用条件得分建模有效朗之万动力学,直接从数据中学习复杂系统的随机简化模型。
Conditional Score-Based Modeling of Effective Langevin Dynamics

- 通过有限时滞转移密度的条件得分,建立系数与滞后相关函数的关系
- 在高维、稀疏采样数据下仍能准确恢复不变统计量和动态相关性
- 无需轨迹微分或反复模拟,适合处理复杂系统建模问题
随机降阶模型广泛用于表征复杂系统的有效动力学,但从未知数据中估计其漂移和扩散系数仍具挑战。传统方法依赖短时轨迹增量、状态空间划分或重复模拟候选模型,在高维系统、粗粒度采样或非均匀采样下变得不可靠或计算昂贵。本文提出一种基于新关系的数据驱动校准方法:将随机降阶模型的系数与有限时滞转移密度的条件得分(即对初态取对数转移密度的梯度)关联起来。该恒等式将滞后相关函数的导数表示为涉及条件得分和未知模型系数的观测滞后对的平稳期望。此形式允许直接从有限滞后统计量约束漂移与扩散结构,无需轨迹微分、状态空间划分或反复积分候选模型,转化为平稳滞后对上的最小二乘拟合问题。我们在三个复杂度递增的系统上验证该方法:一个解析可解的Cox–Ingersoll–Ross扩散、二维非平衡扩散(带仿射乘性噪声)、周期性软自旋随机兰道-利夫希茨链。结果表明,推断模型保持了不变统计量并复现了有限时滞动力学相关性。该框架为从数据学习能复现指定统计与动力学特性的随机降阶模型提供了可扩展路径。
原文摘要 · Abstract (English)
Stochastic reduced-order models are widely used to represent the effective dynamics of complex systems, but estimating their drift and diffusion coefficients from data remains challenging. Standard approaches often rely on short-time trajectory increments, state-space partitioning, or repeated simulation of candidate models, which become unreliable or computationally expensive for high-dimensional systems, coarse temporal sampling, or unevenly sampled data. We introduce a data-driven calibration method based on a novel relationship between the coefficients of a stochastic reduced model and the conditional score of the finite-time transition density, defined as the gradient of the logarithm of the transition density with respect to the initial state. The resulting identity expresses derivatives of lagged correlation functions as stationary expectations over observed lagged pairs involving this conditional score and the unknown model coefficients. This formulation allows the drift and diffusion structure to be constrained directly from finite-lag statistics, without differentiating trajectories, partitioning state space, or repeatedly integrating candidate reduced models during calibration, yielding a least-squares fitting problem over stationary lagged pairs. We validate the approach on three systems of increasing complexity: an analytically tractable Cox--Ingersoll--Ross diffusion, a two-dimensional nonequilibrium diffusion with affine multiplicative noise, and a periodic soft-spin stochastic Landau--Lifshitz chain. Across these tests, the inferred models preserve the invariant statistics while reproducing finite-lag dynamical correlations. The framework provides a scalable route for learning stochastic reduced-order models from data that reproduce prescribed statistical and dynamical properties.
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