Probabilistic Simulation of Gas-Liquid Interfaces with Coupled Flow Matching on Spherical Harmonics

Kiet Bennema ten Brinke; Rachna Ramesh; Douwe Orij; Ivo Roghair; Vlado Menkovski

cs.LGOct 12, 2026

Probabilistic Simulation of Gas-Liquid Interfaces with Coupled Flow Matching on Spherical Harmonics

Authors: Kiet Bennema ten Brinke, Rachna Ramesh, Douwe Orij, Ivo Roghair, Vlado Menkovski

Organizations: Department of Mathematics and Computer Science, Eindhoven University of Technology · Department of Chemical Engineering and Chemistry, Eindhoven University of Technology

Abstract

Bubbly flows involve strongly coupled bubble motion, deformation, and hydrodynamic interactions, making high-fidelity simulation computationally expensive. We propose a neural surrogate that jointly predicts bubble trajectories and interface deformation in a compact, shared representation, replacing costly mesh-resolved simulation with fast generative inference. Because two-phase flow dynamics is nonlinear and chaotic, we treat forecasting as a conditional generative problem rather than deterministic regression, predicting entire future windows in a single pass. We train and evaluate the surrogate on BubbleSH, a Front-Tracking dataset of rising deformable bubble swarms, assessing performance using trajectory, shape, and distributional metrics over kinematic, morphological, and interaction-based properties. The surrogate reproduces key flow statistics while running several orders of magnitude faster than the underlying DNS solver.

Figures & tables

Figure 1: Snapshot of 32 bubble meshes from the BubbleSH dataset

	$\varepsilon$	Time	Speedup
DNS	$5$	$2.6$ days	$1\times$
DNS	$40$	$0.6$ days	$1\times$
Surrogate	$5$	$0.85$ sec	$260$ , $000\times$
Surrogate	$40$	$0.85$ sec	$58$ , $000\times$

Table 2: Comparison of required time to run 0.1s of simulation.

Figure 2: Probability mass functions of marginal kinematic, shape and interaction properties calculated from predicted and ground truth target windows from experiment ( iii ) . Unit of distance is bubble diameter, unit of time is seconds. See Appendix B.1 for details.

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Model	SH order	Prior $\mu$	Prior $\sigma_{\mathbf{p}},\sigma_{\mathbf{c}}$	Prior $s$	Layers	Epochs	Hidden dim	Dropout
Model $(L=5)$	5	$\langle\Delta\mathbf{x}_{0:T_{c}}\rangle$	$\sigma_{0:T_{c}}\cdot s$	$1$	$3\times$ (MP,U-Net)	400	64	0
Gaussian prior	5	$\mathbf{x}_{T_{c}}$	$0.01,0.05$	-	$3\times$ (MP,U-Net)	400	64	0
Deterministic	5	$\langle\Delta\mathbf{x}_{0:T_{c}}\rangle$	0	-	$3\times$ (MP,U-Net)	100	64	0.05
Independent	5	$\langle\Delta\mathbf{x}_{0:T_{c}}\rangle$	$\sigma_{0:T_{c}}\cdot s$	1	$3\times$ U-Net	400	64	0
Transformer	5	$\langle\Delta\mathbf{x}_{0:T_{c}}\rangle$	$\sigma_{0:T_{c}}\cdot s$	1	$4\times$ (Space,Time)	400	128	0
Model $(L=3)$	3	$\langle\Delta\mathbf{x}_{0:T_{c}}\rangle$	$\sigma_{0:T_{c}}\cdot s$	1	$3\times$ (MP,U-Net)	400	64	0

Appendix

Table 3: Training hyperparameters for the experiments.

Figure 3: Predicted and ground truth probability mass functions of position-based marginal quantities from the $T_{c}+1:T$ frames, from the main experiment (iii) .

Figure 4: Predicted and ground truth probability mass functions of shape-based marginal quantities. From the main experiment (iii) .

Figure 5: Frequency analysis of the spherical harmonic coefficients trajectories $\mathbf{c}$ at frames $T_{c}:T$ . The solid lines are ground truth, the predictions are dotted. From the main experiment (iii) .

Figure 6: Visualizations of centered 32 bubble centroid trajectories. Duration is 200 ms, number of timesteps is 100, filtered to exclude periodic boundary jumps.

Figure 7: Visualizations of the shape of one bubble changing over time, taken from [ 25 ] .

Neural Surrogate Modeling

Neural surrogate modeling uses learned models to approximate costly simulations, enabling faster prediction, design, control, and inference.

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BubbleSH: A Dataset of Rising Bubbles with Deformable Interfaces

Rachna Ramesh, Kiet Bennema ten Brinke, Douwe Orij, Ivo Roghair+1

Department of Mathematics and Computer Science, Eindhoven University of Technology · Department of Chemical Engineering and Chemistry, Eindhoven University of Technology

Generative Modeling Spherical Harmonics Trajectory Prediction Computational Fluid Dynamics

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Bubbly flows exhibit complex multiscale dynamics, with deformable bubbles interacting through the surrounding liquid and giving rise to strongly coupled kinematic and morphological behavior. We present BubbleSH, a bubbly flows dataset consisting of transient, three-dimensional bubble-swarm dynamics obtained from high-fidelity direct numerical simulations of bubbles rising in a periodic domain. The dataset provides time-resolved bubble trajectories, velocities, and shape evolution, with bubble morphology compactly represented using spherical harmonics. Designed to be lightweight yet physically expressive, the dataset enables data-driven modeling of bubbly flow simulators where shape deformation and bubble-bubble interactions play a central role. We characterize the dataset with bubble kinematics, morphology, and interaction patterns, and introduce evaluation metrics for both trajectory and shape prediction. The sensitivity of bubble-swarm dynamics to local perturbations makes BubbleSH particularly well suited to generative models that learn distributions over possible future trajectories. We evaluate a permutationally and translationally equivariant probabilistic emulator on BubbleSH given the proposed metrics. Therefore, we establish a compact, high-fidelity dataset and a benchmark for developing and evaluating data-driven models of deformable, chaotic multiphase systems.

Abstract

Neural Surrogate Modeling

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