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Data-Free Weak-Form Staggered Neural Operators for Magneto-Mechanical Coupling in Finite-Strain Elastomers

Alireza Yazdandousthamedani, Ahmad Moeineddin, Reza Najian Asl, Shahed Rezaei, Michael Kaliske

arXiv:2610.03156v1cs.CE

TL;DR

Finite-strain magneto-mechanical problems in heterogeneous elastomers are costly to solve repeatedly, and strongly coupled operator-learning formulations remain limited. The paper develops a data-free weak-form staggered neural operator that alternately trains magnetic and mechanical fields while retaining their constitutive coupling. It reports accurate predictions across parameterized material and geometry settings, with a neural-initialized Newton correction for cases where direct out-of-distribution mechanical predictions are poor.

  • Problem

    Repeated nonlinear magneto-mechanical simulations are computationally expensive, while physics-informed operator learning for strongly coupled finite-strain problems remains comparatively underdeveloped.

    Source: §1 · p. 2

  • Method

    The WSNO trains separate magnetic and mechanical operators alternately using finite-element weak-form residuals, without labeled solution fields, while retaining constitutive coupling.

    Source: §1 · p. 5, §5 · p. 39

  • Results

    For ten unseen hollow-cylinder geometries, mean relative L2 errors were approximately 0.11% for magnetic scalar potential and 1.36% for the three-dimensional displacement field.

    Source: §5 · p. 40

  • Takeaways & Limitations

    The results support weak-form operator learning for predicting coupled magneto-mechanical responses across heterogeneous material distributions and geometries, with neural-initialized Newton correction when direct out-of-distribution predictions are insufficient.

    Source: §5 · p. 40

    For test morphologies substantially different from training, direct mechanical predictions generalized poorly, although neural-initialized Newton correction converged closely to nonlinear finite-element references.

    Source: §5 · p. 39

Abstract

from arXiv · show

Magneto-active elastomers, as a class of smart materials, exhibit strongly coupled magnetic and mechanical behavior at finite strains. Considering variations in microstructure, material properties, and geometry can lead to computationally expensive analyses. Building on the finite operator learning (FOL) framework, this study develops a data-free, physics-informed operator-learning framework for families of coupled finite-strain magneto-mechanical boundary-value problems. The governing magnetostatic and mechanical equations are enforced through finite-element weak-form residuals, allowing the neural operators to be trained without labeled finite-element solution data. The main contribution of this study is the development of a weak-form staggered neural operator (WSNO) framework for strongly coupled magneto-mechanical saddle-point problems. Separate magnetic and mechanical neural operators are trained alternately using a staggered optimization strategy, while their physical coupling is retained through the constitutive relations and residual evaluations. The resulting framework learns mappings from parameterized material and geometric descriptions to the corresponding coupled magnetic and mechanical solution fields. The proposed framework is investigated across several settings, including heterogeneous random-inclusion microstructures, varying magnetic phase contrast, area-fraction-dependent geometries, strongly out-of-distribution material morphologies, and three-dimensional geometry-parametric problems. In addition, the learned operator is combined with a neural-initialized Newton strategy, in which the nonlinear finite-element solver is initialized using the neural prediction. The results demonstrate that the proposed operator-learning framework can accurately capture coupled magneto-mechanical responses across a broad range of parametric problem settings.

1. Introduction

Magneto-active elastomers pose computationally demanding, mutually coupled magnetic and mechanical problems, especially when material properties are heterogeneous. This study addresses the limited development of operator learning for these finite-strain problems with a data-free, weak-form staggered framework.

1. Introduction

  • Heterogeneous magneto-active elastomers require solving mutually dependent magnetic and mechanical fields, making repeated simulations of nonlinear boundary-value problems computationally expensive.Large contrasts between magnetic particles and the elastomer produce spatially varying properties and localized field gradients.Source: §1 · p. 2
  • Conventional supervised models depend on large representative datasets and do not inherently satisfy governing physical laws, while conventional PINNs face training, weighting, and sharp-gradient challenges.PINNs may also require retraining when geometry, materials, loading, or boundary conditions change.Source: §1 · p. 3
  • Prior physics-informed operator approaches enforce physics through strong-form residuals or variational and finite-discretization formulations, but fully coupled finite-strain magneto-mechanical problems remain less developed.Earlier examples include electromagnetic-only or weakly coupled formulations, as well as methods that use some labeled FEM data.Source: §1 · pp. 3–4
  • Strongly coupled magneto-mechanical saddle-point systems can make monolithic optimization difficult because field residuals have different scales and sensitivities.A coupled formulation must preserve interaction between magnetic and mechanical fields while reducing interference between their optimization pathways.Source: §1 · pp. 4–5
  • The proposed data-free WSNO enforces magnetostatic and mechanical equations with finite-element weak-form residuals and alternately trains separate magnetic and mechanical operators.The complementary field remains in constitutive relations and residual evaluation but is excluded from the active operator’s backpropagation pathway.Source: §1 · p. 5

2. Coupled partial differential equations and loss formulation

The formulation represents finite-strain magneto-mechanical behavior with displacement and magnetic scalar potential as primary unknowns. It derives coupled constitutive responses from an energy function and constructs finite-element weak-form residuals for training.

2.1. Finite-strain magneto-mechanical problem

  • The governing problem uses displacement and magnetic scalar potential as primary unknowns, with magnetostatic equations and mechanical equilibrium posed in the reference configuration.Mechanical body forces are neglected in the numerical examples.Source: §2 · p. 5, §2.1 · pp. 5–6
  • The magnetic scalar potential represents the magnetic field, with boundary conditions divided into Dirichlet and Neumann parts.The magnetostatic equations include a curl-free field condition and a divergence condition on magnetic induction.Source: §2.1 · pp. 5–6

2.2. Magneto-elastic constitutive response

  • The constitutive response derives mechanical stress and magnetic quantities from a shared magneto-elastic energy, coupling deformation and magnetic field.Changes in magnetic field modify mechanical response, while deformation alters magnetic constitutive response.Source: §2.2 · pp. 6–7
  • The deformation-dependent magnetic formulation distinguishes a purely magnetic invariant from a coupled magneto-mechanical invariant, governed by separate material parameters.The isotropic linear magnetic response also uses the local relative permeability.Source: §2.2 · p. 7
  • Heterogeneous material behavior is parameterized by spatially varying Young’s modulus and relative permeability, mapping material-property fields to coupled solution fields.The energy includes elastic and coupling contributions, with the studied response using linear magnetization and deformation-dependent coupling.Source: §2.2 · pp. 6–7

2.3. Weak-form residual formulation

  • The governing strong forms are recast as mechanical and magnetic weak forms using admissible displacement and potential test functions.The mechanical form includes prescribed tractions and body-force density; the magnetic form includes prescribed normal magnetic flux.Source: §2.3 · pp. 7–8
  • Finite-element interpolation and gradient matrices produce element-level field quantities, whose mechanical and magnetic residuals are assembled into global vectors for physics-informed losses.Shape-function matrices interpolate displacement and magnetic scalar potential, while corresponding gradient matrices act with respect to reference coordinates.Source: §2.3 · p. 8

3. Weak-form staggered neural operator framework

WSNO learns coupled magnetic and mechanical solution fields across parameterized magneto-mechanical problems using field-specific neural operators and finite-element weak-form residuals. Alternating optimization keeps the fields physically coupled while avoiding simultaneous updates.

3. Weak-form staggered neural operator framework

  • WSNO learns a family of coupled magnetic and mechanical solution maps from parameterized material and geometry descriptions using two field-specific neural operators.The magnetic operator predicts the scalar potential; the mechanical operator predicts displacement.Source: §3 · pp. 8–9
  • The framework is architecture-independent, allowing different neural-operator models to represent the magnetic and mechanical solution maps.FNO, DeepONets, and iFOL are listed as possible architectures.Source: §3 · p. 9

3.1. Field-specific neural operators

  • The two field-specific operators remain physically coupled through constitutive relations and residual evaluations, rather than representing independent problems.Both magnetic and mechanical residuals depend on the coupled solution fields.Source: §3.1 · p. 10

3.2. Weak-form physics-informed loss

  • Finite-element weak-form residuals train the operators without reference finite-element solution fields or spatial automatic differentiation.Predicted nodal fields are evaluated using finite-element interpolation, gradients, quadrature, and assembly.Source: §3.2 · p. 11

3.3. Staggered training of the coupled operators

  • Because simultaneous optimization exhibits poor convergence, WSNO alternates operator updates while holding the complementary field fixed for each stage.The fixed field still contributes to constitutive equations and residual evaluation.Source: §3.3 · p. 11, §3.3.3 · p. 13

3.3.3. Staggered training sequence

  • Training begins with magnetic pretraining in the undeformed configuration, then alternates mechanical and magnetic predictions in successive cycles.The staggered procedure is reported to provide more stable training than direct monolithic optimization.Source: §3.3 · p. 12, §3.3.3 · p. 13

3.3.1. Mechanical training stage

  • During mechanical training, the current magnetic-potential prediction is held fixed and only the mechanical loss updates mechanical-network parameters.The magnetic prediction still informs the magnetic field and coupled stress response.Source: §3.3.1 · p. 12

3.3.2. Magnetic training stage

  • During magnetic training, the updated displacement is held fixed while only the magnetic loss is minimized, preserving deformation-dependent constitutive coupling.The displacement still enters deformation-gradient, constitutive-response, induction, and magnetic-residual evaluations.Source: §3.3.2 · p. 12

4. Results

The results section describes benchmark studies of WSNO across heterogeneous microstructures, material distributions, and geometry. Independently computed nonlinear finite-element solutions serve as validation references, not training data.

4. Results

  • The studies assess prediction accuracy, generalization, and robustness across random-inclusion microstructures, area-fraction-dependent responses, out-of-distribution configurations, and three-dimensional geometry-parametric problems.The out-of-distribution study also combines neural predictions with Newton correction.Source: §4 · p. 13
  • Neural-operator predictions are compared with independently computed nonlinear finite-element solutions used only for validation, not training.The described two-dimensional heterogeneous examples and three-dimensional geometry-parametric problem use FNO-based operators.Source: §4 · p. 13
  • The condensed training procedure includes magnetic pretraining, staggered mechanical and magnetic stages, and optional coupled fine-tuning.The supplied algorithm fragments identify these stages but do not report their benchmark outcomes.Source: §4 · p. 14

4.1. Coupled magneto-mechanical prediction for random inclusion microstructures

The WSNO predicts coupled magnetic and mechanical fields for heterogeneous random-inclusion microstructures, using staggered training to address the coupled problem’s saddle-point structure. Its predictions capture the overall field distributions, with localized magnetic-field discrepancies associated with gradient-based recovery.

4.1. Coupled magneto-mechanical prediction for random inclusion microstructures

  • The example uses 15 circular inclusions with smooth indicators to represent the heterogeneous layout and interpolate matrix and inclusion properties.The model predicts potential and displacement directly; the magnetic field is recovered from the potential gradient.Source: §4.1 · pp. 15, 17
  • Staggered training alternates magnetic and mechanical optimization because monolithic minimization can be unstable when coupled residuals differ in magnitude and sensitivity.The example trains on 50 randomly generated microstructures and tests on 10 unseen ones.Source: §4.1 · p. 16, Figure 3 · p. 17
  • The WSNO captures the overall coupled field distributions in unseen microstructures, with strong agreement for potential and displacement and localized magnetic-field discrepancies.The magnetic-field discrepancies are consistent with recovering the field from the predicted potential’s gradient.Source: §4.1 · pp. 17, 19, Figure 4 · p. 18
  • Across 10 in-distribution test samples, potential, magnetic-field, and horizontal-displacement errors are evaluated, with magnetic-field error expected to exceed potential error because of gradient recovery.The cited figure compares relative L2 errors for these three outputs.Source: §4.1 · pp. 17–18, Figure 5 · p. 19

4.2. Generalization across magnetic phase contrast

This study varies inclusion permeability alongside random-inclusion geometry to assess WSNO generalization across magnetic phase contrasts. The operator retains predictive capability at unseen intermediate permeability values, and increasing the training set improves magnetic-field and displacement accuracy.

4.2. Generalization across magnetic phase contrast

  • Unlike the fixed-contrast study, this evaluation varies inclusion relative permeability as well as microstructure, while keeping the geometry, discretization, boundary conditions, and mechanical properties unchanged.The matrix relative permeability remains fixed at 1.Source: §4.2 · p. 19
  • Increasing training from 50 to 100 microstructures improves magnetic-field and horizontal-displacement accuracy across the tested permeability contrasts, while potential-error reduction is modest.Potential is already predicted with high accuracy using 50 training samples.Source: §4.2 · p. 20, Figure 6 · p. 20
  • The WSNO retains predictive capability for unseen microstructures at intermediate inclusion permeabilities not explicitly included during training.The evaluated permeability values 4 and 8 are interpolation cases within the training interval; permeability 5 is represented during training.Source: §4.2 · p. 20

4.3. Area-fraction-dependent magneto-mechanical response of a single inclusion

The single-inclusion study tests extrapolation to larger area fractions within the same geometric family. Regional magnetic errors remain below 10% for both magnetic field and induction in the inclusion and matrix, while global errors increase with inclusion size.

4.3. Area-fraction-dependent magneto-mechanical response of a single inclusion

  • The test set uses larger area fractions than training but retains a single centered circular inclusion, so it tests extrapolation within the same geometry family rather than a new topology.Training covers approximately fA ∈ [0.10, 0.249], while testing covers approximately fA ∈ [0.253, 0.30].Source: §4.3 · p. 22, Figure 7 · p. 22
  • Global errors increase with area fraction; potential has the lowest error, while magnetic field and induction errors are larger and more sensitive near the interface.The magnetic field is recovered from the potential gradient, and induction also depends on heterogeneous permeability.Source: §4.3 · pp. 22–23, Figure 8 · p. 23
  • Regional magnetic-field and induction errors remain below 10% in both inclusion and matrix regions for the larger-area-fraction test samples.The model was trained on 45 smaller-area-fraction samples and tested on 15 samples outside the training interval.Source: Figure 9 · p. 23, §4.3 · pp. 23–24
  • At fA = 0.30, predicted potential and horizontal displacement profiles closely follow the finite-element solution, including slope changes at the material interface.Full-field errors are relatively small and localized, with larger deviations near the interface.Source: §4.3 · p. 24

4.4. Out-of-distribution prediction and neural-initialized Newton correction

The study tests WSNO on circular-inclusion microstructures that differ from its Fourier-generated training morphologies, then uses its predictions to initialize a coupled Newton solve. Across area fractions, the initialized solver converges in far fewer iterations and less time than conventional FE.

4.4. Out-of-distribution prediction and neural-initialized Newton correction

  • For this test, magnetic preconditioning and two staggered training cycles use fewer epochs than earlier studies because the goal is an informative Newton initial state, not a highly accurate direct prediction.Each training stage uses 2,000 epochs, compared with 20,000 epochs per stage in the preceding studies.Source: §4.4 · p. 24

4.4.1. Fourier-generated training distributions

  • The model trains on 1,000 smooth, irregular Fourier-generated material distributions, while circular-inclusion microstructures provide the deliberately different test morphology.A reduced set of Fourier coefficients is mapped to a smooth phase-indicator field used to construct material properties.Source: §4.4.1 · pp. 25–26, Figure 11 · p. 26

4.4.2. Circular out-of-distribution test microstructures

  • Thirty out-of-distribution tests use non-overlapping circular inclusions of fixed radius, with ten independently generated microstructures for each target area fraction.The samples cover the specified area fractions and are formed by randomly positioning inclusions in the square domain.Source: §4.4.2 · pp. 26–27, Figure 12 · p. 28

4.4.3. Neural-initialized Newton correction

  • NiN uses the direct WSNO prediction as the initial state for the coupled nonlinear FE equations in one load increment, unlike the zero-state, 100-increment FE reference.Both procedures use the same nonlinear convergence tolerances and linear-solver settings.Source: §4.4.3 · p. 27, §4.4.4 · p. 28

4.4.4. Newton convergence and computational efficiency

  • Across all investigated area fractions, NiN converges in approximately 4–5 Newton iterations within one load increment and below 3 s, versus approximately 400 iterations over 100 increments and 239–261 s for conventional FE.For representative out-of-distribution microstructures at 10% and 30% area fractions, WSNO corrections are more pronounced for ux than Bx.Source: §4.4.4 · p. 29

4.5. Geometry-parametric prediction of magnetostrictive response of hollow magneto-active cylinders

The study varies hollow-cylinder radius, cavity size, and length while retaining a shared finite-element topology, and trains field-specific operators for the geometry-dependent solution. On unseen geometries, WSNO predicts accurately and transfers to finer discretizations without retraining.

4.5.1. Fixed-topology mapping of the hollow-cylinder geometry

  • The geometry map preserves each node’s circumferential and normalized through-wall positions, keeping node numbering, connectivity, and degree-of-freedom ordering fixed as dimensions vary.Mapped physical coordinates also enter the finite-element Jacobians, gradients, quadrature, and assembled residuals.Source: §4.5.1 · p. 32, Figure 14 · p. 33

4.5.2. Geometry-parametric neural operator

  • Separate Fourier neural operators predict magnetic potential and displacement, with geometry descriptors supplied to the operators and used in the physical-coordinate mapping and weak-form residuals.The mechanical operator also receives the predicted magnetic potential; wall thickness is supplied as an additional descriptor.Source: §4.5.2 · pp. 32–33

4.5.3. Training and test geometries

  • Of 100 independently sampled geometries, 75 are used for physics-informed training, while ten held-out geometries are selected for FE-based evaluation.Training and test samples vary the outer radius, inner radius, and cylinder length.Source: §4.5.3 · p. 33, Figure 15 · p. 34

4.5.4. Prediction accuracy

  • Across ten unseen geometries, mean relative error for the complete three-dimensional displacement field is approximately 0.76%; eight cases are below approximately 1.00%, and sample 76 reaches approximately 1.84%.The magnetic scalar potential is reported as consistently highly accurate over the evaluated parameter space.Source: §4.5.4 · p. 34, Table 2 · p. 35
  • At refined resolution, mean relative L2 errors are 0.60% for the magnetic scalar potential and 2.73% for the three-dimensional displacement field; mean inference time is 0.128 s versus 1387 s for FE.The same trained parameters are used without additional training or fine-tuning.Source: §4.5.4 · pp. 36–37, Table 3 · p. 38, Figure 18 · p. 38

5. Conclusion

The data-free WSNO alternates separate magnetic and mechanical operators trained on finite-element weak-form residuals while retaining constitutive coupling. It predicts coupled fields across heterogeneous and geometry-parametric cases, and neural-initialized Newton correction recovers solutions when direct out-of-distribution predictions are insufficient.

5. Conclusion

  • On unseen heterogeneous two-dimensional microstructures, WSNO reproduced magnetic scalar potential and displacement fields, while magnetic-field component errors were larger because they require spatial differentiation of the predicted potential.The results support capturing coupled responses while preserving magnetic-mechanical interaction.Source: §5 · p. 39
  • In area-fraction extrapolation beyond the training interval, mean relative L2 errors were approximately 0.97% for magnetic potential and 1.68% for the dominant displacement component.Regional magnetic errors remained below 10% in both inclusion and matrix phases.Source: §5 · p. 39
  • For smooth-Fourier-to-circular-inclusion morphology shifts, direct mechanical predictions generalized poorly, but neural-initialized Newton correction converged close to finite-element references across tested area fractions.Magnetic quantities retained moderate predictive accuracy in this setting.Source: §5 · p. 39
  • For ten unseen hollow-cylinder geometries, mean relative L2 errors were approximately 0.11% for magnetic scalar potential and 1.36% for three-dimensional displacement.The geometry varies by outer radius, inner radius, and cylinder length while preserving a common finite-element topology.Source: §5 · pp. 39–40
  • Across the investigated settings, weak-form operator learning predicts coupled magneto-mechanical responses, while staggered optimization improves training robustness and neural-initialized Newton correction recovers high-accuracy solutions when direct extrapolation is insufficient.The framework is intended for families of heterogeneous material distributions and geometries.Source: §5 · p. 40
  • Future work should extend the framework to complex three-dimensional heterogeneous microstructures, varying loads and boundary conditions, and constitutive models with anisotropy, magnetic saturation, or rate dependence.The paper also identifies multiscale and optimization integration as a direction for accelerating repeated nonlinear simulations.Source: §5 · p. 40

Appendix A. Hyperparameters and numerical settings

The appendix collects neural-operator architectures, training hyperparameters, and problem-specific numerical settings. It also records the setup for the out-of-distribution study and neural-initialized Newton correction.

Appendix A. Hyperparameters and numerical settings

  • The numerical examples use an FNO-based WSNO framework, with architectures and training hyperparameters summarized in Table A.4.Problem-specific discretization, material, and parametric settings are listed in Table A.5.Source: Appendix A · p. 41, Table A.4 · p. 41, Table A.5 · p. 42
  • Magnetic learning rates follow a 10^-4 to 10^-5 schedule across the listed numerical examples.Table A.4 summarizes the neural-operator training hyperparameters.Source: Appendix A · p. 41, Table A.4 · p. 41
  • In the phase-contrast study, distinct training microstructures are paired with three inclusion relative-permeability values; 50 and 100 microstructures therefore yield 150 and 300 realizations, respectively.Both cases use the same 10 unseen microstructures for evaluation.Source: Appendix A · p. 41
  • The out-of-distribution study trains on 1000 smooth Fourier-generated material distributions and evaluates on circular-inclusion microstructures at 10%, 20%, and 30% area fractions.Ten realizations are considered per area fraction, and direct WSNO predictions initialize the Newton correction; Table A.6 lists solver settings.Source: Appendix A · pp. 41–42, Table A.6 · p. 43

Appendix B. Normalization of physical quantities

The appendix normalizes spatial, mechanical, energetic, and magnetic quantities using characteristic scales. Once the length and stiffness scales are prescribed, dimensional quantities can be recovered from normalized results.

Appendix B. Normalization of physical quantities

  • The numerical examples are formulated in normalized form so results are independent of a particular physical length and stiffness scale.L0 and E0 denote arbitrary characteristic length and mechanical stiffness scales, respectively.Source: Appendix B · p. 43
  • Magnetic quantities are normalized using a characteristic magnetic-field scale, with the scalar-potential scale related to the field through H = −∇Xϕ.The appendix also defines scales for magnetic induction and free-space permeability relative to the mechanical-energy scale.Source: Appendix B · pp. 43–44
  • The coupling energy and its dimensionless constitutive parameters are expressed in normalized form, with normalized coupling parameters equivalently written using the stated normalization condition.The appendix also notes that the magnetic invariants are quadratic in the magnetic field.Source: Appendix B · p. 44
  • After L0 and E0 are prescribed, dimensional coordinates, displacement, modulus, stress, energy density, and potential are recovered from normalized results.The text states that normalized numerical results can be interpreted for different physical length and stiffness scales without changing the computational formulation.Source: Appendix B · p. 45

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