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Predicting The Evolution of Interfaces with Fourier Neural Operators

Paolo Guida, William L. Roberts

TL;DR

This paper tackles the challenge of real-time prediction of liquid-vapour interface dynamics in multiphase flows by employing Fourier Neural Operators (FNOs) as fast surrogate models trained on volume-of-fluid simulation data. The authors design a five-layer 2D FNO with spectral convolutions in Fourier space to learn mappings from input fields (including an initial signed distance or volume fraction representation) to evolving interfaces, achieving accurate predictions while significantly reducing computational cost. Validation across two cases demonstrates strong generalization to unseen initial conditions, with metrics indicating high fidelity (e.g., MSE around 9.72, R^2 around 0.95) and millisecond inference times, suitable for real-time control and digital twin applications. The results suggest that FNO-based surrogates can complement or replace costly CFD in fast-process control, with planned extensions to 3D systems and phase-change phenomena.

Abstract

Recent progress in AI has established neural operators as powerful tools that can predict the evolution of partial differential equations, such as the Navier-Stokes equations. Some complex problems rely on sophisticated algorithms to deal with strong discontinuities in the computational domain. For example, liquid-vapour multiphase flows are a challenging problem in many configurations, particularly those involving large density gradients or phase change. The complexity mentioned above has not allowed for fine control of fast industrial processes or applications because computational fluid dynamics (CFD) models do not have a quick enough forecasting ability. This work demonstrates that the time scale of neural operators-based predictions is comparable to the time scale of multi-phase applications, thus proving they can be used to control processes that require fast response. Neural Operators can be trained using experimental data, simulations or a combination. In the following, neural operators were trained in volume of fluid simulations, and the resulting predictions showed very high accuracy, particularly in predicting the evolution of the liquid-vapour interface, one of the most critical tasks in a multi-phase process controller.

Predicting The Evolution of Interfaces with Fourier Neural Operators

TL;DR

This paper tackles the challenge of real-time prediction of liquid-vapour interface dynamics in multiphase flows by employing Fourier Neural Operators (FNOs) as fast surrogate models trained on volume-of-fluid simulation data. The authors design a five-layer 2D FNO with spectral convolutions in Fourier space to learn mappings from input fields (including an initial signed distance or volume fraction representation) to evolving interfaces, achieving accurate predictions while significantly reducing computational cost. Validation across two cases demonstrates strong generalization to unseen initial conditions, with metrics indicating high fidelity (e.g., MSE around 9.72, R^2 around 0.95) and millisecond inference times, suitable for real-time control and digital twin applications. The results suggest that FNO-based surrogates can complement or replace costly CFD in fast-process control, with planned extensions to 3D systems and phase-change phenomena.

Abstract

Recent progress in AI has established neural operators as powerful tools that can predict the evolution of partial differential equations, such as the Navier-Stokes equations. Some complex problems rely on sophisticated algorithms to deal with strong discontinuities in the computational domain. For example, liquid-vapour multiphase flows are a challenging problem in many configurations, particularly those involving large density gradients or phase change. The complexity mentioned above has not allowed for fine control of fast industrial processes or applications because computational fluid dynamics (CFD) models do not have a quick enough forecasting ability. This work demonstrates that the time scale of neural operators-based predictions is comparable to the time scale of multi-phase applications, thus proving they can be used to control processes that require fast response. Neural Operators can be trained using experimental data, simulations or a combination. In the following, neural operators were trained in volume of fluid simulations, and the resulting predictions showed very high accuracy, particularly in predicting the evolution of the liquid-vapour interface, one of the most critical tasks in a multi-phase process controller.
Paper Structure (12 sections, 24 equations, 4 figures, 2 tables)

This paper contains 12 sections, 24 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Neural operators.
  3. Fourier neural operators.
  4. Validation Metrics
  5. Network architecture.
  6. Optimization.
  7. Hardware.
  8. Results
  9. Case 1: forecasting.
  10. Case 2: learning dynamics.
  11. Performance validation.
  12. Conclusions

Figures (4)

  • Figure 1: Time evolution of the liquid interface in the dam-break scenario. The figure shows three predicted time-steps comparing ground truth with the outcome of the FNO. The model is trained on the first 60% of the time sequence and evaluated on the remaining 40%. The mesh resolution is $164\times164$
  • Figure 2: Training diagnostics and spatial accuracy of the Fourier Neural Operator on the dam-break benchmark. (\ref{['fig:lp-loss']}) The evolution of the $L^p$ loss over 100 training epochs indicates consistent convergence of the surrogate model. (\ref{['fig:mean-evolution']}) Compares the predicted and ground-truth spatial means of the signed-distance field (SDF) across training epochs, showing strong agreement and low variance. (\ref{['fig:error-snapshots']}) Visualizes the spatial distribution of absolute error for three representative frames sampled from the test set, illustrating the model's accuracy in resolving complex interface features and localized discrepancies. These diagnostics highlight the model’s robust training dynamics and generalization performance across unseen dam-break scenarios.
  • Figure 3: Comparison of predicted and ground-truth results at intermediate and final time steps. Each subfigure presents a visual comparison between the initial input, the ground-truth at a given time, and the predicted field from the FNO model. The top row of each panel shows the signed-distance fields (RDF): (left) the initial RDF at $t=0$, (centre) the ground truth at the evaluated time, and (right) the FNO prediction. The bottom row shows the corresponding binary volume fractions reconstructed from each RDF. (\ref{['fig:validation-025']}) Evaluates the model’s prediction at an intermediate time step ($t=0.25\,\mathrm{s}$). (\ref{['fig:validation-050']}) Shows predictions at the final time step ($t=0.50\,\mathrm{s}$), highlighting the model’s ability to extrapolate interface dynamics from initial conditions. The FNO effectively captures the evolving interface's global structure and fine-scale morphology in both cases.
  • Figure 4: Comparison of $L^p$ training and validation loss across architectures. The FNO exhibits superior convergence behavior and consistently achieves the lowest validation loss, underscoring its ability to learn continuous operator mappings in complex flow regimes. While the U-Net performs well early in training, it levels off with a higher validation loss and greater variance. Although more efficient than FNO in training time, the GCN model fails to capture the full spatial-temporal dynamics and saturates at a significantly higher loss. The latter highlights the trade-off between accuracy and training cost: FNO (1257 s) vs. GCN (470 s) vs. U-Net (190 s).