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Coupling of the Finite Element Method with Physics Informed Neural Networks for the Multi-Fluid Flow Problem

Michel Nohra, Steven Dufour

TL;DR

The work addresses accurate multi-fluid flow simulation by coupling finite elements for the Navier–Stokes equations with physics-informed neural networks to advect a level-set interface. It introduces a multi-level PINN architecture to improve free-surface topology handling and compares three time-integration strategies, plus PINN-based reinitialization methods to maintain a sharp, differentiable interface. Key findings show that strong initial-condition imposition and the PINN-R reinitialization yield higher accuracy and better curvature/mass conservation than alternatives, and the approach achieves benchmark-level predictions for rising-bubble dynamics. The combined FEM–PINN framework offers a flexible, differentiable, and potentially more robust pathway for simulating interfacial flows with capillary effects in engineering applications.

Abstract

Multi-fluid flows are found in various industrial processes, including metal injection molding and 3D printing. The accuracy of multi-fluid flow modeling is determined by how well interfaces and capillary forces are represented. In this paper, the multi-fluid flow problem is discretized using a combination of a Physics-Informed Neural Network (PINN) with a finite element discretization. To determine the best PINN formulation, a comparative study is conducted using a manufactured solution. We compare interface reinitialization methods to determine the most suitable approach for our discretization strategy. We devise a neural network architecture that better handles complex free surface topologies. Finally, the coupled numerical strategy is used to model a rising bubble problem.

Coupling of the Finite Element Method with Physics Informed Neural Networks for the Multi-Fluid Flow Problem

TL;DR

The work addresses accurate multi-fluid flow simulation by coupling finite elements for the Navier–Stokes equations with physics-informed neural networks to advect a level-set interface. It introduces a multi-level PINN architecture to improve free-surface topology handling and compares three time-integration strategies, plus PINN-based reinitialization methods to maintain a sharp, differentiable interface. Key findings show that strong initial-condition imposition and the PINN-R reinitialization yield higher accuracy and better curvature/mass conservation than alternatives, and the approach achieves benchmark-level predictions for rising-bubble dynamics. The combined FEM–PINN framework offers a flexible, differentiable, and potentially more robust pathway for simulating interfacial flows with capillary effects in engineering applications.

Abstract

Multi-fluid flows are found in various industrial processes, including metal injection molding and 3D printing. The accuracy of multi-fluid flow modeling is determined by how well interfaces and capillary forces are represented. In this paper, the multi-fluid flow problem is discretized using a combination of a Physics-Informed Neural Network (PINN) with a finite element discretization. To determine the best PINN formulation, a comparative study is conducted using a manufactured solution. We compare interface reinitialization methods to determine the most suitable approach for our discretization strategy. We devise a neural network architecture that better handles complex free surface topologies. Finally, the coupled numerical strategy is used to model a rising bubble problem.
Paper Structure (17 sections, 49 equations, 14 figures, 2 tables)

This paper contains 17 sections, 49 equations, 14 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Governing equations
  3. Adimensional equations
  4. Numerical methods
  5. The finite element method for the Navier-Stokes equations
  6. Physics-Informed Neural Networks
  7. Proposed PINN for the level-set advection problem
  8. Time integration schemes for the advection equation
  9. Neural network architecture
  10. Re-initialization methods for the level-set function
  11. Numerical results
  12. Comparison of the time-integration schemes
  13. Comparison between the multi-level PINN and the strong PINN
  14. Re-initialization methods
  15. The steady-state Laplace problem
  16. ...and 2 more sections

Figures (14)

  • Figure 1: Feedforward neural network
  • Figure 2: A representation of two time intervals, showing the discrete times $t_n$, the discrete velocities $\boldsymbol{u}_n$, the continuous in time interpolation of the velocities $\bar{\boldsymbol{u}}_n$, and the level-set function $F_n$ defined over each time interval.
  • Figure 3: The architecture of the PINN-R used to reinitialize the level-set function.
  • Figure 4: (a) The interface shape obtained at $t=1$ for the comparison of the three time-integration schemes, (b) a zoom-in on the interface shape at $t=1$, (c) the error on the position of the interface, for the problem with a manufactured solution using the three different time integration schemes.
  • Figure 5: (a) The interface shape obtained at time $t=1$ with the strong single-level PINN, and the proposed multi-level PINN, (b) a zoom-in on the interface shapes, (c) the error $E$ on the interface for the proposed methods, for the problem with a manufactured solution.
  • ...and 9 more figures