用随机神经动力学方法实现稀疏观测下的动态场重建与不确定性量化
Sparse-to-Field Reconstruction via Stochastic Neural Dynamic Mode Decomposition
- 基于随机神经微分方程扩展DMD，建模非线性连续时间动态
- 仅需10%观测密度即实现优于基线的重建精度，且能恢复真实动力学结构
- 适用于多实现实验数据，可保留群体变异性而非平均化
许多重要现实系统（如风场、洋流）具有动态特性且难以建模。学习其控制动力学是科学机器学习的核心挑战。动态模态分解（DMD）虽提供简单数据驱动近似，但受限于稀疏/噪声观测、线性假设及缺乏严谨不确定性量化。为此，我们提出随机NODE-DMD，一种可解释的概率化DMD扩展，能建模连续时间非线性动态。该方法可在任意坐标实现时空连续重建，并量化预测不确定性。在四个基准测试、一个合成场景及三个物理流体系统上，仅用10%观测密度训练时，其重建精度超越基线。同时能通过学习到的模态与连续时间特征值对齐真实动力学结构。对于含多个实现实例的数据集，该方法学习到的潜在动力学分布能保留群体变异性，而非跨模式平均。
原文摘要 · Abstract (English)
Many consequential real-world systems, like wind fields and ocean currents, are dynamic and hard to model. Learning their governing dynamics remains a central challenge in scientific machine learning. Dynamic Mode Decomposition (DMD) provides a simple, data-driven approximation, but practical use is limited by sparse/noisy observations from continuous fields, reliance on linear approximations, and the lack of principled uncertainty quantification. To address these issues, we introduce Stochastic NODE-DMD, a probabilistic extension of DMD that models continuous-time, nonlinear dynamics while remaining interpretable. Our approach enables continuous spatiotemporal reconstruction at arbitrary coordinates and quantifies predictive uncertainty. Across four benchmarks, a synthetic setting and three physics-based flows, it surpasses a baseline in reconstruction accuracy when trained from only 10% observation density. It further recovers the dynamical structure by aligning learned modes and continuous-time eigenvalues with ground truth. Finally, on datasets with multiple realizations, our method learns a calibrated distribution over latent dynamics that preserves ensemble variability rather than averaging across regimes. Our code is available at: https://github.com/sedan-group/Stochastic-NODE-DMD
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