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.

翻译：本文提出一种利用条件评分扩散模型进行贝叶斯推断的框架，用于解决一类力学反问题：根据试件在载荷作用下力学响应的噪声测量数据，推断其空间变化的材料属性。条件评分扩散模型是一种生成模型，它通过联合分布样本学习逼近条件分布的评分函数。具体而言，我们使用单一神经网络（即评分网络）逼近对应于多组测量实现的评分函数，随后基于朗之万动力学采用合适的马尔可夫链蒙特卡洛方法对后验分布进行采样。评分网络的训练仅需前向模型模拟，因此该方法能够兼容黑箱前向模型与复杂测量噪声。此外，评分网络一经训练，即可重复用于求解不同测量实现下的反问题。我们通过一系列涉及从噪声测量数据推断非均匀材料属性的高维力学反问题验证了所提方法的有效性。部分算例采用合成数据，其他算例则包含实际弹性成像实验采集的数据。进一步的应用案例表明，所提方法能够处理不同的测量模态、推断量的复杂分布模式、非高斯与非加性噪声模型以及非线性黑箱前向模型。结果表明，该框架能够高效求解大规模基于物理原理的反问题。