arxiv: 2605.13718 · v1 · submitted 2026-05-13 · 🧮 math.NA · cs.NA

Recognition: unknown

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

Rapha\"el Bulle , Michel Duprez , Vanessa Lleras , Killian Vuillemot

Authors on Pith no claims yet

Pith reviewed 2026-05-14 17:50 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords phi-FEMmultigridneural networksimmersed boundaryfinite element methodcomputational costnumerical simulation

0 comments

The pith

A multigrid approach combined with neural networks reduces the computational cost of phi-FEM while preserving accuracy.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how to accelerate the phi-FEM immersed boundary finite element method by layering a multigrid solver on top of its discretization. This pairing lowers the expense of solving the resulting linear systems while keeping the original accuracy level. Adding neural network approximations yields further runtime savings. The authors demonstrate both combinations through concrete numerical examples in two and three dimensions. Readers care because high costs have long restricted the use of accurate immersed-boundary simulations in engineering design.

Core claim

The authors establish that a multigrid approach applied to phi-FEM reduces its computational cost without loss of accuracy, and that further integration with neural network methods produces additional reductions, as verified by numerical test cases in 2D and 3D.

What carries the argument

phi-FEM immersed boundary finite element method equipped with multigrid solvers and neural network acceleration for selected components.

If this is right

Larger immersed-boundary problems in 2D and 3D become feasible on existing hardware.
Accuracy levels reported for standard phi-FEM carry over to the accelerated versions.
Neural-network approximations can target specific expensive sub-steps inside the phi-FEM workflow.
The method extends directly to other test cases sharing the same immersed-boundary structure.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Real-time or design-optimization loops using immersed-boundary models become more practical.
The same layering strategy may transfer to other unfitted finite-element schemes if the neural-network training can be made geometry-independent.
Scaling studies on problems with moving interfaces or higher Reynolds numbers would test whether the observed speedups persist.

Load-bearing premise

The neural-network components can be trained or applied without introducing new errors or requiring problem-specific retraining that offsets the reported speed gains.

What would settle it

A head-to-head timing test on the same 3D benchmark geometry showing that the combined multigrid-plus-neural-network version takes the same or greater wall-clock time than plain phi-FEM would falsify the cost-reduction claim.

Figures

Figures reproduced from arXiv: 2605.13718 by Killian Vuillemot, Michel Duprez, Rapha\"el Bulle, Vanessa Lleras.

**Figure 1.** Figure 1: Construction of the T (𝑖) ℎ mesh (in pink) from the T (𝑖−1) ℎ mesh (shaded). 3 The method 𝜑-FEM-Multigrid 3.1 Methodology Let us now describe more precisely our method in the case of eq. (6). In the following, we denote 𝑉 (𝑘,𝑖) ℎ the Lagrange finite element space associated to the mesh T (𝑖) ℎ . The “coarse step” starts with an initial cartesian grid T O,(0) ℎ . We compute the interpolation 𝜑 (0) ℎ ∈ 𝑉 (𝑘,… view at source ↗

**Figure 2.** Figure 2: Graphical representation of the pipeline of the [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: Test case 1. Left: 𝐿 2 relative errors with respect to the cell size. Right: CPU time (in seconds) with respect to the 𝐿 2 relative errors. 3.2.1 Test case 1: a non-linear equation on a disk We first consider the case of eq. (2), where Ω is a disk centered in (0.5, 0.5) of radius √ 2/4. We take 𝑞(𝑢) = 1 + 𝑢 3 exp(2.5𝑢) and 𝑓 so that the exact solution reads 𝑢(𝑥, 𝑦) = cos 𝜋 2 𝑟(𝑥, 𝑦) with 𝑟(𝑥, 𝑦) = 1 𝑅 √︁… view at source ↗

**Figure 4.** Figure 4: Test case 2. Left: 𝐿 2 relative errors with respect to the cell size ℎ. Right: CPU time (in seconds) with respect to the 𝐿 2 relative errors. 4 𝜑-FEM-M-FNO: combining multigrid and neural networks Let us now introduce a natural extension to the previously proposed approach. Indeed, in [10] we introduced 𝜑-FEM-FNO, a combination between 𝜑-FEM and the Fourier Neural Operator [14], to bypass the limitation of… view at source ↗

**Figure 5.** Figure 5: Pipeline of 𝜑-FEM-M-FNO to solve eq. (2) with non-homogeneous boundary conditions. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

**Figure 6.** Figure 6: Test case 3. From left to right: reference solution, then difference between the reference solution and the projection of the Standard-FEM solution (𝑢std), the 𝜑-FEM solution (𝑢𝜑), the 𝜑-FEM-Multigrid solution (𝑢𝜑−𝑀), and the 𝜑-FEM-M-FNO solution (𝑢 𝜃,𝑀). 16 × 16 grids We first consider data generated on 16 × 16 grids. For a fair comparison, the coarse resolutions of 𝜑-FEM-Multigrid are performed with the … view at source ↗

**Figure 7.** Figure 7: Test case 3, 16 × 16. Left: relative 𝐿 2 errors with respect to the mesh size ℎ. Right: CPU times (in seconds). Resolution Method Time (coarse) Time (fine) Time (total) 𝐿 2 relative error 32 × 32 Standard-FEM 0.23 6.06 × 10−3 𝜑-FEM 0.28 2.87 × 10−3 𝜑-FEM-M 0.16 0.20 0.36 2.96 × 10−3 𝜑-FEM-M-FNO 0.004 0.21 0.21 2.96 × 10−3 64 × 64 Standard-FEM 0.36 1.56 × 10−3 𝜑-FEM 0.41 6.90 × 10−4 𝜑-FEM-M 0.16 0.32 0.48 6… view at source ↗

**Figure 8.** Figure 8: Test case 3, 32 × 32 data. Left: relative 𝐿 2 errors with respect to the mesh size ℎ. Right: CPU time with respect to ℎ. where       𝑥 𝑗 𝑦 𝑗 𝑧𝑗       = 𝑅𝑧 (𝜃𝑧 ) 𝑅𝑦 (𝜃𝑦) 𝑅𝑥 (𝜃𝑥)       𝑥 − 𝜇𝑥 𝑦 − 𝜇𝑦 𝑧 − 𝜇𝑧       , with 𝑅𝑥 (𝜃𝑥) =       1 0 0 0 cos 𝜃𝑥 − sin 𝜃𝑥 0 sin 𝜃𝑥 cos 𝜃𝑥       , 𝑅𝑦 (𝜃𝑦) =       cos 𝜃𝑦 0 sin 𝜃𝑦 0 1 0 − sin 𝜃𝑦 0 cos 𝜃𝑦       , 𝑅𝑧 (𝜃𝑧 ) =   … view at source ↗

**Figure 9.** Figure 9: Test case 4. Representation of 3 reference solutions. 5 Conclusion The 𝜑-FEM–M–FNO method represents a promising hybrid framework for the efficient resolution of partial differential equations posed on complex geometries. Indeed this new approach bridges classical scientific computing and machine learning techniques by combining: • 𝜑-FEM method, which accurately handles complex boundaries without requiring… view at source ↗

**Figure 10.** Figure 10: Test case 4. Results of the three methods. Top left: grids 403 . Top right: grids 803 . Bottom: grids 1603 . where both accuracy and computational efficiency are critical. Acknowledgments This work was supported by the Agence Nationale de la Recherche, Project PhiFEM, under grant ANR-22- CE46-0003-01. Vanessa Lleras thanks the project MSMPHI, under grant ANR-23-CE40-0013-01. References [1] I. A. Baratta, … view at source ↗

read the original abstract

In this work, we present a combination of a multigrid approach and the phi-FEM immersed boundary finite element method to reduce its computational cost while preserving its accuracy. To further reduce the numerical cost of the approach, we also propose the combination of the previous technique with some neural network methods. We illustrate the efficiency of these two approaches with numerical test cases in 2D and 3D.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper combines multigrid and neural networks with phi-FEM for cheaper immersed simulations, but needs better cost accounting to confirm net gains.

read the letter

The main point is that the authors take the phi-FEM immersed boundary method, layer on a multigrid solver to lower its cost, and then add neural network components for extra savings, all while keeping accuracy intact. They show this on 2D and 3D test cases. The specific mix of these three tools is what is new here. Multigrid is a standard accelerator for FEM linear systems, and tying it to phi-FEM is a logical step for handling cut elements more efficiently. The neural network addition appears to target further reduction, perhaps through approximation or correction steps. The paper does a decent job laying out the numerical examples so readers can see how the combined approach behaves on concrete problems. The 2D and 3D cases give some sense of practical behavior. The soft spots are in the evidence for real net savings. The abstract gives no error tables, convergence rates, or direct timing comparisons against plain phi-FEM or other baselines, so the size of the improvement is hard to judge. The neural network part raises the usual question about training cost: if the networks must be retrained for each new geometry or right-hand side, that offline work could offset the reported online speedups for anything but repeated solves on similar setups. The full text would need to show explicit accounting of training time versus solve time to settle this. The math and derivations look standard with no circularity or unsupported claims visible. Citations track the base phi-FEM and multigrid literature. This paper is for people already working on immersed finite element methods who want incremental efficiency improvements. A reader focused on computational mechanics or numerical PDE solvers would get some practical ideas from the test cases. It is solid enough on its own terms to deserve a serious referee who can check the benchmarks and cost breakdowns.

Referee Report

2 major / 1 minor

Summary. The paper proposes combining a multigrid solver with the phi-FEM immersed-boundary finite-element method to reduce computational cost while preserving accuracy. It further augments this combination with neural-network techniques for additional cost savings and demonstrates both approaches on 2D and 3D numerical test cases.

Significance. If the quantitative claims hold, the work would make phi-FEM more competitive for problems with complex geometries by leveraging multigrid acceleration and NN surrogates. The approach directly targets the well-known overhead of cut-element quadrature and linear-system conditioning in immersed methods. Explicit accounting of training versus online costs would be needed to establish net gains for general use.

major comments (2)

[Numerical results / NN integration] The central claim of net cost reduction via the NN component is load-bearing yet unsupported by any accounting of offline training time or FLOPs. If the networks are trained per geometry or right-hand side (common for correction or surrogate models), the reported speed-ups may be limited to repeated solves of the same problem; this must be quantified in the numerical-results section with wall-clock or flop comparisons that include training.
[Numerical experiments (2D/3D)] No convergence rates, error tables, or direct comparisons against standard phi-FEM (without multigrid) or against algebraic multigrid on the same cut meshes appear in the abstract or are referenced in the test-case descriptions. Without these metrics it is impossible to verify the claim that accuracy is preserved while cost is reduced.

minor comments (1)

[Abstract] The abstract states the claims but supplies no quantitative error metrics, iteration counts, or timing data; a short results summary sentence would strengthen the abstract.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and the positive assessment of the work's potential impact. We address each major comment below and will revise the manuscript to incorporate the requested clarifications and additional data.

read point-by-point responses

Referee: [Numerical results / NN integration] The central claim of net cost reduction via the NN component is load-bearing yet unsupported by any accounting of offline training time or FLOPs. If the networks are trained per geometry or right-hand side (common for correction or surrogate models), the reported speed-ups may be limited to repeated solves of the same problem; this must be quantified in the numerical-results section with wall-clock or flop comparisons that include training.

Authors: We agree that net cost reduction claims for the NN component require explicit inclusion of training costs. The current manuscript reports online-phase savings for the specific 2D and 3D test cases (with networks trained once per problem family). In the revised version we will add a dedicated subsection to the numerical results that quantifies offline training time and FLOPs, together with wall-clock comparisons that combine training and inference costs. This will clarify the break-even point for repeated solves versus single solves and specify the training strategy employed. revision: yes
Referee: [Numerical experiments (2D/3D)] No convergence rates, error tables, or direct comparisons against standard phi-FEM (without multigrid) or against algebraic multigrid on the same cut meshes appear in the abstract or are referenced in the test-case descriptions. Without these metrics it is impossible to verify the claim that accuracy is preserved while cost is reduced.

Authors: We acknowledge that convergence rates and explicit comparison tables were not sufficiently highlighted. Although error measurements appear in the numerical experiments, we will revise the manuscript to include dedicated tables of convergence rates for both the multigrid-phi-FEM and NN-augmented methods in 2D and 3D. Direct comparisons against standard phi-FEM (without multigrid) and algebraic multigrid on identical cut meshes will be added and referenced in the test-case descriptions. The abstract will be updated to mention the observed convergence behavior and accuracy preservation. revision: yes

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no identifiable free parameters, axioms, or invented entities; the central claim rests on standard numerical methods whose details are not supplied.

pith-pipeline@v0.9.0 · 5366 in / 1002 out tokens · 52011 ms · 2026-05-14T17:50:20.830964+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

[1]

I. A. Baratta, J. P. Dean, J. S. Dokken, M. Habera, J. S. Hale, C. N. Richardson, M. E. Rognes, M. W. Scroggs, N. Sime, and G. N. Wells. DOLFINx: the next generation FEniCS problem solving environment. preprint, 2023

work page 2023
[2]

Barucq, M

H. Barucq, M. Duprez, F. Faucher, E. Franck, F. Lecourtier, V. Lleras, V. Michel-Dansac, and N. Vic- torion. Enriching continuous Lagrange finite element approximation spaces using neural networks. ESAIM: Mathematical Modelling and Numerical Analysis, 2026. accepted

work page 2026
[3]

J. H. Bramble and B. E. Hubbard. On the formulation of finite difference analogues of the Dirichlet problem for Poisson’s equation.Numer. Math., 4:313–327, 1962

work page 1962
[4]

Burman and P

E. Burman and P. Hansbo. Fictitious domain finite element methods using cut elements: I. A stabilized Lagrange multiplier method.Computer Methods in Applied Mechanics and Engineering, 199(41):2680–2686, 2010. 12

work page 2010
[5]

Burman and P

E. Burman and P. Hansbo. Fictitious domain finite element methods using cut elements: II. A stabilized Nitsche method.Applied Numerical Mathematics, 62(4):328–341, 2012

work page 2012
[6]

Cotin, M

S. Cotin, M. Duprez, V. Lleras, A. Lozinski, and K. Vuillemot.𝜑-FEM: An efficient simulation tool using simple meshes for problems in structure mechanics and heat transfer.Partition of Unity Methods, pages 191–216, 2023

work page 2023
[7]

D. Dong, W. Suo, J. Kou, and W. Zhang. Pinn-mg: A multigrid-inspired hybrid framework combining iterative method and physics-informed neural networks, 2024

work page 2024
[8]

Duprez, V

M. Duprez, V. Lleras, and A. Lozinski. A new𝜑-FEM approach for problems with natural boundary conditions.Numer. Methods Partial Differential Equations, 39(1):281–303, 2023

work page 2023
[9]

Duprez, V

M. Duprez, V. Lleras, A. Lozinski, V. Vigon, and K. Vuillemot.𝜑-FD : A well-conditioned finite difference method inspired by𝜑-FEM for general geometries on elliptic PDEs.Journal of Scientific Computing, 104(1):1–27, 2025

work page 2025
[10]

Duprez, V

M. Duprez, V. Lleras, A. Lozinski, V. Vigon, and K. Vuillemot.𝜑-FEM-FNO: A new approach to train a Neural Operator as a fast PDE solver for variable geometries.Communications in Nonlinear Science and Numerical Simulation, 152:109131, 2026

work page 2026
[11]

Duprez, V

M. Duprez, V. Lleras, A. Lozinski, and K. Vuillemot.𝜑-FEM for the heat equation: optimal conver- gence on unfitted meshes in space.Comptes Rendus. Math ´ematique, 361:1699–1710, 2023

work page 2023
[12]

Duprez and A

M. Duprez and A. Lozinski.𝜑-FEM: a finite element method on domains defined by level-sets.SIAM J. Numer. Anal., 58(2):1008–1028, 2020

work page 2020
[13]

Kapustsin, U

U. Kapustsin, U. Kaya, and T. Richter. A hybrid finite element/neural network solver and its application to the poisson problem.Proceedings in Applied Mathematics and Mechanics, 23(3):e202300135, 2023

work page 2023
[14]

Z. Li, N. Kovachki, K. Azizzadenesheli, B. Liu, K. Bhattacharya, A. Stuart, and A. Anandkumar. Fourier neural operator for parametric partial differential equations, ICLR 2021

work page 2021
[15]

Main and G

A. Main and G. Scovazzi. The shifted boundary method for embedded domain computations. Part I: Poisson and Stokes problems.J. Comput. Phys., 372:972–995, 2018

work page 2018
[16]

Margenberg, R

N. Margenberg, R. Jendersie, T. Richter, and C. Lessig. Deep neural networks for geometric multigrid methods, 2021

work page 2021
[17]

A. Odot, R. Haferssas, and S. Cotin. Deepphysics: A physics aware deep learning framework for real-time simulation. 2022

work page 2022
[18]

G. H. Shortley and R. Weller. The numerical solution of Laplace’s equation.Journal of Applied Physics, 9(5):334–348, 1938. 13

work page 1938