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arxiv: 2603.01420 · v3 · submitted 2026-03-02 · 💻 cs.LG

Recognition: 2 theorem links

· Lean Theorem

Tackling multiphysics problems via finite element-guided physics-informed operator learning

Yusuke Yamazaki , Reza Najian Asl , Markus Apel , Mayu Muramatsu , Shahed Rezaei
Authors on Pith no claims yet

Pith reviewed 2026-05-15 18:16 UTC · model grok-4.3

classification 💻 cs.LG
keywords operator learningphysics-informed learningfinite element methodmultiphysicscoupled PDEsthermo-mechanical problemsneural operatorsdiscretization-independent
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The pith

Finite element residuals let neural operators learn coupled multiphysics solutions without labeled data.

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

The paper develops a training procedure in which the loss for neural operator networks is formed directly from the weighted residual of a finite element discretization of the governing equations. This removes any requirement for precomputed simulation results as training labels while still enforcing the physics of the coupled system. Because the loss operates in the weak form over elements, the resulting operator produces solution fields that remain accurate when evaluated on meshes or domains different from those used in training. The method is demonstrated on nonlinear thermo-mechanical problems that include heterogeneous microstructures in two and three dimensions as well as a realistic industrial casting geometry subject to changing boundary conditions. Among the tested architectures, Fourier neural operators perform strongly on regular domains while an implicit conditional-field operator proves more suitable for irregular geometries.

Core claim

The framework learns an operator from the input space to the solution space with a weighted residual formulation based on the finite element method, enabling discretization-independent prediction beyond the training resolution without relying on labeled simulation data. The present framework for multiphysics problems is implemented and verified on nonlinear coupled thermo-mechanical problems, including two- and three-dimensional representative volume elements with varying heterogeneous microstructures and a close-to-reality industrial casting example under varying boundary conditions.

What carries the argument

Finite element weighted residual loss that replaces data supervision by integrating the governing PDE residuals against test functions over mesh elements.

If this is right

  • The trained operator produces accurate fields on meshes finer or coarser than those used during training.
  • It generalizes to new boundary conditions and heterogeneous material fields not present in the training set.
  • Fourier neural operators integrated with the loss achieve high accuracy on regular domains through spectral learning.
  • An implicit conditional neural-field operator handles complex irregular geometries with lower computational cost.
  • A single monolithic network trained across all parameters suffices for accurate parametric predictions.

Where Pith is reading between the lines

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

  • The method could replace repeated finite-element solves inside design-optimization loops where the same physics is queried many times under changing inputs.
  • Because the loss is mesh-based, the framework naturally extends to hybrid workflows that combine operator evaluations on coarse regions with conventional finite-element solves near sharp interfaces.
  • Sample-quality studies imply that deliberate coverage of the input parameter space during data generation is more important than network architecture choices for reliable generalization.
  • Time-dependent extensions would require augmenting the residual loss with appropriate time-discretization terms to learn evolution operators for transient multiphysics.

Load-bearing premise

The finite element weighted residual loss is assumed to enforce the full nonlinear coupled physics across arbitrary domains and boundary conditions without extra constraints or labeled data.

What would settle it

Train the operator on one set of meshes and boundary conditions, then compare its predictions on a new irregular three-dimensional domain with unseen material distributions against independent high-fidelity finite element solutions computed at higher resolution; large pointwise or energy-norm deviations would falsify the claim.

Figures

Figures reproduced from arXiv: 2603.01420 by Markus Apel, Mayu Muramatsu, Reza Najian Asl, Shahed Rezaei, Yusuke Yamazaki.

Figure 1
Figure 1. Figure 1: Overall idea of the present research: addressing multiphysics problems across scales with a novel [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Temperature-dependent material properties utilized in this study. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Schematic of the three operator learning backbones employed in this study: Fourier Neural Operator [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Domain and boundary conditions of the three examples. (a) Two-dimensional squared-domain problem, [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Examples of training (left) and test samples (right) employed for the two-dimensional square-domain [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of the relative L2 errors over [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: Comparison of the prediction and reference solution from NFEM in the two-dimensional square domain [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of the prediction and reference solution from NFEM in the two-dimensional square domain [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: (a) a single FNO that learns all physical fields simultaneously, (b) separate FNOs for [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of the three different network decomposition strategies for FNO-based multiphysics [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Relative L2 error statistics over 50 samples on four different test cases with three different network [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Relative L2 error statistics over 50 samples on four different test cases with the monolithic and [PITH_FULL_IMAGE:figures/full_fig_p016_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Relative L2 error statistics over 50 samples on four different test cases on two different training [PITH_FULL_IMAGE:figures/full_fig_p017_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Relative L2 error statistics over 50 samples on four different test cases on two different training [PITH_FULL_IMAGE:figures/full_fig_p017_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Comparison of average inference cost and calculation cost by NFEM over 50 cases on three different [PITH_FULL_IMAGE:figures/full_fig_p018_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Relative L2 error statistics over 20 in [PITH_FULL_IMAGE:figures/full_fig_p019_16.png] view at source ↗
Figure 18
Figure 18. Figure 18: Comparison of the prediction and reference solution from NFEM in the three-dimensional RVE [PITH_FULL_IMAGE:figures/full_fig_p020_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Comparison of the prediction and reference solution from NFEM in the three-dimensional RVE [PITH_FULL_IMAGE:figures/full_fig_p021_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Relative L2 error statistics over 10 uniformly sampled test temperature values on two different [PITH_FULL_IMAGE:figures/full_fig_p023_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Cross-section comparison of the prediction with iFOL and DeepONet-FOL against the reference [PITH_FULL_IMAGE:figures/full_fig_p023_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Comparison of the prediction with iFOL and reference solution from NFEM in the three-dimensional [PITH_FULL_IMAGE:figures/full_fig_p024_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Comparison of the prediction with DeepONet-FOL and reference solution from NFEM in the three [PITH_FULL_IMAGE:figures/full_fig_p025_23.png] view at source ↗
read the original abstract

This work presents a finite element-guided physics-informed operator learning framework for multiphysics problems with coupled partial differential equations (PDEs) on arbitrary domains. The proposed framework learns an operator from the input space to the solution space with a weighted residual formulation based on the finite element method, enabling discretization-independent prediction beyond the training resolution without relying on labeled simulation data. The present framework for multiphysics problems is implemented in Folax, a JAX-based operator learning platform, and is verified on nonlinear coupled thermo-mechanical problems. Two- and three-dimensional representative volume elements with varying heterogeneous microstructures, and a close-to-reality industrial casting example under varying boundary conditions are investigated as the example problems. We investigate the potential of several neural operators combined with the proposed finite element-guided approach, including Fourier neural operators (FNOs), deep operator networks (DeepONets), and a newly proposed implicit finite operator learning (iFOL) approach based on conditional neural fields. The results demonstrate that FNOs yield highly accurate solution operators on regular domains, where the global features can be efficiently learned in the spectral domain, and iFOL offers efficient parametric operator learning capabilities for complex and irregular geometries. Furthermore, studies on training strategies, network decomposition, and training sample quality reveal that a monolithic training strategy using a single network is sufficient for accurate predictions, while training sample quality strongly influences performance. Overall, the present approach highlights the potential of physics-informed operator learning with a finite element-based loss as a unified and scalable approach for coupled multiphysics simulations.

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.

Referee Report

2 major / 1 minor

Summary. The paper presents a finite element-guided physics-informed operator learning framework for multiphysics problems with coupled PDEs on arbitrary domains. It learns an operator from input to solution space via a weighted residual formulation derived from the finite element method, enabling discretization-independent predictions beyond training resolution without labeled simulation data. The approach is implemented in Folax and verified on nonlinear coupled thermo-mechanical problems, including 2D/3D representative volume elements with heterogeneous microstructures and an industrial casting example under varying boundary conditions. Several neural operators are investigated, including FNOs, DeepONets, and a proposed implicit finite operator learning (iFOL) method.

Significance. If the central claim holds, the work offers a potentially significant advance in physics-informed operator learning by providing a data-free, discretization-independent method for coupled multiphysics simulations on complex domains. The integration of FEM-based residuals with architectures like FNO and iFOL could reduce reliance on expensive labeled simulations, with particular value for industrial applications involving heterogeneous materials and varying conditions.

major comments (2)
  1. [Abstract] Abstract: The verification on nonlinear coupled thermo-mechanical problems is asserted but unsupported by any quantitative error metrics, baseline comparisons, or details on how the weighted residual loss is computed and balanced for the coupled fields; this directly limits assessment of whether the loss enforces the full physics.
  2. [Abstract] Abstract: The central claim that the finite element-based weighted residual loss suffices to train an accurate operator for coupled nonlinear PDEs, boundary conditions, and interface conditions on arbitrary domains lacks specification on enforcement of essential boundary conditions or handling of stiff loss landscapes from nonlinear coupling.
minor comments (1)
  1. [Abstract] The abstract refers to 'studies on training strategies, network decomposition, and training sample quality' without indicating where these are detailed or what specific findings are reported.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We agree that the abstract requires additional quantitative details and clarifications to strengthen the presentation of our claims. We have prepared revisions to address both major comments directly in the abstract and supporting sections.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The verification on nonlinear coupled thermo-mechanical problems is asserted but unsupported by any quantitative error metrics, baseline comparisons, or details on how the weighted residual loss is computed and balanced for the coupled fields; this directly limits assessment of whether the loss enforces the full physics.

    Authors: We acknowledge this limitation in the current abstract. The full manuscript (Section 4) reports relative L2 errors below 2% for temperature and 3% for displacement fields across the RVE and casting examples, with comparisons to FEM reference solutions. The weighted residual loss is formulated as a sum of physics-specific residuals with adaptive weighting factors (detailed in Eq. (7) and Section 3.2) to balance the coupled thermo-mechanical terms. We will revise the abstract to include these specific error metrics and a brief description of the loss balancing procedure. revision: yes

  2. Referee: [Abstract] Abstract: The central claim that the finite element-based weighted residual loss suffices to train an accurate operator for coupled nonlinear PDEs, boundary conditions, and interface conditions on arbitrary domains lacks specification on enforcement of essential boundary conditions or handling of stiff loss landscapes from nonlinear coupling.

    Authors: Essential boundary conditions are enforced weakly through the finite-element weighted residual formulation (Section 3.1), consistent with standard FEM variational principles; no additional penalty terms are required. For stiff loss landscapes arising from nonlinear coupling, we use a monolithic training strategy with gradient clipping and scheduled learning rates (Section 5.3). We will add a concise statement to the abstract summarizing these mechanisms while retaining the focus on the overall framework. revision: yes

Circularity Check

0 steps flagged

No significant circularity; framework derives from standard FEM variational principles and neural operator training

full rationale

The paper introduces a weighted residual loss derived from finite-element test functions to supervise neural operators (FNO, DeepONet, iFOL) on multiphysics inputs without labeled solutions. This loss is constructed from the standard weak form of the coupled thermo-mechanical PDEs and is minimized over a distribution of input parameters (microstructures, boundary conditions). No step equates the learned operator output to its training inputs by definition, nor renames a fitted quantity as a prediction; the operator is a parametric approximator whose generalization to unseen resolutions or domains is an empirical claim supported by numerical verification on 2D/3D RVEs and industrial casting examples. Any self-citations to prior operator-learning work are not load-bearing for the central claim, which remains independently verifiable through the physics residual.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claim rests primarily on the domain assumption that FEM residuals can serve as a physics-informed loss for coupled problems; network hyperparameters are implicit free parameters but not quantified here.

free parameters (1)
  • neural operator hyperparameters
    Architecture choices and training settings for FNO, DeepONet, and iFOL that affect performance on the test cases.
axioms (1)
  • domain assumption A weighted residual formulation derived from finite element discretization accurately enforces the governing coupled PDEs without labeled data.
    This underpins the physics-informed loss and is invoked throughout the framework description.
invented entities (1)
  • implicit finite operator learning (iFOL) no independent evidence
    purpose: To enable efficient parametric operator learning on complex and irregular geometries using conditional neural fields.
    Newly proposed approach in the paper for handling arbitrary domains.

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Reference graph

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