We study transient nonequilibrium dynamics in Fisher-regularized Wasserstein gradient flows and identify a sign-changing cross-dissipation mechanism generated by the coupling between transport dissipation and Fisher-information geometry. Using the Ornstein--Uhlenbeck Fokker--Planck system as an analytically tractable setting, we derive an exact reduced variance dynamics on the Gaussian manifold, \[ \dot{u}=2(1-u)+\frac{\varepsilon}{u}, \] where \(u(t)=σ^2(t)\) is the variance and \(\varepsilon>0\) is the Fisher regularization strength. The reduced dynamics reveal distinct transient regimes induced by the interaction between transport relaxation and information-geometric curvature. The associated cross-dissipation term changes sign at the critical scale \(σ=1\), separating cooperative acceleration for localized states with \(σ<1\) from transient interference at larger variance scales. In the subcritical regime, Fisher curvature accelerates the descent of the baseline free energy; beyond the critical transition, it partially opposes the Ornstein--Uhlenbeck pullback and generates transient overshoot toward a displaced Fisher-regularized equilibrium. We also establish a bounded transient-acceleration-window result, showing that the cooperative acceleration phase has finite duration with an upper bound depending only on the Fisher regularization strength. Finite-difference simulations support the analytical predictions and suggest that qualitatively similar sign-transition behavior may persist beyond Gaussian closure for non-Gaussian initial conditions, including bimodal and Laplace distributions. Overall, the results provide a transient dynamical perspective on Fisher-regularized dissipative systems and show how information-geometric curvature can reorganize intermediate-time Wasserstein relaxation while preserving the globally dissipative structure of the flow.

翻译：我们研究了Fisher正则化Wasserstein梯度流中的瞬态非平衡动力学，并识别出一种由输运耗散与Fisher信息几何耦合产生的变号交叉耗散机制。以Ornstein--Uhlenbeck Fokker--Planck系统作为可解析处理的框架，我们在高斯流形上推导出精确的约化方差动力学方程：\[ \dot{u}=2(1-u)+\frac{\varepsilon}{u} \]，其中\(u(t)=σ^2(t)\)为方差，\(\varepsilon>0\)为Fisher正则化强度。该约化动力学揭示了由输运弛豫与信息几何曲率相互作用诱导的不同瞬态区域。对应的交叉耗散项在临界尺度\(σ=1\)处发生符号变化，将局域化态（\(σ<1\)）中的协同加速与大尺度方差下的瞬态干涉区分开来。在亚临界区域，Fisher曲率加速了基准自由能的下降；超越临界转变后，它部分抵消Ornstein--Uhlenbeck回拉效应，并产生指向偏移后Fisher正则化平衡态的瞬态过冲。我们还建立了有界瞬态加速窗口定理，表明协同加速阶段具有有限持续时间，其上界仅取决于Fisher正则化强度。有限差分模拟支持了分析预测，并表明类似符号转变行为在非高斯初始条件（包括双峰分布和拉普拉斯分布）下可能超越高斯封闭性而持续存在。总体而言，这些结果为Fisher正则化耗散系统提供了瞬态动力学视角，揭示了信息几何曲率如何在保持流全局耗散结构的同时重新组织中等时间尺度的Wasserstein弛豫过程。