Polyconvex neural-network surrogates with learned paired directional features for anisotropic hyperelastic homogenization

S. Ares de Parga; A. Cornejo; J. A. Hernández; R. Rossi

physics.comp-phOct 8, 2026

Polyconvex neural-network surrogates with learned paired directional features for anisotropic hyperelastic homogenization

Authors: S. Ares de Parga, A. Cornejo, J. A. Hernández, R. Rossi

Organizations: Centre Internacional de Mètodes Numèrics en Enginyeria (CIMNE), Barcelona, Spain · Department of Aeronautics and Astronautics, Stanford University, Stanford, CA 94305, USA · Universitat Politècnica de Catalunya, Department of Civil and Environmental Engineering (DECA), Barcelona, Spain · Universitat Politècnica de Catalunya, Departament de Resistència de Materials i Estructures a l’Enginyeria, E.S. d’Enginyeries Industrial, Aeroespacial i Audiovisual de Terrassa (ESEIAAT), Terrassa, Spain

Abstract

Approximating the effective anisotropic hyperelastic response of a representative volume element (RVE) requires constitutive expressiveness, mechanical consistency, and computational efficiency. This work develops a polyconvex physics-augmented neural network (PANN) construction based on paired directional features whose orientations, stretch powers, and determinant exponents adapt to the RVE response. The features are composed with convex, componentwise nondecreasing cores, realized as an input-convex neural network (ICNN) or an input-convex Kolmogorov-Arnold network (ICKAN). The complete energies satisfy objectivity, an exactly stress-free reference, two-dimensional polyconvexity, and prescribed growth conditions by construction, and nonnegative energy is certified separately for the selected models. On a multicavity and a single-cavity periodic cell, these energies are compared with an Unconstrained energy, trained identically without the curvature and growth constraints, and with three hyperreduced projection-based reduced-order models (HPROMs), which serve as physics-based references. On the first cell, learning the features lowers the stress error sevenfold, and six features suffice on both. The constrained energies remain below one percent in stress error and in the displacement error of a tensile-coupon FE $^2$ application with a full-order reference, whereas the Unconstrained energy and the HPROMs are more accurate inside the data. Beyond the data, however, the Unconstrained energy loses rank-one convexity, and under confined compression it stops below half of the 500 MPa that the constrained energies reach. These results quantify what the guarantees cost in accuracy and what they secure beyond the data.

Figures & tables

Panel of Figure 1 — Figure 1: Overview of the proposed constitutive construction. Sampled strain states are evaluated with the periodic RVE model to supply energy and stress data. These responses train learnable paired directional features and a monotone convex ICNN or ICKAN core. Analytic reference and growth terms complete the stored energy. Differentiation then supplies the work-conjugate stress and constitutive tangent. Criteria (C1)–(C6), defined in Section 3.1 , follow from the complete construction rather than from sampled numerical checks.

Panel of Figure 2 — Figure 2: Multilayer monotone cores mapping the conditioned paired features $\bm{y}$ to the scalar contribution $\widehat{H}(\bm{y})$ . (a) The ICNN learns nonnegative linear maps and applies fixed softplus activations at hidden nodes. (b) The ICKAN learns one convex, nondecreasing scalar function per edge and sums them at nodes. Purple dashed paths denote nonnegative maps of the complete input, and ellipses denote intervening layers. Glyph counts indicate where each operation acts, not deployed widths or parameter counts. Unrestricted biases are omitted.

Panel of Figure 3 — Figure 3: Integrated-hat parameterization of an ICKAN edge. (a) A localized nonnegative curvature hat integrates to a nonnegative slope and a convex $C^{2}$ basis with a zero left tail and a linear right tail. (b) Nonnegative learned coefficients combine several fixed bases into a nonnegative edge curvature, and integration gives a nondecreasing slope and a convex, nondecreasing connection. The curves illustrate the analytic construction, not the deployed grid.

	Unconstrained energy	ICNN	ICKAN
Role	Flexible hyperelastic reference	Guaranteed paired-feature energy	Guaranteed paired-feature energy
Inputs	Objective measures from $(\bm{C},J)$	Paired features (fixed or learned)	Paired features (fixed or learned)
Learned scalar map	Smooth energy network	Nonnegative weights, fixed activations	Convex, nondecreasing edge functions
(C1)–(C4)	By construction	By construction	By construction
(C5)–(C6)	Not imposed	By construction	By construction

Table 1: Constitutive constructions carried into the numerical assessment. Criteria (C1)–(C6) are defined in Section 3.1 , and the guarantees refer to the constructions of this section.

Panel of Figure 4 — Figure 4: Intrusive HPROMs used in the comparison. FOM snapshots provide either an affine state representation or a primary–secondary partition in which the ANN predicts the secondary coordinates. Online, HPROM and HPROM–ANN solve projected equilibrium with fixed and adaptive cubature, respectively, whereas D-HPROM–ANN evaluates the nonlinear decoder directly. All three retain microscopic constitutive evaluations and return the same stress–tangent interface.

Model	State representation	Equilibrium unknowns	Integration	Learned component
FOM	$\bm{u}$	$N$	Full mesh	None
HPROM	$\widetilde{\bm{u}}=\bm{u}_{\rm ref}+\bm{V}_{\rm tra}\bm{q}_{\rm tra}$	$n_{\rm tra}$	Fixed ECM (residual and stress)	None
HPROM–ANN	$\widetilde{\bm{u}}(\bm{\xi})=\bm{u}_{\rm ref}+\bm{V}\bm{A}_{m}\bm{\xi}+\overline{\bm{V}}\widehat{\mathcal{N}}(\bm{\xi})$	$n$	Adaptive ECM (residual and stress)	Secondary coordinates; residual and stress weights
D-HPROM–ANN	$\widetilde{\bm{u}}_{D}(\bm{e})=\bm{u}_{\rm ref}+\bm{V}\bm{A}_{m}\bm{e}+\overline{\bm{V}}\widehat{\mathcal{N}}(\bm{e})$	None	Adaptive ECM (stress only)	Same closure and stress weights as HPROM–ANN

Table 2: HPROMs used in the accuracy–cost comparison and their FOM reference. All HPROMs retain microscopic constitutive evaluations.

Figure 5: FOM microscopic fields at the common diagnostic state

(E_{11},E_{22},2E_{12})=(18,-4,8)\%

, shown on the actual deformed cells without displacement amplification. Top: magnitude of the periodic displacement fluctuation

\bm{w}

of Eq. ( 5 ), normalized by the cell side

\ell

. Bottom: local three-dimensional Cauchy von Mises stress, including the plane-strain out-of-plane stress, with Gauss-point values averaged to the nodes for visualization. Each panel uses its own color limits to resolve the pattern within its cell, so colors do not compare magnitudes across cells.

Figure 6: MC-RVE cavity geometry (left) and quadratic mesh (right). Dashed lines identify cavity major axes.

Figure 7: Constitutive sampling domains, in percent. Left: MC-RVE representation benchmark. Right: SC-RVE deployment benchmark, with fitting and validation states on a Cartesian grid and independent-test states off it. Wireframes outline the fitting boxes, and colors identify fitting, validation, and independent-test states.

Figure 8: MC-RVE validation error, the relative squared stress error of Section 5.1 , versus the number

m

of paired directional features. Lines show the median over three seeds and bands span their minimum–maximum range. The vertical guide marks

m=6

, used for the independent comparison.

Model	Stress [%]	Energy [%]	Tangent [%]
ICNN, fixed	0.3282	0.1096	1.9247
ICNN, learned	0.0445	0.0079	0.3851
ICKAN, fixed	0.3014	0.0943	1.7600
ICKAN, learned	0.0406	0.0066	0.3622
Unconstrained energy	0.0073	0.0011	0.0857

Table 3: MC-RVE independent-test errors at

m=6

. Each entry is the relative error norm of the stress

\bm{s}

, the energy

W

, or the tangent

\bm{D}

of Eq. ( 3 ) over the 512 test states, as the median over three seeds. For every quantity, each learned seed is more accurate than each fixed seed. Adam stopped at a validation plateau for the fixed variants, whereas five of the six learned runs and the Unconstrained energy, which has no paired features, reached the 200 000-step cap.

Figure 9: MC-RVE energies beyond the data along equibiaxial compression,

\bm{F}=\sqrt{J}\,\bm{I}

, for the learned ICNN and ICKAN energies and the Unconstrained energy. Lines show the median of three seeds and shaded bands their range. (a) Minimum rank-one curvature over unit directions. (b) Energy density toward volume collapse.

Figure 10: SC-RVE cavity geometry (left) and quadratic mesh (right). The dashed line identifies the cavity major axis.

Figure 11: Relative truncation norm

\epsilon_{\rm POD}

of Eq. ( 46 ) for the 4950 SC-RVE displacement snapshots versus the number of modes. The 39 retained modes meet the tolerance of

10^{-6}

, and their rotation into primary and secondary coordinates leaves this span unchanged.

Figure 12: Element supports of the three HPROMs. The pale full-cell mesh provides location context only. Colors distinguish residual-only, stress-only, and shared elements.

Model	$e_{S}^{\rm test}$ [%]	$e_{W}^{\rm test}$ [%]
HPROM	0.0169	–
HPROM–ANN	0.2177	–
D-HPROM–ANN	0.0959	–
ICNN	0.6214	0.1929
ICKAN	0.6359	0.1900
Unconstrained energy	0.0052	0.0010

Table 4: SC-RVE independent-test constitutive errors of the HPROMs and the selected energies, in percent.

Figure 13: Coupon problem. Undeformed macro mesh with equal-and-opposite resultants of 100 kN per end, which balance each other, so the three point constraints only remove rigid-body motion. Every integration point queries the SC-RVE, solved by the FOM or replaced by an HPROM or a learned energy.

Figure 14: Nested FOM–FE 2 reference at 100 kN per end, shown on the deformed mesh without displacement amplification. Top: displacement magnitude. Bottom: in-plane von Mises equivalent of the macroscopic Cauchy stress, $\sigma_{\rm eq}=(\sigma_{11}^{2}-\sigma_{11}\sigma_{22}+\sigma_{22}^{2}+3\sigma_{12}^{2})^{1/2}$ . Gauss-point values are averaged to the nodes for visualization.

Model	$e_{u}$ [%]	$e_{E}$ [%]	$e_{S}$ [%]	Time [s]	Speed-up
FOM	–	–	–	4268.68	–
HPROM	0.0057	0.0165	0.0010	67.35	63.38
HPROM–ANN	0.0520	0.1497	0.0373	30.14	141.61
D-HPROM–ANN	0.0232	0.0485	0.0097	3.79	1126.90
ICNN	0.4066	0.6785	0.0739	1.04	4105.14
ICKAN	0.4197	0.6914	0.0732	1.95	2187.66

Table 5: Coupon simulations: final-field errors of Eq. ( 61 ) with respect to the nested FOM, online wall time, and speed-up over the FOM.

Figure 15: Confined compression beyond the data with the SC-RVE energies. (a) Setup for compression in

y

, with the homogenized block of 128 quadratic triangles evaluating the SC-RVE energy at every integration point. For compression in

x

, the traction acts on a vertical face. (b, c) Applied pressure versus shortening, with one marker per increment of 25 MPa and a cross at the increment in which Newton does not converge. (d, e) Minimum rank-one curvature over the 384 Gauss points of the block at each converged increment.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Model	Parameters	Stress [%]	Energy [%]	Tangent [%]
ICNN, $m=6$	975	0.0445	0.0079	0.3851
ICNN, wide	26 919	0.0398	0.0069	0.2962
ICKAN, $m=6$	1 659	0.0406	0.0066	0.3622
ICKAN, wide ∗	24 847	0.0744	0.0104	1.8311
Unconstrained, small	981	0.0126	0.0019	0.1579
Unconstrained	25 473	0.0073	0.0011	0.0857

Appendix

Table 6: MC-RVE capacity controls: independent-test errors, in percent.

Model	Parameters	$e_{S}^{\rm test}$ [%]	$e_{u}$ [%]
ICNN, $m=6$	975	0.6214	0.4066
ICNN, wide	26 919	0.6066	0.4162
ICNN, $m=32$	2 379	0.5370	0.3433
ICKAN, $m=6$	1 659	0.6359	0.4197
ICKAN, wide	24 847	0.6540	0.4322
ICKAN, $m=32$	3 999	0.6177	0.3469

Appendix

Table 7: SC-RVE capacity and feature-count controls, in percent.

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Universal neural network approximation of symmetric polyconvex energies and envelopes

Florian Gammer

Faculty of Mathematics, University of Regensburg, Universitätsstraße 31 93053 Regensburg, Germany

Neural Network Approximation Theory Input Convex Neural Networks

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Symmetric polyconvexity provides a sufficient condition for weak lower semicontinuity in small-strain elasticity and a tractable lower bound for the relaxation of non-(quasi)convex energies. Since classical algorithms for computing such functions can be computationally expensive, neural network approximations have recently emerged as an alternative. A central difficulty with establishing such neural network based approaches for symmetric polyconvexity is the additional monotonicity property. As we show, the corresponding weight restrictions on ReLU-based input-convex networks do not yield universal approximation of the lifted representers. Here, we introduce symmetric polyconvex neural networks (SPCNNs) for constitutive modeling and approximation of symmetric polyconvex envelopes. The deep architecture combines convexity with the monotonicity required of a representer in the space of minors. We prove a universal approximation theorem for symmetric polyconvex energies, without requiring isotropy, and quantify the number of affine pieces attainable by the lifted network and derive reduced architectures for isotropic energies. Numerical examples illustrate stress-based regression and approximation of the symmetric polyconvex envelope of a double-well potential. Symmetric polyconvexity is guaranteed by construction, while the accuracy of the fitted energies and envelopes is assessed numerically.

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