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

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

arXiv:2610.12801v1physics.comp-phcs.LGcs.NAmath.NA

TL;DR

Repeated nonlinear RVE solves make effective constitutive evaluation costly. The paper builds polyconvex energy surrogates from adaptive paired directional features, which retain low in-data errors while constrained models remain solvable beyond the data where the unconstrained model does not.

  • Problem

    Repeated nonlinear RVE solves make online structural evaluations costly, motivating surrogates that reduce evaluation cost without losing constitutive behavior.

    Source: §1 · p. 1

  • Method

    The paper composes adaptive paired directional features with convex, componentwise nondecreasing cores to construct polyconvex energies with mechanical properties built in.

    Source: §6 · p. 30

  • Results

    The constrained energies remain near 0.6% stress error and 0.4% coupon displacement error, while beyond-data compression reaches 500 MPa versus below half that load for the Unconstrained energy.

    Source: §6 · p. 30

  • Takeaways & Limitations

    The examples indicate that constrained energies trade some in-data accuracy for solvability beyond the data, while HPROMs retain the microscopic state at higher online cost.

    Source: §6 · p. 30

    The accuracy cost remains to be lowered on cells such as the SC-RVE, and the construction has yet to be extended to three dimensions.

    Source: §6 · p. 30

Abstract

from arXiv · show

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.

1 Introduction

The paper seeks efficient surrogates for anisotropic homogenized response that retain mechanical consistency and representational flexibility. It develops adaptive paired directional features for polyconvex energies and uses hyperreduced projection-based models as physics-based references.

1 Introduction

  • Repeated nonlinear RVE solves make structural FE2 costly, motivating surrogates that reduce online evaluation cost without losing constitutive behavior.The paper contrasts direct constitutive learning, which dispenses with microscopic solves online, with intrusive reduced models that retain approximate microscopic states and projected equilibrium.Source: §1 · pp. 1–2

1.1 Energy-based constitutive learning and polyconvexity

  • Learning a stored-energy function makes stress conservative and its tangent major-symmetric, while polyconvexity supplies a sufficient condition for rank-one convexity.The paper notes that potential learning alone leaves finite-strain curvature uncontrolled, and that polyconvexity is sufficient rather than necessary for homogenized materials.Source: §1.1 · p. 2

1.2 Directional features and representational expressiveness

  • Adaptive directional features must distinguish anisotropic responses while controlling their reference derivatives so stress normalization remains compatible with polyconvexity.The paired construction addresses the risk that learned anisotropic contributions produce nonzero reference stress or that arbitrary corrections violate polyconvexity or material symmetry.Source: §1.2 · pp. 2–3
  • The same paired features can be combined with either a smooth ICNN or an integrated-hat ICKAN, keeping directional information and mechanical guarantees fixed while changing the scalar core.The ICKAN composition requires convex, nondecreasing edge functions; feature ordering can otherwise obstruct representation regardless of network width.Source: §1.2 · pp. 2–3

1.3 Projection-based reduced-order models

  • HPROMs preserve an approximate microscopic state and constituent law, but retain online equilibrium or stress evaluations and lack the effective-potential guarantees of the learned energies.The three references progressively reduce online microscopic work; the direct D-HPROM–ANN variant removes the online equilibrium solve.Source: §1.3 · pp. 3–4

1.4 Contribution and outline

  • The proposed energies use trainable orientations, stretch powers, and determinant exponents with convex monotone cores to establish objectivity, exact reference stress, two-dimensional polyconvexity, and growth by construction.A sufficient bound separately certifies nonnegative energy for the six-feature models studied.Source: §1.4 · p. 4
  • The assessment compares constrained energies with an identically trained unconstrained model and HPROMs across constitutive tests and structural FE2 examples, with conclusions limited to two cells, plane strain, and two loadings.The FOM–FE2 reference measures surrogate error at fixed discretizations within the adopted homogenization model, not error against explicitly resolved microscopic structure.Source: §1.4 · p. 4

2 Finite-strain homogenization and the reference problem

The reference problem maps an imposed in-plane deformation to the homogenized cell response by solving periodic microscopic equilibrium. Constitutive evaluators then supply stress and tangent through a shared interface to the macroscopic structural solver.

2.1 Kinematics and work-conjugate variables

  • The model uses plane-strain kinematics and a three-component engineering Green-strain input work-conjugate to the effective second Piola–Kirchhoff stress.The engineering shear component is γ12 = 2E12 and is paired with S12 without an additional factor of two in the stress vector.Source: §2.1 · p. 5

2.2 Periodic microscopic equilibrium

  • Periodic microscopic equilibrium combines the imposed macroscopic deformation with an unknown periodic displacement fluctuation, with the internal void surface traction-free.One fluctuation coordinate is fixed to remove arbitrary translation, and the microscopic deformation must remain orientation-preserving.Source: §2.2 · p. 5

2.3 Effective energy and stress

  • The homogenized energy and stress are obtained from the equilibrium branch, with Hill–Mandel work consistency relating averaged microscopic work to macroscopic stress and energy derivatives.Newton convergence establishes stationarity on the followed branch, not global minimization or uniqueness of the cell response.Source: §2.3 · p. 6

2.4 Consistent constitutive tangent

  • The consistent tangent differentiates stress through the strain-dependent microscopic equilibrium and includes the relaxation correction from changing fluctuation coordinates.Sensitivities are computed from linear systems using the converged microscopic tangent, and the reference implementation checks the linearization against finite differences.Source: §2.4 · p. 6

2.5 Macroscopic equilibrium and the constitutive interface

  • Every constitutive evaluator supplies stress and tangent to the same macroscopic equilibrium interface, while learned energies remove the online microscopic solve and HPROMs retain an approximate microscopic state.Reported surrogate errors are at fixed macroscopic and microscopic discretizations; homogenization, boundary-layer, and discretization errors remain outside the comparison.Source: §2.5 · pp. 6–7

3 Learnable paired-feature polyconvex energies

This section presents complete polyconvex energy constructions that combine learnable paired directional features with monotone convex ICNN or ICKAN cores, reference corrections, and analytic growth terms. It also specifies a monotone integrated-hat ICKAN core and explains that learned features preserve the stated mechanical admissibility guarantees by construction.

3 Learnable paired-feature polyconvex energies

  • The proposed energy replaces repeated online microscopic solves with a learned effective stored energy whose common potential supplies stress and constitutive tangent.The construction passes from RVE data through paired features and a convex core to analytic reference and growth terms.Source: §3 · p. 7, Figure 1 · p. 8

3.1 Admissibility criteria and scope of the guarantees

  • The complete ICNN and ICKAN energies satisfy six admissibility criteria by construction, while global nonnegative energy is checked separately for trained parameters.The criteria cover potential structure, objectivity, symmetric stress with anisotropic response, reference normalization, polyconvexity, and growth; the guarantees are two-dimensional.Source: §3.1 · p. 7

3.2 Unconstrained energy reference

  • The Unconstrained reference model enforces zero reference energy and stress but leaves learned curvature unrestricted, imposing neither polyconvexity nor growth.Its affine-in-strain correction fixes the reference conditions without changing the raw network Hessian.Source: §3.1 · p. 7, §3.2 · p. 10

3.3 Feature design: expressiveness and admissibility

  • Because the core is componentwise nondecreasing, feature design shapes deformation-state ordering; adding features can enlarge admissible responses but cannot guarantee inclusion of the target.The paired construction couples directional stretch to volume through determinant exponents while retaining convexity in independent (F, J).Source: §3.3 · p. 10

3.3.1 Paired directional features

  • Each paired feature combines perpendicular directional stretches with determinant coupling, using constrained exponents and learned orientations to preserve objectivity and convexity in (F, J).The paired contributions yield an isotropic reference gradient, allowing a scalar determinant correction to remove reference stress without sacrificing convexity.Source: §3.3.1 · p. 11

3.4 Monotone ICNN and integrated-hat ICKAN cores

  • Composing the convex feature map with a convex, componentwise nondecreasing core preserves convexity of the learned contribution; the paper provides ICNN and ICKAN alternatives.The ICNN uses nonnegative linear maps and softplus activations, while the ICKAN sums convex, nondecreasing learned edge functions.Source: §3.4 · p. 11

3.4.1 Monotone ICNN

  • The ICNN uses nonnegative input and hidden-layer maps with softplus activations to make its scalar core convex, componentwise nondecreasing, and C2.C2 smoothness supports a continuous constitutive tangent.Source: §3.4.1 · p. 12

3.4.2 Monotone integrated-hat ICKAN

  • The ICKAN constructs each edge from nonnegative curvature bases, whose integration yields convex, nondecreasing connections that are summed at each node.Integrated-hat bases use nonnegative triangular curvature functions on a fixed uniform knot grid.Source: §3.4.2 · p. 13, Figure 3 · p. 13
  • Integrated-hat ICKAN edges are convex and nondecreasing, with curvature localized to a fixed knot grid and linear tails outside their combined support.The fixed grid and finite number of bases limit where an individual edge can learn curvature.Source: §3.4.2 · pp. 13–14

3.5 The complete energy and its admissibility guarantees

  • Unlike Linden et al., who need extra invariant terms for anisotropic energies, paired features make reference stress isotropic, so one affine determinant correction suffices and preserves convexity for either sign.The correction is zero in value at J = 1.Source: §3.5 · p. 14
  • Wlearn and Wcorr set the learned response and reference state, while Wvol and Wdist prescribe growth toward admissible-domain boundaries; the analytic terms can still affect reference stiffness.The analytic terms vanish with their first derivatives at the reference, but contribute to reference stiffness because J is nonlinear in e.Source: §3.5 · p. 15
  • The C2 energy is objective and two-dimensional polyconvex on GL+(2), has W(I)=0 and s(0)=0, and diverges under collapse, unbounded expansion, and unbounded distortion at bounded positive area.These properties hold for either of the specified convex, componentwise nondecreasing cores when α, β, ε > 0.Source: §3.5 · p. 15
  • Learning feature parameters changes the representation while leaving the stated guarantees intact.The guarantees apply to the constructions summarized for the numerical assessment.Source: §3.5 · p. 15

4 Projection-based reduced-order models

HPROMs reduce the microscopic equilibrium problem using snapshot-based displacement representations and hyperreduction, while retaining the constituent law as physics-based references for the learned energies.

4.1 Periodic microscopic reference

  • Periodic coordinates and a strain-dependent lifting reconstruct physical displacements that satisfy the prescribed periodic constraints, providing the state representation used for reduction.The snapshots retain the affine displacement information used in strain-informed identification.Source: §4.1 · p. 16

4.2 POD approximation and the affine PROM

  • POD modes compress the independent displacement coordinates into an affine reduced basis, with dimension selected by snapshot reconstruction error rather than a stress-error bound.The affine approximation still has nonlinear constitutive behavior and reduced equilibrium equations; evaluating them visits every microscopic element before hyperreduction.Source: §4.2 · p. 16

4.3 Nonlinear reduction through latent-space closure

  • Latent-space closure retains a rich POD span but uses three input-informed primary coordinates and an ANN to predict secondary displacement modes.The decoder maps primary coordinates to microscopic displacement; PROM–ANN solves for coordinates satisfying projected equilibrium rather than imposing ξ = e.Source: §4.3 · pp. 16–17

4.4 Hyperreduction of the projected equations

  • Hyperreduction sparsifies element contributions to projected equilibrium and homogenized stress, using separate integration rules because their integrands and accuracy requirements differ.The selected elements retain ordinary within-element quadrature, and the two tasks may use distinct supports and weights.Source: §4.4 · p. 17, §4.4.1 · p. 17

4.4.1 Fixed elementwise cubature

  • Fixed elementwise cubature compresses sampled element contributions by truncated SVD, then selects a sparse support with nonnegative multipliers to reproduce retained integration modes.The residual and stress rules use fixed supports and weights online.Source: §4.4.1 · p. 17

4.4.2 Manifold-adaptive weights

  • Manifold-adaptive weights retain a common pruned support but learn separate state-dependent weight maps for equilibrium residuals and stress.The adaptive residual derivative includes element stiffness, decoder curvature, and weight variation; only stiffness enters the microscopic modified-Newton iteration.Source: §4.4.2 · pp. 17–18

4.5 Constitutive output

  • HPROMs return average first Piola–Kirchhoff stress and a consistent tangent of the implemented stress map; D-HPROM–ANN instead evaluates the decoder directly with ξ = e.The iterative variants obtain the tangent by differentiating the implicit reduced equilibrium state.Source: §4.5 · p. 18

4.6 Mechanical structure retained by the HPROMs

  • Because equilibrium and stress use separate cubature rules and adaptive weights add a gradient term, the HPROMs have no claimed exact effective potential or polyconvexity certificate.Their consistent tangent is the derivative of the implemented stress map and need not be symmetric.Source: §4.6 · p. 18

5 Numerical examples

The numerical examples compare constrained and unconstrained learned energies with HPROMs on two periodic cells and in coupon FE². They assess in-data accuracy and, beyond the fitting data, admissibility and compression behavior.

5.1 Training protocol

  • Both convex cores and the Unconstrained model were fitted to FOM energies and stresses plus the undeformed tangent, with stress weighted most heavily and independent test states evaluated after checkpoint selection.Models used three seeds; Adam training was followed by L-BFGS refinement, and the lowest-validation-error checkpoint was retained.Source: §5.1 · pp. 20–21

5.2.1 Cell, data, and controlled comparison

  • Mesh refinement to 8961 elements changed MC-RVE homogenized energy, stress, and tangent by less than 0.1% over 64 audited states; at E11 = 0.20, SC-RVE peak microscopic stress norm changed 0.486%, average axial stress less than 0.001%.The SC-RVE has a single inclined cavity and is the microscopic model propagated through the structural FE2 calculations.Source: §5 · p. 20, §5.2.1 · p. 21, §5.3.1 · p. 24

5.2.2 Number of paired features

  • The MC-RVE feature sweep found learned models entered a low-error regime at m = 6, while fixed-feature models improved gradually through m = 32.Fixed and learned variants start from the same features and share other training ingredients, isolating feature adaptation; SC-RVE features were selected anew rather than transferred between geometries.Source: §5.2.2 · p. 22, §5.2.1 · p. 22, §5.3.1 · p. 25

5.2.3 Effect of learning the features

  • At m = 6 on the MC-RVE, learning reduced median stress error from 0.328% to 0.0445% for ICNN and 0.301% to 0.0406% for ICKAN, about 7.4-fold in either core.Energy and tangent errors improved by factors of about 14 and 5; even 32 fixed features remained about three times less accurate than learned six-feature models in relative stress error.Source: §5.2.3 · pp. 22–23, Table 3 · p. 23, Figure 8 · p. 23

5.2.4 Admissibility beyond the data

  • Beyond the MC-RVE fitting box, learned constrained fits remained above 220 MPa rank-one curvature, while every Unconstrained fit changed sign between J = 0.71 and 0.73.Toward volume collapse, constrained energies diverged while Unconstrained energies leveled off at 397–440 MPa; the six learned fits also received a sufficient-bound certificate of nonnegative energy.Source: §5.2.4 · pp. 23–24, Figure 9 · p. 24

5.3.2 Construction of the HPROMs

  • The SC-RVE HPROMs use a 39-mode snapshot basis; affine HPROM reduces the mesh to 183 elements, HPROM–ANN to 19, and D-HPROM–ANN evaluates only 10 stress-rule elements.HPROM–ANN solves for three primary coordinates using a closure for 36 secondary coordinates; D-HPROM–ANN instead prescribes the primary coordinates from macroscopic strain.Source: §5.3.2 · pp. 25–26

5.3.3 Constitutive accuracy

  • On 400 off-grid SC-RVE states, ICNN/ICKAN stress errors were 0.621%/0.636%, versus 0.0052% for Unconstrained and 0.0169%, 0.0959%, 0.218% for affine HPROM, D-HPROM–ANN, HPROM–ANN.At matched capacity, the constraint cost is 76 to 126 on the SC-RVE; the HPROMs evaluate the constituent law on reduced meshes rather than learning the stress response.Source: §5.3.3 · pp. 26–27, Table 4 · p. 27
  • On the SC-RVE, rank-one curvature stayed above 176 MPa for ICNN and ICKAN across a fourfold radial expansion, while Unconstrained showed negative curvature at about 13% of states.The deployed constrained models were certified nonnegative; derivative checks and vanishing work around a closed strain cycle supported the potential structure of all three energies.Source: §5.3.3 · p. 27

5.3.4 Coupon FE2: accuracy and online cost

  • In the coupon, HPROM displacement errors are within 0.06% with speed-ups of 63–1127; the affine, iterative, and direct models have errors of 0.0057%, 0.0520%, and 0.0232%, respectively.The affine HPROM takes 67.35 s, the iterative HPROM–ANN 30.14 s, and the direct D-HPROM–ANN 3.79 s; all three HPROMs can reconstruct approximate microscopic displacement fields.Source: Table 5 · p. 28, §5.3.4 · p. 28
  • The six-feature ICNN and ICKAN give coupon displacement errors of 0.407% and 0.420%, versus 0.0022% for the Unconstrained energy.All coupon states remain inside the fitting box, where the three energies have positive rank-one curvature; the constrained energies run in the common solver at comparable cost, while their guarantees matter beyond the data.Source: §5.3.4 · pp. 28–29

5.3.5 Confined compression beyond the data

  • Both constrained energies reach 500 MPa in x and y, while the Unconstrained energy loses rank-one convexity and Newton convergence before 250 MPa in x and 225 MPa in y.The constrained energies keep minimum rank-one curvature above 194 MPa at every Gauss point and increment; the Unconstrained energy's minimum curvature turns negative at its last converged increment.Source: Figure 15 · p. 29, §5.3.5 · p. 29

6 Conclusions

The learned energies enforce mechanical admissibility by construction, with six paired features sufficient in both tested cells. They trade some in-data accuracy for better behavior beyond the data, where the constrained energies preserve solvability under compression.

6 Conclusions

  • Objectivity, an exactly stress-free reference, two-dimensional polyconvexity, and prescribed growth hold by construction, while a sufficient bound certifies nonnegative energy for every six-feature model studied.The affine determinant correction normalizes stress without tying its coefficient to volumetric barrier stiffness.Source: §6 · p. 30
  • Under confined compression beyond the data, constrained energies reach 500 MPa, while Newton with the Unconstrained energy stops below half that load as rank-one curvature turns negative.The constrained energies remain below 1% in stress error and coupon displacement error; the HPROMs and Unconstrained energy are more accurate inside the data.Source: §6 · p. 30
  • On the MC-RVE, six learned paired features reduce held-out stress error sevenfold to about 0.04%; six features also suffice on the SC-RVE.The selected constrained energies have about 0.6% SC-RVE stress error and about 0.4% coupon displacement error.Source: §6 · p. 30
  • The accuracy cost of admissibility varies by microstructure, from about six times the Unconstrained error on the MC-RVE to about 120 times on the SC-RVE.Wider cores and more features do not remove this cost, consistent with the monotone-core ordering restriction.Source: §6 · p. 30
  • HPROMs retain an approximate microscopic state and constituent law at higher online cost, whereas learned energies use direct evaluations and constrained versions provide guarantees the HPROMs lack by construction.On this benchmark, constitutive rankings match coupon displacement-error rankings and all six models preserve nested-reference Newton convergence.Source: §6 · p. 30
  • The demonstrated construction is two-dimensional; future work proposes lowering its accuracy cost, extending it to three-dimensional cells, and using HPROMs to generate training states.Source: §6 · p. 30

A Rank-one paths and finite-strain energy curvature

Rank-one convexity requires nonnegative curvature along every rank-one perturbation, a condition implied by the paper’s polyconvex representation. Convexity in strain alone does not guarantee it, because the stress contribution can make finite-strain curvature negative under sufficiently strong compression.

A Rank-one paths and finite-strain energy curvature

  • The Legendre–Hadamard condition tests the second derivative along rank-one paths, rather than convexity in strain variables alone.Source: A Rank-one paths and finite-strain… · p. 30
  • Polyconvexity implies nonnegative curvature along rank-one paths because the determinant is affine there and convexity of the polyconvex representation bounds the energy on each segment.The condition is expressed as nonnegative second derivative along every rank-one perturbation.Source: A Rank-one paths and finite-strain… · pp. 30–31
  • For an energy expressed in strain, the chain-rule curvature contains both a tangent contribution and a stress-weighted curvature-of-strain-path contribution.The latter can be negative in compression even when the constitutive tangent is positive semidefinite.Source: A Rank-one paths and finite-strain… · p. 31
  • For W(E) = µE : E with µ > 0, the energy is strictly convex in engineering strain, yet a rank-one perturbation at F = λI yields curvature 2µλ² + µ(λ² − 1).The example uses H = diag(1, 0) and keeps the determinant positive for t > −λ.Source: A Rank-one paths and finite-strain… · p. 31
  • At λ = 1/2, the strain-convex energy has rank-one curvature −µ/4, violating the Legendre–Hadamard condition; this counterexample requires λ < 1/3.The negative stress term offsets the positive tangent term under sufficiently strong compression.Source: A Rank-one paths and finite-strain… · p. 31

B Proofs for the paired features and complete energy

The proofs of Propositions 1 and 2 rely on establishing convexity of a single stretch–determinant contribution first.

B Proofs for the paired features and complete energy

B.1 Joint convexity of the directional building block

The directional building block is jointly convex in its vector and determinant inputs under explicit exponent bounds. These bounds follow from positive semidefiniteness of its radial–determinant Hessian block and transverse eigenvalues.

B.1 Joint convexity of the directional building block

  • Joint convexity of the stretch–determinant contribution reduces to checking the radial–determinant Hessian block and transverse directions.The building block depends on x through t = ∥x∥, with t > 0.Source: §B.1 · pp. 31–32
  • The radial–determinant Hessian block is positive semidefinite when its diagonal entries and determinant are nonnegative.The diagonal entries are nonnegative for a ≥ 1 and b ≥ 0.Source: §B.1 · p. 32
  • The determinant factor requires b ≤ a − 1; violating this bound gives the block curvatures of opposite sign.Source: §B.1 · p. 32
  • The full Hessian is positive semidefinite for a ≥ 1 and 0 ≤ b ≤ a − 1, yielding p ≥ 1/2 and 0 ≤ b ≤ 2p − 1 after substituting a = 2p.The same argument with q and c establishes the corresponding bounds in Eq. (25); convexity extends continuously to x = 0.Source: §B.1 · p. 32

B.2 Proof of Proposition 1

The proof establishes convexity and objectivity of the paired directional contributions, then verifies their reference value and strain derivative. At the reference, the directional dependence cancels in the combined gradient.

Convexity and objectivity

  • Each paired directional contribution is convex in independent (F, J) and objective under left proper rotations.The directions are fixed with respect to deformation, making the directional maps linear.Source: §B.2 · p. 32

Reference value and derivative

  • At E = 0, both directional invariants and J equal one, and the combined reference gradient loses its directional dependence.The engineering-strain gradient is ρ_i(1, 1, 0)^T.Source: §B.2 · p. 32
  • The derivative calculation uses ∂E I4(di) = 2di ⊗di and ∂EJ|0 = I, with the perpendicular term following the same form.The product rule is applied to the first contribution before adding its perpendicular counterpart.Source: §B.2 · p. 32

B.3 Proof of Proposition 2

The proof verifies the energy’s potential structure, objectivity, reference normalization, polyconvexity, and prescribed growth conditions. Its symmetry argument uses left rotations and therefore does not impose a material symmetry group.

Potential structure, objectivity, and stress symmetry (C1)–(C3)

  • The complete energy is objective, differentiating it gives symmetric material stress and a major-symmetric tangent, and it is smooth on GL+(2).The proof uses left rotations only, so it does not impose a material symmetry group.Source: §B.3 · p. 32

Reference normalization (C4)

  • Wlearn sets the reference energy, Wcorr cancels its reference stress, and the growth terms have zero value and strain gradient there.Together these establish both reference conditions for the complete energy.Source: §B.3 · p. 32

Polyconvexity (C5)

  • The energy is polyconvex because its independent-variable representation is convex in (G, j), and W(F) is obtained by setting j = det F.The learned term is convex up to a constant subtraction; the correction is affine and the growth terms are convex.Source: §B.3 · pp. 32–33

Growth (C6)

  • As J approaches zero or infinity, or ∥F∥F grows with J in a compact positive range, the analytic growth terms diverge.The logarithmic terms control J → 0+, the positive quadratic term controls J → ∞, and ε∥F∥2_F/2 controls large ∥F∥F at bounded J.Source: §B.3 · p. 33

C Sufficient lower bound for nonnegative energy

The sufficient lower bound reduces nonnegative-energy certification to a scalar bound in J. Interval and tail inequalities then establish the required bound for every J > 0 for the trained parameters.

Feature lower bound

  • Hadamard’s determinant inequality gives tu ≥ J2, yielding the featurewise bound zi ≥ kiJ^ηi without requiring simultaneous minima across directions.Because each feature increases in t and u, its minimum at fixed J occurs on tu = J2.Source: C Sufficient lower bound for… · p. 33

Scalar lower bound

  • The resulting scalar lower bound combines feature terms with logarithmic, quadratic, and deformation-norm growth contributions.Convexity of the resulting function is checked with sufficient inequalities; when those bounds are inconclusive, a sharper interval bound is used.Source: C Sufficient lower bound for… · p. 33

Interval audit

  • Positive margins from a 4000-interval audit establish the sufficient bound for trained parameters, up to floating-point accuracy, over every J > 0.The endpoint and tail inequalities cover the full domain, not only the grid points.Source: C Sufficient lower bound for… · p. 33

D Latent-space closure details

The latent-space construction maps retained POD coefficients to strain-informed primary coordinates, then deploys a decoder using primary and secondary modes. Its displacement error separates closure error from POD truncation, while the reduced tangent includes decoder curvature.

Input-informed construction

  • The input-informed construction finds a least-squares map from POD coefficients to strain and, assuming rank three, defines a three-column orthonormal primary block.The corresponding coordinate transformation maps the coefficients into primary modal amplitudes.Source: D Latent-space closure details · p. 34

Deployed decoder

  • The secondary block complements the primary block within the retained POD span, and the two blocks form the deployed decoder through an orthogonal basis change.The resulting basis spans the same subspace as the retained POD basis.Source: D Latent-space closure details · p. 34

Displacement error

  • The displacement error separates closure error within the retained span from POD truncation error, which increasing network capacity cannot recover.The two terms correspond to distinct approximation mechanisms.Source: D Latent-space closure details · p. 34

Projected residual tangent

  • Differentiating the projected residual adds a decoder-manifold curvature contribution that need not vanish at reduced equilibrium.The reduced tangent therefore includes a term beyond the projected FOM stiffness contribution.Source: D Latent-space closure details · p. 34

E Elementwise cubature conventions

The section specifies elementwise cubature conventions and how the residual and stress rules are assembled, fitted, and differentiated for the reduced models.

E Elementwise cubature conventions

  • Residual cubature samples element contributions across training states, stress cubature uses four stress components, and a truncated SVD compresses the sampled integrand matrix.Each element occupies a row of the sampled matrix.Source: E Elementwise cubature conventions · p. 34
  • The fitting system enforces the multiplier sum without duplicating a constant component already represented by retained modes, adding exact reproduction when needed.If the constant lies in the retained span, the final row is redundant; otherwise it enforces exact reproduction.Source: E Elementwise cubature conventions · p. 34
  • After pruning, neural fitting targets the full-mesh sum from uncompressed contributions, using scaled-softmax residual and stress losses initialized from sampled cubature weights.The final objective is integration accuracy because reproducing weights are not unique.Source: E Elementwise cubature conventions · p. 35
  • Tangent implementations differ by model: affine HPROM uses linearized equilibrium directions, HPROM–ANN differences partial-derivative blocks, and D-HPROM–ANN differences the complete direct stress map.The iterative variants avoid a new nonlinear microscopic solve for each perturbed strain.Source: E Elementwise cubature conventions · p. 35

F Capacity and feature-count controls

Capacity controls show that widening constrained cores changes their errors little, while an unconstrained model remains more accurate at matched capacity; 32-feature fits improve errors but fail the sufficient bound, so six-feature models remain deployed.

F Capacity and feature-count controls

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