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Physics-informed neural networks for solving two-phase flow problems with moving interfaces

This paper develops and rigorously analyzes a meshfree Physics-Informed Neural Network (PINN) framework for solving two-phase flow problems with moving interfaces under both prescribed and solution-driven motion scenarios, providing comprehensive error estimates and practical training guidelines validated through numerical experiments.

Original authors: Qijia Zhai, Pengtao Sun, Xiaoping Xie, Xingwen Zhu, Chen-Song Zhang

Published 2026-04-02

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Qijia Zhai, Pengtao Sun, Xiaoping Xie, Xingwen Zhu, Chen-Song Zhang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict how two different liquids, like oil and water, will mix and move inside a container. But here's the catch: the boundary between them isn't a static wall; it's a living, breathing, dancing line that changes shape, moves, and twists every single second.

This is the problem of Two-Phase Flow with Moving Interfaces. It's crucial for everything from designing microchips that cool themselves with liquid to understanding how blood flows through veins.

Traditionally, scientists solve this by drawing a digital "net" (a mesh) over the container. As the liquids move, they have to constantly redraw this net to fit the new shape of the boundary. It's like trying to take a photo of a dancing partner by constantly reshaping your camera frame to fit their movements. It's slow, computationally expensive, and prone to errors when the dance gets too wild.

The Paper's Big Idea: The "Physics-Informed" Neural Network

This paper introduces a smarter way using Physics-Informed Neural Networks (PINNs). Think of a neural network not as a rigid net, but as a super-smart, shape-shifting artist.

Here is the breakdown of their solution in simple terms:

1. No More Nets (Meshfree)

Instead of drawing a grid, the researchers use a "cloud" of points scattered throughout the space. The neural network is like a student who learns the rules of the universe (the laws of physics, specifically the Navier-Stokes equations) by looking at these points.

Analogy: Imagine teaching a child to draw a circle. Instead of giving them a stencil (the mesh), you tell them, "A circle is a shape where every point is the same distance from the center." The child (the AI) learns the concept and can draw a perfect circle anywhere, even if the paper is crumpled or the shape is changing.

2. The "Two-Team" Strategy

Since the two liquids (Phase 1 and Phase 2) have different properties (like oil is slippery, water is sticky), the researchers use two separate neural networks working side-by-side.

The Metaphor: Imagine two painters working on a canvas that is being split down the middle by a moving line.
- Painter A knows the rules for the "Oil" side.
- Painter B knows the rules for the "Water" side.
- They don't just paint randomly; they are constantly checking with each other at the moving line (the interface) to make sure the colors blend correctly and the forces balance out. If the oil pushes the water, the water must push back equally.

3. The "Teacher" (The Loss Function)

How does the AI know if it's doing a good job? It doesn't need a teacher with a red pen checking every answer. Instead, it has an internal Teacher (The Loss Function) that checks its work against the laws of physics.

The Game: The AI tries to guess the speed and pressure of the fluids. The Teacher checks:
- Did you conserve mass? (Did water disappear?)
- Did you balance the forces? (Did the oil push too hard?)
- Did you respect the boundary? (Did the two liquids mix where they shouldn't?)
- Did you match the starting conditions?
- The Goal: The AI adjusts its "brain" (parameters) to minimize the mistakes (residuals) until it satisfies all the physics laws simultaneously.

4. The Two Types of Dances

The paper tackles two specific scenarios:

The Choreographed Dance (Prescribed Motion): The boundary moves according to a pre-written script (e.g., "The circle will rotate and stretch like this"). The AI just needs to learn the fluid flow around this known path.
The Improv Dance (Solution-Driven Motion): This is the hard part. The boundary moves because of the fluid flow. The fluid pushes the boundary, and the boundary's new shape changes the fluid flow. It's a chicken-and-egg problem.
- The Solution: The researchers built a clever loop. The AI guesses the flow, uses that guess to move the boundary, then re-evaluates the flow based on the new boundary, and repeats this until everything settles into a consistent story.

5. Why This Matters (The "Aha!" Moment)

The researchers didn't just build a cool tool; they proved why it works mathematically.

The Insight: They discovered that the biggest errors don't come from the middle of the fluids, but from the boundaries (the moving line, the walls, and the start time).
The Advice: If you want your AI to be super accurate, don't just throw random points everywhere. Put more points right on the moving boundary. It's like focusing your camera lens on the dancer's feet rather than the empty stage.

Summary

This paper presents a mesh-free, AI-driven method to simulate complex fluid interactions where the boundary between liquids is constantly changing.

Old Way: Constantly reshaping a digital net (slow and hard).
New Way: Using two AI "painters" who learn the laws of physics and constantly check their work at the moving boundary.
Result: A faster, more flexible way to solve problems that were previously too difficult for traditional computers, especially when the fluids have very different properties (like high-contrast viscosity).

It's a step toward letting computers "understand" the physics of moving liquids without needing to force them into a rigid grid, making simulations of complex real-world flows (like blood flow or oil spills) much more accurate and efficient.

1. Problem Statement

The paper addresses the numerical solution of two-phase flow problems with moving interfaces, a fundamental challenge in computational fluid dynamics (CFD) with applications in microfluidics, biomedical engineering, and environmental science.

Key Challenges:

Moving Interfaces: The interface between two immiscible fluids evolves over time, either via a prescribed motion (known velocity) or solution-driven motion (velocity determined by the fluid flow itself).
Discontinuous Properties: Material properties (density $\rho$ , viscosity $\mu$ ) exhibit sharp discontinuities across the interface, leading to singularities in the solution.
Coupling Complexity: The problem involves a bidirectional coupling between fluid dynamics (Navier-Stokes equations) and interface evolution (kinematic conditions).
Limitations of Traditional Methods: Mesh-based methods (e.g., FEM, FVM) require frequent mesh adaptation and remeshing for moving interfaces, which is computationally expensive and complex, especially in 3D. Meshfree methods often lack rigorous theoretical error estimates for time-dependent domains.

2. Methodology

The authors propose a Physics-Informed Neural Network (PINN) framework that is meshfree and utilizes piecewise deep neural networks to handle the discontinuous nature of the problem.

A. Mathematical Formulation

Governing Equations: The incompressible Navier-Stokes equations are solved in two time-dependent subdomains, $\Omega_1(t)$ and $\Omega_2(t)$ , separated by an interface $\Gamma(t)$ .
Interface Conditions:
- Kinematic: Continuity of velocity (or prescribed jump).
- Dynamic: Balance of forces (stress continuity or prescribed jump).
Loss Function (Least-Squares): The problem is reformulated as a minimization of a spatiotemporal least-squares loss functional $R(\tilde{u}_1, \tilde{u}_2)$ $R (\tilde{u}_{1}, \tilde{u}_{2})$ . This functional aggregates residuals from:
1. PDE Residuals: Inside each fluid subdomain ( $\Omega_i$ ).
2. Interface Conditions: On the moving interface $\Gamma(t)$ .
3. Boundary Conditions: On fixed external boundaries.
4. Initial Conditions: At $t=0$ .
Piecewise Architecture: Separate neural networks are trained for each fluid subdomain ( $NN_1$ for $\Omega_1$ , $NN_2$ for $\Omega_2$ ) to naturally accommodate different solution characteristics and discontinuities across the interface.

B. Handling Moving Interfaces

Prescribed Motion: The interface position $\Gamma(t)$ is known a priori. Sampling points are distributed in the known subdomains.
Solution-Driven Motion: The interface velocity is unknown and equals the fluid velocity ( $w = v_1 = v_2$ $w = v_{1} = v_{2}$ ). The authors propose an iterative nonlinear algorithm (Algorithm 1):
1. Initialize the interface.
2. Train PINNs to predict velocity fields.
3. Update the interface position by integrating the predicted velocity over time.
4. Regenerate training points based on the new interface geometry.
5. Repeat until convergence.

C. Training Strategy

Spatiotemporal Sampling: Training points are distributed in $(d+1)$ -dimensional space (space + time).
Adaptive Weighting: To balance the different orders of magnitude in residuals (PDE vs. boundary vs. interface), an adaptive weighting strategy is employed during training.
Observation Points: For cases with high-contrast coefficients (large jumps in $\rho$ or $\mu$ ), specific observation points with known exact pressure values are added near the interface to stabilize the training and improve accuracy.

3. Key Contributions

A. Rigorous Theoretical Analysis

Error Estimation: The paper provides a comprehensive energy error estimate for the PINN approximation in evolving domains.
Reynolds Transport Theorem: A novel application of the Reynolds transport theorem is used to handle the time derivative of the kinetic energy term in moving domains, a critical step often overlooked in PINN literature.
Convergence Guarantees: The derived error bounds (Theorem 4.1) show that the total error consists of:
1. Optimization Error: Arising from the neural network's inability to minimize the loss perfectly.
2. Quadrature Error: Arising from the Monte Carlo integration of the loss function.
  Crucially, the analysis reveals that quadrature errors from interface, boundary, and initial conditions play a dominant role, necessitating higher sampling density in these regions.

B. Unified Framework

The method handles both prescribed and solution-driven interface motions within a single theoretical framework, addressing the complex coupling in the latter case.

C. Practical Guidance

The theoretical analysis provides actionable guidance for practitioners: to achieve optimal accuracy, computational resources (sampling points) must be strategically allocated, with increased density on the moving interface and boundaries, not just in the interior domain.

4. Numerical Results

The authors validated the method through three numerical experiments using PyTorch:

Example 1 (Prescribed Motion, No Jumps):
- A deforming, translating, and rotating elliptical interface with continuous physical properties.
- Result: High accuracy ( $L^2$ error $\approx 10^{-2}$ ) with convergence rates consistent with theory as sampling points increased.
Example 2 (Prescribed Motion, High Contrast):
- A complex five-fold modulation interface with a 1000:1 jump in density and viscosity.
- Result: Without intervention, errors were high ( $\approx 10^{-1}$ ) due to poor pressure approximation at the jump. By adding 50 observation points for pressure on the interface, errors dropped significantly ( $\approx 10^{-2}$ ), validating the necessity of data augmentation for high-contrast problems.
Example 3 (Solution-Driven Motion):
- The most challenging case where the interface evolves solely based on the fluid velocity solution.
- Result: The iterative tracking algorithm successfully captured the complex deformation of the interface. The method maintained accuracy comparable to the prescribed cases, demonstrating robustness in handling the bidirectional coupling between flow and geometry.

5. Significance and Impact

Meshfree Advantage: Eliminates the need for complex mesh generation and adaptation, making it highly suitable for problems with rapidly changing or topologically complex interfaces.
Theoretical Rigor: Fills a gap in the PINN literature by providing rigorous error bounds for time-dependent, multi-phase problems, moving beyond empirical success to theoretical guarantees.
Engineering Applicability: Offers a viable alternative to traditional CFD for microfluidics, biomedical flows, and environmental modeling where interface tracking is difficult.
Scalability: The approach is generalizable to higher dimensions and multiphysics coupling, offering a pathway for solving previously intractable fluid-structure interaction problems.

In summary, this paper establishes a robust, theoretically grounded, and practically effective PINN framework for two-phase flows with moving interfaces, successfully bridging the gap between deep learning methods and rigorous numerical analysis for complex fluid dynamics.