Calibration of neural viscoelastic models via full-field data

Brain M. Riemer; Markus Kästner; Karl A. Kalina

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Calibration of neural viscoelastic models via full-field data

Brain M. Riemer, Markus Kästner, Karl A. Kalina

arXiv:2609.03645v1Sep 3, 2026cs.CE

TL;DR

The paper addresses calibration of thermodynamically consistent neural viscoelastic models when stress-strain tuples are unavailable from full-field experiments. It proposes an unsupervised GSM-based PANN calibrated from reaction forces and surface displacements, and reports close agreement with reference behavior on synthetic ideal and noisy data. The framework is subject to limitations from incomplete invariant representations and possible NaN failures during optimization.

Problem
Most neural viscoelasticity models are supervised and rely on stress-strain tuples derived from standard experiments, limiting calibration directly from full-field measurements.
Method
The framework calibrates a small-strain viscoelastic PANN using global reaction forces and surface displacements, GSM potentials, invariant representations, EGM constraints, implicit integration, and backward adjoint gradients.
Results
The selected PANNs agree very well with reference stresses and internal variables across the training datasets, while validation reports negligible stress differences despite requiring around 6 to 10 global Newton iterations.
Takeaways & Limitations
The framework demonstrates calibration of a thermodynamically consistent viscoelastic PANN using quantities measurable in real experiments and is presented as transferable to different constitutive settings.
Takeaways & Limitations
The DDP invariants do not form a functional basis because mixed invariants involving both A and ε are missing, and unfavorable parameters can produce NaN values during training.

Abstract

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We propose an unsupervised learning framework for calibrating a physics-augmented neural network (PANN) for small-strain viscoelasticity via full-field data. It only requires quantities that are directly accessible in real experiments for training, namely global reaction forces and surface displacements. The underlying PANN is embedded in the generalized standard materials theory, in which two scalar-valued potentials render the constitutive model thermodynamically consistent by construction, while invariant-based representations of the free energy and the dual dissipation potential additionally ensure material symmetry. Considering a thin specimen under the plane stress assumption, we formulate a constrained optimization problem based on the equilibrium gap method in combination with quasi-Newton optimizers and automatic differentiation. Thereby, the unknown out-of-plane strain follows from the plane stress condition and the evolution of the internal variables is captured by an implicit time integration scheme. The resulting system of nonlinear equations is solved via a local Newton iteration at quadrature point and time step. To drastically reduce the computational cost of training, the backward adjoint method is employed to compute the gradient of the target loss, instead of backpropagating through all Newton iteration steps. The proposed framework is demonstrated for synthetic data, including noisy displacements and forces, showing excellent agreement across a wide range of deformation rates and load paths.

1 Introduction

Neural constitutive models offer flexibility for complex material responses, while full-field calibration methods seek to identify models from experimentally accessible global forces and displacement fields. This work develops a thermodynamically consistent PANN framework for small-strain viscoelasticity calibrated without stress-strain tuples.

Motivation: Classical constitutive models can be too restrictive for highly nonlinear materials, motivating neural-network approaches that flexibly capture complex responses.Neural networks can also automate aspects of constitutive modeling.
Neural constitutive modeling: Physics-augmented neural models embed physical knowledge strongly through architectures or weakly through problem-specific loss functions, with reported improvements in extrapolation capability.The literature uses several related labels, including physics-informed, mechanics-informed, and thermodynamics-based.
Neural constitutive modeling: Neural inelasticity models differ in how they enforce thermodynamic consistency, ranging from loss-based enforcement to GSM or potential-based formulations that guarantee non-negative dissipation by construction.Some elasto-plastic approaches solve return-mapping problems during optimization without prescribing internal variables a priori.
Full-field calibration: Full-field experiments provide spatially resolved displacement fields but do not directly provide stress-strain pairs, creating a need for inverse calibration methods such as FEMU, VFM, and EGM.EGM and VFM use full-field displacements directly to compute deformation measures and stresses, avoiding forward boundary-value simulations but potentially increasing noise sensitivity.
Objectives and contributions: The proposed framework calibrates a small-strain viscoelastic PANN from global reaction forces and surface displacements using GSM potentials, EGM constraints, plane stress, implicit integration, and adjoint gradients.The formulation solves the local nonlinear equations by Newton iteration at each time step and quadrature point.

2 Fundamentals

The fundamentals formulate small-strain continuum mechanics and a GSM-based viscoelastic constitutive model. Thermodynamic consistency is obtained from scalar potentials, while invariant representations impose material symmetry.

2.1 Continuum mechanics: The continuum setting assumes small strains and infinitesimal rigid-body rotations in a three-dimensional reference domain.The displacement field generates the linearized strain tensor.
2.1 Continuum mechanics: The symmetric Cauchy stress is work-conjugate to strain, and its traction on a boundary with negligible volume and inertial loads satisfies the momentum-balance setting.The boundary is partitioned into complementary, disjoint Dirichlet and Neumann parts.
2.2 Viscoelasticity: The model restricts viscoelasticity to small strains with one strain-type internal variable q.The internal variable is a symmetric second-order tensor field.
2.2 Viscoelasticity: Within GSM theory, the Helmholtz free energy yields stress and the driving force conjugate to the internal variable, while a dual dissipation potential governs its evolution.The dual dissipation potential is taken without explicit dependence on the internal variable in this work.
2.2.1 Generalized standard materials: Convexity, normalization, and stationarity properties of the dissipation formulation ensure non-negative dissipation rates D≥0 by construction.The free-energy derivatives are normalized to vanish at ε=0 and q=0.
2.2.2 Material symmetry: The free energy and dual dissipation potential are expressed through scalar invariants that remain unchanged under transformations from the material symmetry group.For isotropic materials, the symmetry group is G=O(3).

3 Physics-augmented neural network model

The PANN represents viscoelastic behavior through invariant-based neural potentials embedded in generalized standard materials theory, enforcing symmetry and thermodynamic structure by construction. Its FICNN architectures impose convexity and monotonicity, while the selected Maxwell-type formulation remains a restricted model class.

Potential-based constitutive structure: Two scalar-valued potentials represent the Helmholtz free energy and dual dissipation potential within the generalized standard materials framework.The potentials are constructed using neural networks and support thermodynamic consistency.
Invariant representations: Isotropic invariants of strain and elastic strain provide symmetry-aware inputs for the equilibrium and non-equilibrium free-energy parts.The listed invariants form a functional basis for symmetric second-order tensors under O(3).
Neural architectures: Component-wise non-decreasing FICNNs with skip connections define the potential networks and enforce convexity in their prescribed inputs.Correction terms make the equilibrium energy and stress contribution vanish in the undeformed state.
Model scope: The constitutive model is of Maxwell type because only the elastic strain p = ε − q enters the non-equilibrium invariants.The authors state that this is not the most general viscoelastic model, even within the considered setting.
Dual dissipation potential: The dual dissipation potential uses convex invariant inputs in A and ε, but its invariant set is not a functional basis because mixed invariants are omitted.This representation therefore does not span the most general invariant dependence of the pair (A, ε).
Model scope: A PICNN could provide greater flexibility for the dual dissipation potential, but the selected FICNN-only model uses fewer trainable variables and was sufficient for the examples considered.Convexity in strain is not strictly required by the GSM framework.

4 Equilibrium gap method

The equilibrium gap method calibrates the PANN from full-field surface displacements and global reaction forces for thin specimens modeled under plane stress. Finite-element discretization, implicit evolution equations, and local Newton solves connect measured data to an equilibrium-based optimization loss.

Method formulation: The EGM identifies PANN parameters unsupervised by minimizing deviations from the discretized weak form of linear momentum balance.Equilibrium is enforced approximately as a soft constraint through loss terms on assembled nodal forces.
Experimental data and assumptions: The method uses dense displacement measurements and global reaction forces that are accessible in experiments, restricting the setup to thin two-dimensional samples because DIC observes surfaces.The region of interest is represented as a two-dimensional domain under a plane-stress assumption.
Finite-element discretization: In-plane displacements are interpolated with continuous piecewise-linear CST elements, yielding element-wise constant strains and one Gauss point per element.The resulting numerical integration is exact for the element-wise constant strain and internal variable fields.
Finite-element discretization: Element stresses determine local nodal forces through numerical integration, which are then assembled globally according to mesh connectivity.The constitutive stress calculation requires the strain tensor and internal variable at each time step.
Local constitutive update: Implicit time integration determines internal variables by solving residual equations, while the plane-stress condition determines the unknown out-of-plane strain.Both residuals are solved monolithically by Newton iteration at each quadrature point.
Optimization objective: Measurement noise in displacements is amplified by spatial differentiation into strains and can compromise the local-equilibrium loss, whereas the force loss enforces equilibrium against global reactions.This distinction motivates the separate treatment of free and force-related loss contributions.

5 Numerical examples

The numerical studies evaluate unsupervised full-field calibration under ideal and noisy data, using constrained neural models and staged optimization. Training includes convexity-preserving architectures, stringent solver tolerances, multiple initializations, and safeguards against overfitting and numerical instability.

5 Numerical examples: The framework is evaluated on several numerical examples using ideal and noisy synthetic data sets.The examples illustrate unsupervised training of viscoelastic PANNs via full-field data.
5 Numerical examples: Each FICNN uses one hidden layer with two neurons and softplus activation, with non-negativity constraints enforcing element-wise non-decreasingness and convexity.Skip connections are included, and constraints are imposed on hidden-layer and skip-connection weights.
5 Numerical examples: Noisy-data calibration can overfit when an overly parameterized model achieves nearly perfect nodal-force equilibrium by learning noise instead of material behavior.This motivates distinguishing equilibrium fit from recovery of the underlying material response.
5 Numerical examples: Calibration uses two stages: 50 initial time steps followed by the complete dataset, with 2000 and 1000 function evaluations, respectively.The SLSQP function tolerance is set to 10^-32, so the evaluation limits effectively control optimization.
5 Numerical examples: The combined Newton iteration uses a tolerance of 10^-12 and allows at most 20 iterations.These settings are used to compute the coupled local quantities during training.
5 Numerical examples: Ten initializations are trained for each example, after which the model with the least training loss is selected and validated.The workflow is implemented in Python, TensorFlow, and SciPy and executed on an HPC cluster at TU Dresden.
5 Numerical examples: SLSQP is described as comparatively robust to NaN values caused by parameter choices that drive stresses toward infinity, except when NaNs occur in the first training epoch.For the reported cases, restricting initial weights to [0, 0.1] was sufficient to avoid this initial failure mode.

5.1 Generation of data

Synthetic full-field training data are generated with finite element simulations of specimens designed to produce diverse strain and stress states. A separate biaxial experiment is then used to validate the trained PANN against a linear viscoelastic reference model.

5.1 Generation of data: The equilibrium-gap calibration uses synthetic in-plane displacements and global reaction forces normal to specimen boundaries generated by finite element simulations.The simulations use an in-house finite element code.
5.1 Generation of data: The ground-truth material is a classical linear viscoelastic Maxwell model with one internal variable.Its specified parameters include K = 5 MPa, G = 3 MPa, K0 = 10 MPa, G0 = 7 MPa, η_sph = 4 MPa s, and η_dev = 2 MPa s.
5.1 Generation of data: Uniaxial and biaxial specimens contain two elliptical cutouts to generate a wide range of strain and stress states, with initial thickness h = 1 mm.Uniaxial edge displacements are prescribed as random walks to produce many strain rates during the synthetic experiment.
5.1 Generation of data: Validation uses a synthetic biaxial experiment excluded from training, with quadratic triangular elements and a trained PANN embedded in a finite element code.The resulting stresses are compared with those from the linear viscoelastic reference model.

5.2 Training with ideal data

Ideal data from uniaxial, biaxial, and combined experiments were used to calibrate the PANN, with selected models closely reproducing the reference material response in training and validation.

Training setup: Three ideal datasets—uniaxial, biaxial, and combined experiments—were used to calibrate the PANN without artificial noise.The datasets supplied displacements and global reaction forces computed from the finite-element reference model.
Training results: All three datasets and ten random initializations produced successful SLSQP termination, although calibration quality varied between initializations.For each dataset, the calibration with the smallest final loss was selected for subsequent evaluation.
Training results: The selected PANNs agreed very well with ground-truth stresses, while representative stress components over time were fitted almost perfectly.Figure 4 reports the stress comparisons and R2 values; Figure 5 shows the time histories at a representative quadrature point.
Training results: Internal variables and other thermodynamic quantities were predicted with comparable accuracy across all three training datasets.The reported agreement extended beyond stress components to internal variables and thermodynamic driving forces.
Validation results: Validation on an unseen biaxial specimen showed negligible stress differences between the PANN and the linear viscoelastic ground-truth model.The trained PANNs were embedded in a finite-element code, and the comparison included the three best-performing calibrations.
Validation results: The PANN required around 6 to 10 global Newton iterations versus one for the linear viscoelastic reference model because its FICNN activations introduce nonlinearity.Despite the additional iterations, remaining stress components, internal variables, and displacements remained in very good agreement with the reference model.

5.3 Training with noisy data

Artificial noise was added to full-field displacements and reaction forces to emulate realistic experiments, then used to calibrate PANNs from three noisy datasets. Successful models retained reasonably accurate stress predictions, although noisy-data optimization sometimes failed and errors increased near boundaries.

5.3.1 Generation of artificial noise: Noise was added to in-plane displacements and global reaction forces to mimic realistic experimental measurements.Displacement noise used Gaussian random fields, while reaction-force noise used multiplicative uniform perturbations.
5.3.1 Generation of artificial noise: The displacement-noise amplitude was η = 5 × 10^-5, with influence lengths of 100 mm for uniaxial and 200 mm for biaxial specimens.The assembled nodal-force regularization used α_infl = 0.8.
5.3.2 Results: Ten initializations were attempted for each noisy uniaxial, noisy biaxial, and combined dataset.Training could terminate early when the Newton forward-pass coefficient matrix became ill-conditioned.
5.3.2 Results: Five of ten noisy-uniaxial calibrations and eight of ten noisy-biaxial and combined calibrations completed successfully.The reported success counts reflect optimization robustness under noisy data rather than prediction accuracy alone.
5.3.2 Results: Noisy-data PANNs showed reasonably accurate overall stress predictions, with largest differences near boundaries and best agreement for combined noisy uniaxial-biaxial training.The authors suspected that the ordering among datasets was influenced more by initialization than by training-data choice.

6 Conclusion

The paper presents an unsupervised, thermodynamically consistent framework for calibrating small-strain viscoelastic PANNs from experimentally measurable full-field data. It achieves strong agreement on synthetic data, including noisy measurements, while future work targets real experiments, finite strains, and other inelastic materials.

6 Conclusion: The framework calibrates small-strain viscoelastic PANNs from global reaction forces and surface displacements using constrained equilibrium-gap optimization.It combines plane-stress constraints, internal-variable evolution equations, GSM potentials, invariant representations, and backward-adjoint gradients.
6 Conclusion: Thermodynamic consistency and material symmetry are enforced by construction through GSM-based scalar potentials and invariant representations.The backward adjoint method reduces training cost by evaluating target-loss gradients without backpropagating through every Newton step.
6 Conclusion: The framework shows excellent agreement with synthetic data and reliably recovers material response despite added artificial noise.The conclusion characterizes this as calibration using data measurable in real experiments.
6 Conclusion: Future extensions include calibration with real experimental data, finite-strain formulations, and transfer to further inelastic material classes.Elasto-plasticity is named as one prospective additional material class.

CRediT authorship contribution statement

The authors’ contributions span conceptualization, methodology, software, validation, visualization, writing, supervision, resources, and funding acquisition.

CRediT authorship contribution statement: Brain M. Riemer contributed to conceptualization, formal analysis, investigation, methodology, software, validation, visualization, and writing.Riemer is listed for both original drafting and review and editing.
CRediT authorship contribution statement: Markus Kästner contributed resources, writing, review and editing, and funding acquisition.
CRediT authorship contribution statement: Karl A. Kalina contributed to supervision, conceptualization, formal analysis, investigation, methodology, software, validation, writing, and funding acquisition.Kalina is listed for both original drafting and review and editing.

A Notation

The notation section defines tensor operations, projectors, index conventions, tensor spaces, and superscripts used for spatial points, time steps, iterations, and elements.

A Notation: The Nabla operator is defined as ∇ := ∂/∂x_k e_k, and blackboard-bold domains distinguish integers, reals, and constrained subsets such as Z_≥0 and R_≥0.
A Notation: The notation specifies tensor spaces for real hyper-matrices and tensors, including fully symmetric, skew-symmetric, non-deviatoric, and positive-definite subsets.
A Notation: Superscripts s, i, and e identify discrete time steps, iterations, and elements, respectively, while spatial and temporal functional dependencies are often omitted.The associated counts are n_steps, n_iter, and n_el.
A Notation: Indices a, b, c, d range over {1, 2}, indices k, l, …, r range over {1, 2, 3}, and Einstein summation applies to the lower-case Latin indices.Greek indices and the indices e, i, and s are excluded from the summation convention.
A Notation: The symbols · and : denote single and double tensor contractions, while tr, sym, and (·)^⊤ denote trace, symmetric part, and transposition.The text also defines repeated single contractions through powers such as A^3.
A Notation: The symmetric, spherical, and deviatoric projectors are denoted by I_sym, I_sph, and I_dev, with I_dev = I_sym − I_sph.The orthogonal groups O(3) and SO(3) and the number of global nodes are also introduced.

B Convex functional basis for a symmetric non-deviatoric 2nd order tensor and the orthogonal group

This section establishes invariant-based functional bases for symmetric second-order tensors under the orthogonal group, distinguishing non-deviatoric and deviatoric cases. It shows that some invariant sets fail to separate group orbits, while augmented sets provide valid bases.

The invariants Ī1, K̄, and J̄ form a functional basis for symmetric tensors under O(3).
The set Ī1, K̄, and Ī3 fails to form a functional basis because it does not separate distinct group orbits.
The alternative invariants I1, K, and J also separate O(3) group orbits for symmetric tensors.
The invariants Ī1, K̄, and J̄ separate group orbits for deviatoric tensors, whereas Ī1, Ī2, and Ī3 do not.
For non-deviatoric symmetric tensors, I1, I2, and I3 constitute a functional basis under O(3).

C Backward adjoint method

The backward adjoint method computes loss gradients without backpropagating through every Newton iteration. It uses information from the final iteration and is advantageous when neural networks have many trainable parameters.

The backward adjoint method avoids backpropagating through all Newton iterations by using residual information from the last iteration of each time step.
In this implementation, five unknowns are solved iteratively at each time step.
Adjoint parameters make the method scale with the Newton-solved unknown vector rather than the number of trainable neural-network parameters.
The method computes the loss gradient through sequential linear-system solves followed by the gradient expression.
The approach assumes Newton convergence when neglecting the residual derivative, but training treats the final iteration as converged even when convergence is not guaranteed.