Generating samples from limited information is a fundamental problem across scientific domains. Classical maximum entropy methods provide principled uncertainty quantification from moment constraints but require sampling via MCMC or Langevin dynamics, which typically exhibit exponential slowdown in high dimensions. In contrast, generative models based on diffusion and flow matching efficiently transport noise to data but offer limited theoretical guarantees and can overfit when data is scarce. We introduce Moment Guided Diffusion (MGD), which combines elements of both approaches. Building on the stochastic interpolant framework, MGD samples maximum entropy distributions by solving a stochastic differential equation that guides moments toward prescribed values in finite time, thereby avoiding slow mixing in equilibrium-based methods. We formally obtain, in the large-volatility limit, convergence of MGD to the maximum entropy distribution and derive a tractable estimator of the resulting entropy computed directly from the dynamics. Applications to financial time series, turbulent flows, and cosmological fields using wavelet scattering moments yield estimates of negentropy for high-dimensional multiscale processes.

翻译：从有限信息中生成样本是跨科学领域的一个基本问题。经典的最大熵方法通过矩约束提供了原则性的不确定性量化，但需要通过MCMC或朗之万动力学进行采样，这些方法在高维空间中通常呈现指数级减速。相比之下，基于扩散和流匹配的生成模型能高效地将噪声传输至数据，但理论保证有限，且在数据稀缺时容易过拟合。本文提出矩引导扩散方法，融合了两种方法的优势。基于随机插值框架，MGD通过求解在有限时间内将矩引导至预设值的随机微分方程来采样最大熵分布，从而避免了基于平衡态方法的缓慢混合问题。在大波动极限下，我们严格证明了MGD向最大熵分布的收敛性，并推导出可直接从动力学计算所得熵的可处理估计量。通过小波散射矩在金融时间序列、湍流和宇宙学场中的应用，实现了对高维多尺度过程负熵的估计。