PINNs4Drops: Video-conditioned physics-informed neural networks for two-phase flow reconstruction

TL;DR

This work employs a specialized optical technique that encodes droplet surface information through color-coded glare points, enabling enhanced reconstruction of gas-liquid interfaces, and introduces video-conditioned physics-informed neural networks (VcPINNs), which integrate experimental observations with governing fluid dynamics equations.

Abstract

Two-phase flow phenomena underpin critical technologies such as hydrogen fuel cells, spray cooling, and combustion, where droplet dynamics govern performance and efficiency. Conventional optical diagnostics, including shadowgraphy and particle image velocimetry, provide valuable insights but are limited to two-dimensional projections of inherently three-dimensional flows. We employ a specialized optical technique that encodes droplet surface information through color-coded glare points, enabling enhanced reconstruction of gas-liquid interfaces. To interpret these measurements, we introduce video-conditioned physics-informed neural networks (VcPINNs), which integrate experimental observations with governing fluid dynamics equations. This hybrid framework leverages the strengths of both data-driven learning and physical constraints, allowing accurate volumetric flow reconstruction from limited input images. Applied to droplet impingement experiments, our method yields highly resolved and physically consistent 3D interface and flow dynamics. The combined imaging and PINN reconstruction strategy provides a powerful platform for advancing multiphase-flow analysis, with broad potential impact across energy, cooling, and propulsion applications.

Figures (21)

• Figure 1: Componential overview of the PINNs4Drops framework for prediction of the three-dimensional gas-liquid interface, as well as the velocity and pressure distributions. (A) Experimental setup of the glare-point shadowgraphy technique consisting of a blue backlight, green and red lateral light sources, and a high-speed RGB camera. The impingement of liquid droplets on solid substrates is recorded as an image sequence. (B) A sequence visualizing the droplet dynamics during impingement on a hydrophobic substrate obtained by direct numerical simulation. Physics-based rendering is employed to generate synthetic glare-point shadowgraphy images from the simulated gas-liquid interface geometries. (C) Schematics of the proposed video-conditioned PINNs (VcPINNs). Initially, glare-point shadowgraphy images are processed using a convolutional hourglass network, which extracts pixel-aligned features from the input image at the pixel location $(x, y)$ on the image plane. Subsequently, temporal correspondences are extracted from the sequence of spatial features. The resulting spatio-temporal features, along with the temporal coordinate $t^*$ and the spatial coordinates $\mathbf{x}^*$, are provided as inputs to an MLP. The MLP predicts the phase distribution $\phi$, the three components of the dimensionless velocity vector $\mathbf{u}^*=(u^*,v^*,w^*)^T$, and the dimensionless pressure $p^*$ at the spatio-temporal coordinates $(x^*,y^*,z^*,t^*)$. The loss function $\mathcal{L}$ comprises data loss terms for $\phi$, $\mathbf{u}^*$, and $p^*$, as well as physics-informed loss terms enforcing the Navier-Stokes equations, the continuity equation, and the advection equation for the phase distribution.
• Figure 2: Volumetric interface reconstruction accuracy of IcPINNs during training. We measure the volumetric accuracy by the average 3D-IOU on the validation dataset at the end of each training epoch and compare the IcPINNs variants with the data-driven baseline IcNet. VoF-IcPINNs reach a higher reconstruction accuracy in comparison to IcNet, while the best-performing variant of PF-IcPINNs does not reach the accuracy of IcNet.
• Figure 3: Phase distribution predicted by VoF-IcPINNs and PF-IcPINNs along the center planes of the droplet. Shown are the predictions by VoF-IcPINNs (top) and PF-IcPINNs with an initial diffuse-interface width of $\epsilon_0 = 0.01$ (bottom) in the in-plane and out-of-plane directions. The predictions (left) are shown in comparison to the ground truth simulation data (middle) and the spatial distribution of the absolute error between the prediction and the ground truth (right). Both VoF-IcPINNs and PF-IcPINNs predict a sharp interface in the in-plane direction as indicated by the narrow error distribution (cp. (A) and (C)), while in the out-of-plane direction PF-IcPINNs predicts a sharper interface than VoF-IcPINNs (cp. (B) and (C)).
• Figure 4: Gas-liquid interface reconstruction by VoF-IcPINNs and IcNet for droplet impingement experiments on the structured PDMS substrate. Temporal evolution of (A) the dimensionless droplet height $h^*$ and (B) the in-plane spreading factor $\xi_{in}$, obtained from the reconstructed gas-liquid interface geometries and compared with experimental measurements. We obtained the experimental data of $h^*$ and $\xi_{in}$ from the shadowgraphy contours of the droplet in the input images. (C) Development of the out-of-plane spreading factor $\xi_{out}$ over time. (D) Out-of-plane reconstruction by IcNet (left half) and VoF-IcPINNs (right half) for three different recordings from the experiments. The droplet geometry reconstructed by VoF-IcPINNs is mirrored vertically to allow for a direct comparison of the reconstructed contours. (E) Temporal evolution of the normalized integral volume of the droplet $V^*$; the black reference line represents mass-conservative reconstruction results.
• Figure 5: Velocity and pressure inference of VoF-VcPINNs for one snapshot of the validation dataset. (A) Synthetic input image rendered from a simulated droplet at $t=1.05$ ms after impingement. (B) Streamline visualization of the inferred velocity field and pressure contours along the center plane of the droplet in the in-plane direction in comparison to (C) the ground truth simulation data. The contour of the gas-liquid interface is indicated by the black solid line. The PINNs complete the flow field in a physically consistent way beyond the boundary of the training data domain, which is indicated by the red box.