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A multigrid and neural network approach to reduce the computational cost of phi-FEM

This paper proposes a hybrid approach combining multigrid methods and neural networks to significantly reduce the computational cost of the phi-FEM immersed boundary technique while maintaining its accuracy, as demonstrated through 2D and 3D numerical experiments.

Original authors: Raphaël Bulle, Michel Duprez, Vanessa Lleras, Killian Vuillemot

Published 2026-05-14

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CC BY 4.0

Original authors: Raphaël Bulle, Michel Duprez, Vanessa Lleras, Killian Vuillemot

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 paint a picture of a complex, curvy shape (like a kidney or a cloud) on a grid of square tiles. This is what scientists do when they simulate physical phenomena, like heat flow or stress on a structure, using a method called Finite Element Method (FEM).

The problem? To make the picture accurate, the square tiles have to fit the curvy shape perfectly. This means you have to cut the tiles into tiny, jagged pieces to match the edges. It's like trying to fit a square peg in a round hole by shaving off pieces of the peg. This process is incredibly slow and computationally expensive, especially for 3D objects.

This paper proposes a clever "shortcut" to speed things up without losing the quality of the picture. They combine three tools: a smart way to handle the grid, a "zoom-in" strategy, and a neural network (a type of AI).

Here is how their new approach works, broken down into simple steps:

1. The "Ghost" Grid (𝜑-FEM)

Instead of cutting the tiles to fit the shape, the authors use a method called 𝜑-FEM.

The Analogy: Imagine you have a large, perfect grid of square tiles covering your whole table. You draw your curvy shape on top of it. Instead of cutting the tiles, you just tell the computer: "Ignore the parts of the tiles that are outside the shape."
The Magic: They use a mathematical "level-set function" (think of it as a digital contour map) to define the boundary. The math handles the messy edges automatically. This saves the time of having to cut and reshape the tiles, but solving the math on this "ghost" grid is still slow.

2. The "Zoom-In" Strategy (Multigrid)

To solve the math faster, they use a Multigrid approach.

The Analogy: Imagine you are trying to find a specific street in a huge city.
- The Old Way: You start by walking down every single street in the city, one by one, to find your destination. (This is the standard, slow method).
- The New Way: You first look at a tiny, blurry map of the whole city to get the general direction. Then, you zoom in to a medium map to get closer. Finally, you zoom in to the high-definition street view to find the exact house.
How it helps: The computer solves the problem on a very coarse (low-resolution) grid first. It uses that rough answer as a "head start" to solve the problem on the finer, more detailed grids. This skips the tedious work of starting from scratch every time.

3. The "Crystal Ball" (Neural Networks / FNO)

This is the newest and most exciting part. They combine the "Zoom-In" strategy with a Fourier Neural Operator (FNO).

The Analogy: Think of the Neural Network as a student who has studied thousands of previous maps.
- In the old "Zoom-In" method, the computer still has to do some math on the first, blurry map to get a rough idea.
- In this new method, the AI acts as a crystal ball. Before the computer even starts solving the math on the coarse grid, the AI looks at the shape and the problem and predicts the rough answer instantly.
The Result: The computer doesn't waste time solving the easy, low-resolution part. It jumps straight to the high-resolution part, using the AI's prediction as a perfect starting point.

What Did They Find?

The authors tested this "AI + Zoom + Ghost Grid" combo on 2D shapes (like circles) and 3D shapes (like spheres and complex blobs).

Speed: Their new method was significantly faster than the traditional methods. In some cases, it was 10 to 20 times faster to reach the same level of accuracy.
Accuracy: Despite using the "shortcut" of the AI, the final results were just as accurate as the slow, traditional methods.
Complexity: It worked well even for very complex 3D geometries where other methods struggle.

The Bottom Line

The paper presents a hybrid engine for solving complex physics problems. It uses a smart grid system to avoid messy geometry, a "zoom-in" technique to solve problems efficiently, and a trained AI to skip the boring first steps. The result is a way to get high-precision scientific answers much faster, which is crucial for simulations that usually take hours or days to run.

Note: The paper focuses strictly on the mathematical and computational efficiency of this method. It does not claim to have solved specific medical diagnoses or industrial products yet, but rather provides a faster tool that could be used for such things in the future.

Technical Summary: A Multigrid and Neural Network Approach to Reduce the Computational Cost of $\varphi$ -FEM

Problem Statement

The Finite Element Method (FEM) is a standard tool for solving partial differential equations (PDEs), but it faces significant limitations when applied to complex geometries, such as those found in biomedical simulations. Maintaining accuracy typically requires conforming meshes, which are difficult and computationally expensive to generate for irregular boundaries. While non-conforming methods like CutFEM, the Shifted Boundary Method, and the $\varphi$ -FEM (phi-FEM) have been introduced to bypass this, they often incur high computational costs, particularly when high-resolution meshes are required.

Previous attempts to accelerate these methods include:

Multigrid approaches: Efficient for Cartesian grids but traditionally difficult to couple with immersed boundary methods on complex domains.
Neural Networks (e.g., FNO): Fourier Neural Operators (FNO) have been combined with $\varphi$ -FEM ( $\varphi$ -FEM-FNO) to handle complex geometries. However, on high-complexity problems (e.g., non-linear elasticity), these methods often sacrifice accuracy for speed, failing to reach the precision of FEM-based solvers.

The core problem addressed in this work is how to reduce the computational cost of $\varphi$ -FEM while preserving its high accuracy and handling complex geometries, without the accuracy degradation seen in pure neural operator approaches.

Methodology

The paper proposes two distinct strategies to accelerate $\varphi$ -FEM: a pure numerical multigrid approach and a hybrid approach combining multigrid with neural networks.

1. The $\varphi$ -FEM Framework

The method utilizes a level-set function $\varphi$ to describe the domain $\Omega = \{\varphi < 0\}$ and its boundary $\Gamma = \{\varphi = 0\}$ . The unknown $u$ is rewritten as $u = \varphi w$ , where $w$ is the new unknown. The method operates on a background Cartesian grid, constructing submeshes that intersect the domain and the boundary. The scheme solves the PDE by finding $w_h$ in a finite element space such that the weak form includes stabilization terms on the cut cells.

2. $\varphi$ -FEM-Multigrid

This approach leverages the Cartesian structure of the background mesh to implement a multigrid strategy:

Coarse Step: An initial solution is computed on a coarse Cartesian grid. The level-set function is interpolated, and the $\varphi$ -FEM scheme is solved on the resulting coarse submesh.
Fine Steps: The mesh is iteratively refined. The solution from the previous (coarser) level is interpolated onto the new, finer finite element space and used as an initial guess for the iterative solver.
Constraint: The method assumes the exact domain $\Omega$ is a subset of the initial coarse domain $\Omega^{(0)}_h$ to ensure no part of the geometry is lost during refinement.

3. $\varphi$ -FEM-M-FNO (Hybrid Approach)

To further reduce costs, the authors propose replacing the expensive "coarse step" of the multigrid method with a pre-trained Fourier Neural Operator.

Training: A $\varphi$ -FEM-FNO is trained to map problem data (source term, boundary conditions, and level-set function) to the solution $w$ on a coarse resolution.
Inference Pipeline:
1. Coarse Prediction: The trained FNO predicts a coarse approximation of the solution.
2. Discretization: This matrix/tensor prediction is converted into a finite element function (nodal values).
3. Refinement: The coarse solution is interpolated onto the fine mesh.
4. Fine Resolution: An iterative solver (e.g., BiConjugate Gradient Stabilized) refines the solution on the fine mesh, initialized with the FNO prediction.
Advantages: This reduces the training data requirements (coarse grids are cheaper to generate) and eliminates the need to solve the PDE on the coarse grid numerically, significantly lowering the total computational time.

Key Contributions

Multigrid Integration for $\varphi$ -FEM: The paper successfully adapts multigrid techniques to the $\varphi$ -FEM framework, demonstrating that starting with a coarse solution and refining it preserves accuracy while accelerating convergence.
Hybrid Neural-Multigrid Solver: The authors introduce $\varphi$ -FEM-M-FNO, a novel coupling where a neural operator replaces the coarse grid solve. This approach avoids the accuracy trade-offs of direct neural solvers by using the neural network only for initialization, relying on the rigorous $\varphi$ -FEM solver for the final high-precision result.
Numerical Validation: The methods are validated on both 2D and 3D problems, including non-linear equations and complex, randomly generated geometries defined by level-set functions.

Results

The paper presents four numerical test cases comparing Standard-FEM, $\varphi$ -FEM, $\varphi$ -FEM-Multigrid, and $\varphi$ -FEM-M-FNO:

Test Case 1 (2D Non-linear Disk): Both multigrid variants achieved precision comparable to classical $\varphi$ -FEM but significantly faster than Standard-FEM and standard $\varphi$ -FEM.
Test Case 2 (3D Sphere): In 3D, where direct solvers are prohibitively expensive, the multigrid approach outperformed the other methods in terms of the error-vs-time trade-off, achieving better precision faster.
Test Case 3 (2D Complex Geometries with Non-homogeneous BCs):
- Using FNO trained on $16 \times 16$ grids, $\varphi$ -FEM-M-FNO was consistently the fastest method across fine resolutions ($32$ to $256$), with errors comparable to other $\varphi$ -FEM variants.
- Using FNO trained on $32 \times 32$ grids, the method maintained superior speed while achieving high accuracy (tolerance $10^{-9}$ ).
- The "coarse step" time for the FNO approach was negligible (approx. 0.004s) compared to the numerical coarse solve (approx. 0.16s).
Test Case 4 (3D Complex Geometries): On 3D grids up to $160^3$ , $\varphi$ -FEM-M-FNO was systematically much faster than both $\varphi$ -FEM and $\varphi$ -FEM-Multigrid, while maintaining or improving accuracy. The method successfully handled random, complex geometries defined by level-set functions.

Significance and Claims

The paper claims that the $\varphi$ -FEM-M-FNO method represents a "promising hybrid framework" that bridges classical scientific computing and machine learning. Its primary significance lies in:

Preserving Accuracy: Unlike pure neural operator approaches, this method retains the theoretical convergence and high precision of FEM-based solvers by using the neural network only for initialization.
Efficiency: It significantly reduces computational time by offloading the coarse-grid resolution to a fast neural inference and leveraging multigrid acceleration.
Scalability: The approach is shown to be effective in both 2D and 3D, handling complex, non-conforming geometries without the mesh generation bottlenecks of traditional FEM.

The authors modestly note that challenges remain, specifically the computational cost of the FNO training phase and the lack of a theoretical convergence analysis for the coupled numerical/learning framework. However, they position this work as an innovative foundation for multi-scale and industrial applications where both accuracy and efficiency are critical.