cs.LGApr 9, 2026

Towards Rapid Constitutive Model Discovery from Multi-Modal Data: Physics Augmented Finite Element Model Updating (paFEMU)

Authors: Jingye Tan, Govinda Anantha Padmanabha, Steven J. Yang, Nikolaos Bouklas

Organizations: University of Southern California · Cornell University · Ecole Polytechnique Federale de Lausanne (EPFL) · Pasteur Labs

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Abstract

Recent progress in AI-enabled constitutive modeling has concentrated on moving from a purely data-driven paradigm to the enforcement of physical constraints and mechanistic principles, a concept referred to as physics augmentation. Classical phenomenological approaches rely on selecting a pre-defined model and calibrating its parameters, while machine learning methods often focus on discovery of the model itself. Sparse regression approaches lie in between, where large libraries of pre-defined models are probed during calibration. Sparsification in the aforementioned paradigm, but also in the context of neural network architecture, has been shown to enable interpretability, uncertainty quantification, but also heterogeneous software integration due to the low-dimensional nature of the resulting models. Most works in AI-enabled constitutive modeling have also focused on data from a single source, but in reality, materials modeling workflows can contain data from many different sources (multi-modal data), and also from testing other materials within the same materials class (multi-fidelity data). In this work, we introduce physics augmented finite element model updating (paFEMU), as a transfer learning approach that combines AI-enabled constitutive modeling, sparsification for interpretable model discovery, and finite element-based adjoint optimization utilizing multi-modal data. This is achieved by combining simple mechanical testing data, potentially from a distinct material, with digital image correlation-type full-field data acquisition to ultimately enable rapid constitutive modeling discovery. The simplicity of the sparse representation enables easy integration of neural constitutive models in existing finite element workflows, and also enables low-dimensional updating during transfer learning.

Figures & tables

Figure 1: Outline of the paFEMU framework and corresponding multi-modal transfer learning scheme
 FNN [ 1 ]ICNN [ 45 ]MNN [ 49 ]PCNN
fl(⋅)f_{l}(\cdot)UnconstrainedConvex, nondecreasingNondecreasingConvex, nondecreasing
Wl\bm{W}_{l}Unconstrained≥0\geq\bm{0}≥0\geq\bm{0}≥0\geq\bm{0}
Wl(I1,I2)\bm{\mathcal{W}}_{l}^{(I_{1},I_{2})}UnconstrainedUnconstrained≥0\geq\bm{0}≥0\geq\bm{0}
Wl(J)\bm{\mathcal{W}}_{l}^{(J)}UnconstrainedUnconstrainedUnconstrainedUnconstrained
bl\bm{b}_{l}UnconstrainedUnconstrainedUnconstrainedUnconstrained
Polyconvex guaranteeNot sufficientNecessary but not sufficientNecessary but not sufficientSufficient but not necessary
Table 1: Comparison of feed-forward neural (FNN) variants in the context of constitutive model discovery using invariant-based formulations.
Figure 2: Illustration of the proposed physics-augmented neural network framework integrating an input convex neural network (ICNN) architecture with physics-based augmentation layers.
Figure 3: The invariant space showing the convex hull and the selected invariant triplets used for synthetic data generation.
Figure 4: Sampled principal stress points as functions of invariants for synthetic data generated using the Gent model.
Figure 5: Sampled training data and validation paths in stress–deformation space for constrained uniaxial, biaxial, and simple shear tests.
ParameterValueParameterValueParameterValue
c10c_{10}1.302c20c_{20}0.261c30c_{30}0.246
c01c_{01}0.668c02c_{02}0.245c03c_{03}0.143
κ\kappa0.831  
Table 2: Parameters for the generalized Ogden model
Figure 6: Discretized 2D domain of the digital image correlation speciment, displacement is fixed at y=0y=0 , a prescribed positive incremental displacement is applied uniformly at y=5y=5 . The lateral sides and the internal holes are not constrained in any way.
Set 1: θipolyc\theta^{\mathrm{polyc}}_{i}Set 2: θjrIC\theta^{\mathrm{rIC}}_{j}Set 3: θkuNN\theta^{\mathrm{uNN}}_{k}
ParamValueParamValueParamValue
θ1polyc\theta^{\mathrm{polyc}}_{1}0.999θ1rIC\theta^{\mathrm{rIC}}_{1}0.905θ1uNN\theta^{\mathrm{uNN}}_{1}0.782
θ2polyc\theta^{\mathrm{polyc}}_{2}5.227θ2rIC\theta^{\mathrm{rIC}}_{2}0.506θ2uNN\theta^{\mathrm{uNN}}_{2}0.695
θ3polyc\theta^{\mathrm{polyc}}_{3}-0.696θ3rIC\theta^{\mathrm{rIC}}_{3}1.674θ3uNN\theta^{\mathrm{uNN}}_{3}3.124
θ4polyc\theta^{\mathrm{polyc}}_{4}0.149θ4rIC\theta^{\mathrm{rIC}}_{4}-0.211θ4uNN\theta^{\mathrm{uNN}}_{4}-0.192
θ5polyc\theta^{\mathrm{polyc}}_{5}1.577θ5rIC\theta^{\mathrm{rIC}}_{5}1.429θ5uNN\theta^{\mathrm{uNN}}_{5}1.520
Table 3: Pre-trained model parameter sets (rounded to three decimals, see appendix for full precision). Here, polyc is polyconvex NN, rIC is relaxed ICNN, and uNN is unconstrained NN.
Figure 7: Evaluation of the polyconvexity indicator inequalities over training data. Positive values indicate satisfaction of the indicator inequalities.
Figure 8: Generalization performance under constrained uniaxial, equibiaxial, and simple shear tests. The top row shows the polyconvex model, the middle row the relaxed model, and the bottom row the unconstrained NN model. Solid lines denote the ground-truth stresses, while dashed lines represent model predictions. Green dotted lines indicate the limits of the training regime used during learning. Here, polyc is polyconvex NN, rIC is relaxed ICNN, and uNN is unconstrained NN.
Figure 9: Validation of the learned strain energy functions along canonical loading paths over an extended range. The polyconvex, relaxed, and unconstrained NN models are shown in the top, middle, and bottom rows, respectively. Solid curves denote the ground-truth potential energy, while dashed curves represent the model predictions. Here, polyc is polyconvex NN, rIC is relaxed ICNN, and uNN is unconstrained NN.
Figure 10: Uniaxial test for examining performance of the models in a nonlinear solution scheme. Top row: polyconvex; middle row: relaxed; bottom row: unconstrained NN models. Here, polyc is polyconvex NN, rIC is relaxed ICNN, and uNN is unconstrained NN.
Figure 11: Comparison of transfer learning targets to the pretrained models as well as the pretrain truth at the strain points of which DIC data is generated through FEM. The markers on the Ogden and Neo-Hookean targets indicate where the data points collected for transfer learning. At each data collection point, the scalar-valued reaction force and the full-field displacement are collected for the Ogden and Neo-Hookean targets. For references, the displacement contours of the targets at the last collection point, as well as that of the Gent truth, are shown here. Here, PCNN is polyconvex NN, rICNN is relaxed ICNN, and uNN is unconstrained NN.
Figure 12: Loss components for transfer learning scheme, utilizing the Neo-Hookean synthetic DIC dataset
Figure 13: Transfer learning to Neo-Hookean DIC dataset: validation of learned models to the analytical Neo-Hookean model in a uniaxial tension setting via Newton-Raphson.
Figure 14: Polyconvexity indicator inequality evaluations, for polyconvex NN (left), reduced ICNN (middle) and unconstrained NN (right) models with the Neo-Hookean target.
Figure 15: Transfer to Neo-Hookean Target, displacement error at 50%50\% applied strain
Figure 16: Loss components for transfer learning scheme, utilizing the Ogden synthetic DIC dataset
Figure 17: Transfer learning to Generalized Ogden DIC dataset: Validation of learned models to the analytical generalized Ogden model in a uniaxial tension setting via Newton-Raphson.
Figure 18: Polyconvexity indicator inequality evaluations, for polyconvex NN (left), reduced ICNN (middle), and unconstrained NN (right) models with the generalized Ogden target.
Figure 19: Transfer to Generalized Ogden, displacement error at 50%50\% applied strain
Figure 20: Stress error in the deployment of the transferred model in 3D torsion simulation with a maximum twist angle of 458∘458^{\circ} compared to ground truth. The overall relative error at this state is 8.6%8.6\%
Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Polyconvex NNRelaxed ICNNUnconstrained NN
ParamValueParamValueParamValue
θ1\theta_{1}0.998645544052124θ1\theta_{1}0.905308246612549θ1\theta_{1}0.782256841659546
θ2\theta_{2}5.22694253921509θ2\theta_{2}0.506476998329163θ2\theta_{2}0.694582641124725
θ3\theta_{3}-0.695928037166595θ3\theta_{3}1.67390620708466θ3\theta_{3}3.12386536598206
θ4\theta_{4}0.149173066020012θ4\theta_{4}-0.210611954331398θ4\theta_{4}-0.192448511719704
θ5\theta_{5}1.57681846618652θ5\theta_{5}1.42899298667908θ5\theta_{5}1.51982772350311
Appendix
Table 4: Pretrained model parameter sets (full precision).
Polyconvex NNRelaxed ICNNUnconstrained NN
ParamValueParamValueParamValue
θ1\theta_{1}0.8049869853289970401θ1\theta_{1}0.7789940295960968708θ1\theta_{1}0.6182716140982345010
θ2\theta_{2}5.195067564321128373θ2\theta_{2}0.3324961120417712079θ2\theta_{2}0.4267721052272835380
θ3\theta_{3}-0.2838953915400435069θ3\theta_{3}1.661755726450315995θ3\theta_{3}3.140349947502798944
θ4\theta_{4}0.1072488259756032430θ4\theta_{4}-0.003102093041861720031θ4\theta_{4}-0.006083654397070120678
θ5\theta_{5}1.571875317538424355θ5\theta_{5}1.329829265689388640θ5\theta_{5}1.388819097573269490
Appendix
Table 5: Calibrated model parameter sets for model transfer to NeoHookean material (full precision).
Polyconvex NNRelaxed ICNNUnconstrained NN
ParamValueParamValueParamValue
θ1\theta_{1}1.157275360655964258e+00θ1\theta_{1}5.645962189730805436e-01θ1\theta_{1}7.928536877065486266e-01
θ2\theta_{2}5.028109292028598354e+00θ2\theta_{2}1.982058889749746644e+00θ2\theta_{2}1.564181103306155896e+00
θ3\theta_{3}-8.059457928952151740e-01θ3\theta_{3}4.711667943240553935e-01θ3\theta_{3}2.906866912983426587e+00
θ4\theta_{4}3.782093258970941063e-01θ4\theta_{4}7.098786078326054794e-01θ4\theta_{4}7.722878520017870119e-02
θ5\theta_{5}1.579239498975667289e+00θ5\theta_{5}3.000210769605224481e+00θ5\theta_{5}2.988771051719846916e+00
Appendix
Table 6: Calibrated model parameter sets for model transfer to Ogden material (full precision).
Figure A.1: Losses and R2 score for Polyconvex PCNN model
Figure A.2: Losses and R2 score for Relaxed ICNN model
Figure A.3: Losses and R2 score for Unconstrained NN model
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