PIVONet: A Physically-Informed Variational Neuro ODE Model for Efficient Advection-Diffusion Fluid Simulation

TL;DR

PIVONet presents a physically informed framework that blends continuous normalizing flows (CNFs) with variational stochastic differential equations (VSDEs) to efficiently simulate advection-diffusion fluids and quantify uncertainty. The CNF backbone captures mean transport while a trajectory-conditioned VSDE controller adds stochastic corrections, trained via a pathwise ELBO augmented with physics-based losses. Across three CFD-flow regimes, the approach yields consistent improvements in trajectory fidelity and reproduces realistic velocity-space statistics, while guard rails ensure stability during inference. This method enables fast, uncertainty-aware surrogate modeling of complex fluid dynamics with potential real-time applicability and configurable physical regularization.

Abstract

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).

Figures (11)

• Figure 1: Generated flow trajectories overlaid onto ground truth: (a) target-matching under vortex shearing, (b) sparse particle regions, and (c) flow separation.
• Figure 2: Langevin Dynamics: Particle Motion in Fluid Flow. The Langevin equation $dx_t = u(x_t)dt + \sqrt{2D}dW_t$ governs particle motion in fluid flows. (a) Poiseuille velocity profile. (b) Single particle trajectory. (c) Multiple stochastic realizations from the same initial condition.
• Figure 3: Continuous Normalizing Flow for Particle Trajectory Prediction . The CNF transforms a simple Gaussian base distribution $p$ (a) through intermediate distributions (b) via learned velocity fields $u_t(x)$ (c) to match the complex trajectory distribution $q$. During sampling (d), particles follow stochastic paths from initial position $x_0$ to predicted position $x_1$, capturing both deterministic flow and diffusive behavior.
• Figure 4: The VSDE framework couples a frozen CNF backbone with learned posterior controls and physics-informed losses.
• Figure 5: Inference results of the ODE-only CNF model on the incompressible cylinder flow. Predicted trajectories are overlaid on the ground truth flow realizations. While the model accurately captures the overall mean flow structure and produces smooth, stable trajectories, the predictions collapse toward a deterministic solution. This behavior highlights the model’s inability to represent stochastic variations and multi-modal flow behavior, resulting in limited diversity compared to the ground truth dynamics.