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：本文提出了一种利用条件分数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：即从试件在载荷作用下力学响应的含噪测量中，推断其空间变化的材料属性。条件分数扩散模型是一种生成模型，它通过学习联合分布的样本，来近似条件分布的分数函数。具体而言，我们使用一个单一的神经网络（即分数网络）来近似对应于多次测量实现的分数函数；随后，该网络被用于基于朗之万动力学的适当马尔可夫链蒙特卡罗方案中，对后验分布进行采样。训练分数网络仅需模拟正演模型，因此所提方法能够兼容黑箱正演模型与复杂的测量噪声。此外，一旦分数网络训练完成，即可重复用于解决不同测量实现的反问题。我们通过一系列涉及从含噪测量中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，另一些则包含实际弹性成像实验采集的数据。进一步，我们的应用实例表明，所提方法能够处理不同的测量模态、推断量中的复杂模式、非高斯与非加性噪声模型，以及非线性黑箱正演模型。结果表明，该框架能够高效求解大规模基于物理的反问题。