}}$, which requires multiple evaluations of the forward model $\mathcal{F}$. Step 2 involves the training of the score network, which progressively learns to predict the noise in realizations of $\tilde{{\bm{X}}}$ conditioned on ${\bm{Y}}$ for all levels of noise $\sigma$. Finally, Annealed Langevin dynamics is used to sample from the target posterior distribution in Step 3
• Figure 2: Five typical realizations of ${\bm{X}}$ and ${\bm{Y}}$ sampled from the joint distribution for the synthetic quasi-static elastography example and belong to the training dataset. The first row shows the shear modulus field, while the second row shows the corresponding noisy vertical displacement field used as measurements. In this figure, the standard deviation of the Gaussian noise $\sigma_\eta = 0.025$. Moreover, realizations of ${\bm{X}}$ are ${\bm{Y}}$ are min-max normalized to $[0,1]$ scale
• Figure 3: Posterior statistics -- pixel-wise posterior mean and standard deviation -- estimated using cSDMs and MCS for the synthetic quasi-static elastography example on two test samples. In subfigures (a) and (b), the three rows show the results corresponding to three different levels of measurement noise $\sigma_\eta$ but for the same ground truth. All values are normalized to $[0,1]$ scale
• Figure 4: Posterior statistics -- pixel-wise posterior mean and standard deviation -- for test sample 1 estimated using cSDMs trained assuming measurement noise with standard deviation $\sigma_{\eta} = 0.05$. The reference MC statistics are also estimated with the misspecified measurement noise model. All values are normalized to $[0,1]$ scale
• Figure 5: A realization of the ONH sampled from the parametric prior distribution showing the (1) peripapillary sclera, (2) lamina cribrosa, (3) pia matter, (4) optic nerve, and (5) retina. This figure also appears in ray2023solution

Theorems & Definitions (2)

• Remark 1
• Remark 2