We present PIVONet (Physically-Informed Variational ODE Neural Network), a unified framework that integrates Neural Ordinary Differential Equations (Neuro-ODEs) with Continuous Normalizing Flows (CNFs) for stochastic fluid simulation and visualization. First, we demonstrate that a physically informed model, parameterized by CNF parameters θ, can be trained offline to yield an efficient surrogate simulator for a specific fluid system, eliminating the need to simulate the full dynamics explicitly. Second, by introducing a variational model with parameters φ that captures latent stochasticity in observed fluid trajectories, we model the network output as a variational distribution and optimize a pathwise Evidence Lower Bound (ELBO), enabling stochastic ODE integration that captures turbulence and random fluctuations in fluid motion (advection-diffusion behaviors).

翻译：我们提出PIVONet（物理信息变分常微分方程神经网络），这是一个将神经常微分方程与连续归一化流相结合的统一框架，用于随机流体模拟与可视化。首先，我们证明一个由CNF参数θ参数化的物理信息模型可通过离线训练，为特定流体系统生成高效的代理模拟器，从而无需显式模拟完整动力学。其次，通过引入一个参数为φ的变分模型来捕捉观测流体轨迹中的潜在随机性，我们将网络输出建模为变分分布，并优化路径证据下界，从而实现能捕捉流体运动（平流-扩散行为）中湍流与随机波动的随机ODE积分。