利用扩散模型估计解析延拓中的不确定性
Using Diffusion Models to Estimate Uncertainties in Analytic Continuation
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中文总结 AI 辅助
该研究提出基于扩散模型的解析延拓框架，可量化解析延拓的不确定性，还能评估逆问题固有难度，在液态仲氢模拟数据应用中取得了良好效果。
中文摘要 AI 辅助
逆问题在物理学、化学和工程学中普遍存在，这类问题需要从间接测量中重构隐藏的物理量，一个关键例子是虚时间关联函数（iTCFs）向实频域的解析延拓。该过程需要进行逆拉普拉斯变换，其本质是不适定的，且对输入的微小变化高度敏感。近期基于神经网络（NN）的方法通过学习从虚时间到实频谱的映射取得了良好效果，通常优于最大熵等传统技术。然而，由于该问题不适定，存在多个频谱可拟合同一iTCF，基于回归的方法仅输出单一解，近似于真实解空间的平均值，因此无法捕捉所有可能功率谱的完整分布。为解决这一问题，我们提出一种基于扩散的解析延拓框架，该框架学习与给定iTCF一致的频谱分布，具有两个关键优势：一是可直接从学习到的分布中量化不确定性；二是通过分析该分布的离散程度和结构，可定量评估每个逆问题的固有难度，我们用新指标“不确定性伪体积”来衡量该难度。将该框架应用于液态仲氢的路径积分分子动力学模拟得到的iTCF，我们获得了带误差棒的自扩散系数，并标记出一个次级高频峰可能是虚假伪影。与以往的不确定性量化尝试不同，我们的生成方法基于具体的概率基础，为重构功率谱提供了更具理论依据的置信度度量。
英文摘要
Inverse problems are ubiquitous in physics, chemistry, and engineering, arising when reconstructing hidden quantities from indirect measurements. A key example is the analytic continuation of imaginary-time correlation functions (iTCFs) to the real-frequency domain. This process requires an inverse Laplace transform, which is inherently ill-posed and highly sensitive to small input variations. Recent neural network (NN)-based methods have shown promising results by learning mappings from imaginary-time to real-frequency spectra, often outperforming traditional techniques such as maximum entropy. However, because the problem is ill-posed, many spectra fit the same iTCF. Regression-based approaches output a single solution, which approximates an average over the true solution space, and therefore fail to capture the full distribution of plausible power spectra. To address this issue, we introduce a diffusion-based framework for analytic continuation that learns the distribution of spectra consistent with a given iTCF. It offers two key advantages. First, it quantifies uncertainty directly from the learned distribution. Second, by analyzing the spread and structure of this distribution, we can quantitatively assess the intrinsic hardness of each inversion problem. We measure this hardness with a new metric, the uncertainty pseudo-volume. Applying the framework to an iTCF from a path-integral molecular dynamics simulation of liquid parahydrogen, we obtain the self-diffusion coefficient with an error bar and flag a secondary high-frequency peak as a possible spurious artifact. In contrast to previous attempts at uncertainty quantification, our generative approach rests on a concrete probabilistic basis, providing a more theoretically grounded measure of confidence in the reconstructed power spectra.
