用残差方法直接学习概率分布演化轨迹，无需复杂计算。
Wasserstein Residuals: Learning Gradient Flows from Population Dynamics
- 以非负损失函数强制连续性方程，统一多种现有方法
- 新方法 stitching 在轨迹推断任务上达到当前最佳性能
- 无需模拟、抗观测间隔大，适合真实数据场景
重构群体动态是物理与数据科学中的核心问题。通常，动态被建模为沃瑟斯坦梯度流（WGF）：由能量泛函驱动的概率分布曲线。尽管存在多种数学刻画方式，主流算法仍依赖乔丹-金德尔勒-奥托（JKO）格式。然而，JKO方法对时间离散敏感且需求解代价高昂的最优传输问题。本文提出残差视角，通过非负损失函数强制连续性方程，其最小值即为WGF。结合数据拟合散度，构建单一全局目标函数。该框架统一了多种已有方法，并导出一种新型粒子法——stitching，完全无需模拟，对观测间巨大间隔具有鲁棒性。实验表明，stitching在多个轨迹推断基准测试中达到最先进水平。代码见 github.com/BasisResearch/wasserstein-residuals。
原文摘要 · Abstract (English)
Reconstructing population dynamics is a central problem in the physical and data sciences. Often, the dynamics are modeled as a Wasserstein gradient flow (WGF): a curve of distributions driven by an energy functional. Though there are multiple mathematical characterizations of a WGF, the dominant algorithmic approach relies on the Jordan--Kinderlehrer--Otto (JKO) scheme. JKO-based methods are inflexible to time discretisation and require solving costly optimal transport problems. We take a residual approach, enforcing the continuity equations via a non-negative loss function whose minimum is the WGF. Combined with a data-fitting divergence, this gives a single global objective. This perspective unifies several existing methods and leads to a new particle-based method, stitching, that is simulation-free and robust to large gaps between observations. We demonstrate that the stitching method achieves state-of-the-art performance across trajectory inference benchmarks. For code see github.com/BasisResearch/wasserstein-residuals.
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