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.

我们提出了一个框架，使用条件分数扩散模型执行贝叶斯推理来解决机械学中涉及由加载的噪声测量推断出具有空间变化的材料属性的一类反问题。

机械领域中解决反问题的条件评分扩散模型

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

This study is a pioneering endeavor to investigate the capabilities of Large
Language Models (LLMs) in addressing conceptual questions within the domain of
mechanical engineering with a focus on mechanics. Our examination involves a
manually crafted exam encompassing 126 multiple-choice questions, spanning
various aspects of mechanics courses, including Fluid Mechanics, Mechanical
Vibration, Engineering Statics and Dynamics, Mechanics of Materials, Theory of
Elasticity, and Continuum Mechanics. Three LLMs, including ChatGPT (GPT-3.5),
ChatGPT (GPT-4), and Claude (Claude-2.1), were subjected to evaluation against
engineering faculties and students with or without mechanical engineering
background. The findings reveal GPT-4's superior performance over the other two
LLMs and human cohorts in answering questions across various mechanics topics,
except for Continuum Mechanics. This signals the potential future improvements
for GPT models in handling symbolic calculations and tensor analyses. The
performances of LLMs were all significantly improved with explanations prompted
prior to direct responses, underscoring the crucial role of prompt engineering.
Interestingly, GPT-3.5 demonstrates improved performance with prompts covering
a broader domain, while GPT-4 excels with prompts focusing on specific
subjects. Finally, GPT-4 exhibits notable advancements in mitigating input
bias, as evidenced by guessing preferences for humans. This study unveils the
substantial potential of LLMs as highly knowledgeable assistants in both
mechanical pedagogy and scientific research.

该研究探究了大型语言模型（LLM）在机械工程领域特别是力学方面解答概念性问题的能力。通过对比三种 LLM（ChatGPT（GPT-3.5），ChatGPT（GPT-4）和 Claude（Claude-2.1））在不同力学主题下的回答表现，发现 GPT-4 在除连续介质力学外的各力学领域问题中表现优异，这表明 GPT 模型在处理符号计算和张量分析方面具有潜在的未来改进空间。研究还发现，通过事先给出解释的提示，LLM 的性能都得到了显著提升，突出了提示工程的重要作用。值得注意的是，GPT-3.5 在涵盖更广泛领域的提示下表现出改进的表现，而 GPT-4 在专注于特定学科的提示下表现出色。最后，GPT-4 在减少输入偏差方面取得了显著进展，从人类猜测偏好的情况来看。该研究揭示了 LLM 作为高水平知识助手在机械工程教学和科学研究中的巨大潜力。