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Dimensional hyperreduction of nonlinear finite element models via empirical cubature with manifold-adaptive weights

Joaquín A. Hernández, S. Ares de Parga, Riccardo Rossi

arXiv:2609.03068v1math.NAcs.CEcs.NA

TL;DR

Fixed-weight hyperreduction uses one weight vector across a nonlinear solution manifold, although adaptive weights may reduce the required integration points. The paper introduces MAW–ECM and input-informed manifold construction; across two benchmarks, manifold models reduce sampled points relative to linear ROMs, with further pruning supported by adaptive weights.

  • Problem

    Standard sampling-and-weighting methods require one weight vector to satisfy conditions across training states, including their full linear span, while the state lies on a nonlinear manifold.

    Source: §1.1 · p. 2, §1.2 · p. 2

  • Method

    MAW–ECM starts from a feasible fixed-weight rule and greedily prunes points through convex quadratic weight redistribution with local equality and positivity constraints.

    Source: §1.3 · p. 3

  • Results

    Across both benchmarks, manifold representations reduce integration points by factors of approximately 25 in the metamaterial problem and 8.6 in the damage problem relative to corresponding linear models.

    Source: §8 · p. 44

  • Takeaways & Limitations

    Graph regularization supports reliable regression of adaptive weight fields, while local redistribution removes most points in the inexpensive regime.

    Source: §8 · p. 44

    Both examples have two independent latent coordinates; extension to higher-dimensional spaces and to genuinely multidimensional loading histories remains to be investigated.

    Source: §8 · p. 45

Abstract

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Nonlinear-manifold reduced-order models for parametrized finite element problems can achieve substantial compression both in the number of generalized (latent) coordinates and, through sampling-and-weighting hyperreduction, in the number of sampled elements/integration points. Yet current sampling-and-weighting approaches employ weights that remain fixed over the solution manifold. We contend that this restriction leaves hyperreduction potential untapped: allowing the weights to vary continuously and nonlinearly with the latent coordinates can further decrease the number of sampled spatial entities. To exploit this possibility, we propose the Manifold-Adaptive-Weight Empirical Cubature Method (MAW-ECM). Starting from a feasible fixed-weight ECM rule, a greedy pruning strategy removes sampled entities through convex quadratic weight-redistribution problems enforcing local conditions and positivity. The method is assessed on two nonlinear benchmarks: homogenization of a metamaterial unit cell exhibiting negative incremental stiffness, and a history-dependent continuum-damage problem. In both cases, the nonlinear manifold is constructed from an initial linear compression followed by an input-informed identification of the latent coordinates as general linear combinations of the retained modal amplitudes, incorporating graph information when relevant to seek the intrinsic dimensionality of the solution manifold. We show that combining the nonlinear-manifold representation with MAW-ECM reduces the number of sampled integration points by more than two orders of magnitude relative to the corresponding standard linear reduced model. Relative to the fixed-weight manifold models alone, the adaptive weights eliminate approximately 80% of the remaining points in the homogenization benchmark and more than 97% in the damage benchmark, while essentially preserving their accuracy.

1. Introduction

Manifold HROMs pair nonlinear state representations with sampling-and-weighting hyperreduction, but existing rules use one weight vector across all training states. This section motivates allowing weights to vary over the manifold, presents MAW–ECM and input-informed latent-coordinate construction, and states the study’s scope.

1.1. Manifold HROMs using standard sampling-and-weighting hyperreduction

  • Fixed-weight hyperreduction requires one weight vector to satisfy conditions across training states, including their full linear span, even when some combinations do not lie on the solution manifold.The authors argue that this global linear construction does not fully exploit the nonlinear-manifold structure.Source: §1.1 · p. 2

1.2. Motivation for using adaptive weights

  • Adaptive weights vary with latent coordinates; a one-dimensional quadrature example shows one point can integrate every state when its weight adapts, versus at least (P + 1)/2 fixed-weight points.For f_q(x) = x^q on [0, 1], the adaptive weight at x = 1 is 1/(q + 1).Source: §1.2 · p. 2

1.3. Manifold-adaptive weights ECM

  • MAW–ECM extends SAW–ECM from discrete subspaces to a continuously parameterized manifold, using greedy pruning and graph regularization to redistribute positive weights smoothly.Each tentative point removal leads to a convex quadratic program with linear equality and positivity constraints; graph regularization is activated when needed to preserve positivity.Source: §1.3 · p. 3

1.4. Input-informed construction of the nonlinear manifold

  • The decoder first compresses states linearly, then identifies latent coordinates as linear combinations of retained modal coefficients informed by problem inputs and, when relevant, graph structure.The two benchmark-specific constructions identify the intrinsic dimensionality of their solution manifolds; the damage case uses loading-history graph information.Source: §1.4 · p. 4

1.5. Scope

  • The two benchmark-specific POD-based decoder constructions are not claimed to generalize as manifold-learning procedures, and the study restricts its assessment to low-dimensional parameter and latent spaces.Scalability of latent-graph construction to substantially higher-dimensional spaces is outside the study’s scope.Source: §1.5 · p. 4

2. Full-order finite element model

The full-order model expresses constrained finite-element equilibrium through internal and external forces, with history dependence carried incrementally. The section expands internal forces into Gauss-point contributions, identifies the terms targeted by cubature, and notes that the same rule can approximate volumetric outputs.

2. Full-order finite element model

  • The model solves for independent displacement degrees of freedom under affine kinematic constraints, with history-dependent responses also depending on previously converged internal variables.The external actions are assumed independent of the unknown displacement.Source: §2 · pp. 4–5, §2.2 · p. 6

2.1. Internal forces

  • The displacement-dependent nodal internal force is the hyperreduction target and is written as a sum of element and Gauss-point contributions.Each Gauss point has a positive physical integration weight and an associated internal-force density.Source: §2.1 · p. 5

2.2. External actions

  • External actions are assumed independent of displacement and are not hyperreduced; their constrained-space projection is expressed as a linear function of the input parameters.The input parameters control the amplitudes of the external forces.Source: §2.2 · p. 6

2.3. Volumetric outputs of interest

  • When a volumetric output is present, the hyperreduction cubature rule is also applied to its integral, using the same reduced integration-point set as for internal forces.For the metamaterial benchmark, the output is the volumetric average of the first Piola–Kirchhoff stress.Source: §2.3 · p. 6

3. Manifold hyperreduced-order model

The model combines a nonlinear decoder with tangent-space equilibrium projection and fixed-weight empirical cubature. It first compresses the state, then seeks a sparse positive integration rule that reproduces projected contributions at training states.

3.1. Decoder and encoder

  • The decoder maps latent coordinates to the state, and its approximate inverse encoder returns coordinates; the decoder’s Jacobian supplies the tangent space used in projected equilibrium.The explicit projection-based encoder is an exact left inverse on the approximation manifold.Source: §3.1 · p. 7, §3.2 · p. 8

3.2. Factorized decoder and identification of latent coordinates

  • The factorized decoder combines a low-rank linear basis with a nonlinear coefficient map, while latent coordinates are identified as linear combinations of modal coefficients rather than only selected leading modes.The decoder must be twice differentiable: its first derivative defines the tangent space and its second derivative enters the consistent Newton–Raphson linearization.Source: §3.2 · p. 7, §3.1 · p. 7

3.3. Tangent-space projection of the equilibrium equations

  • Tangent-space projection removes ambient-dimension online operations, leaving the full Gauss-point summation as the remaining mesh-dependent cost addressed by hyperreduction.The factorized decoder allows relevant quantities to be precomputed offline; the fixed-weight ECM rule also initializes the later adaptive rule.Source: §3.3 · p. 9

3.4. Fixed-weight cubature approximation

  • The fixed-weight cubature rule selects integration points and positive weights to reproduce projected internal forces at training states, and can also impose output-integral conditions.The rule may target equilibrium, an output alone, or both together using a stacked integrand.Source: §3.4 · pp. 9–10

3.5. Construction of the fixed-weight ECM rule

  • ECM constructs a sparse nonnegative rule by finding weights that integrate a low-dimensional basis of sampled integrands, rather than solving the NP-hard sparsity objective directly.ECM augments the basis to integrate domain volume exactly and produces a rule with a number of positive weights equal to the basis-column count.Source: §3.5 · pp. 10–11
  • The ECM training matrix is assembled from reconstructed displacement snapshots, their encoded latent coordinates, and Gauss-point quantities evaluated from those states.Sequential randomized SVD is used in the examples to build the basis from column blocks while reducing basis-construction cost.Source: §3.5 · p. 11

4. Cubature with manifold-adaptive weights

MAW cubature keeps integration-point locations fixed while assigning each retained point a positive weight field that varies over the nonlinear manifold. Its construction starts from a feasible fixed-weight ECM rule and greedily prunes points while preserving local integration conditions and regularity.

4.1. Problem statement and requirements

  • For history-dependent constitutive models, snapshots must remain in training-trajectory order because each constitutive update uses preceding-state internal variables.Stress reconstruction therefore proceeds in trajectory order.Source: §4.1 · p. 11
  • MAW assigns each retained Gauss point a positive weight field over the latent space while keeping its reference-configuration location fixed.The adaptive rule is intended to reproduce the fixed-weight rule’s integration conditions with the same accuracy.Source: §4.1 · pp. 11–12
  • The rule seeks the smallest support, exact state-wise integration conditions, nonnegative weights that preserve total volume, and smooth weight fields.These requirements are formulated as a constrained optimization problem.Source: §4.1 · p. 12

4.2. Initialization and local integration constraints

  • At each sampled latent state, local equality constraints are initialized from the fixed-weight ECM rule and combine projected integrands with invariant conditions.The invariant conditions include the all-ones vector to enforce the volume constraint at every state.Source: §4.2 · p. 13

4.3. Objective function

  • The objective combines the number of retained integration points with a regularizer that penalizes irregular variation of weight fields over the sampled manifold.The regularizer includes the latent graph Laplacian.Source: §4.3 · p. 14

4.4. Optimization problem and solution strategy

  • Pruning starts from the feasible fixed-weight ECM rule and removes points while redistributing weights to preserve local equalities, positivity, and regularity.The final common support is identified by the nonzero rows of the weight matrix.Source: §4.4 · p. 14

5. Greedy pruning strategy

This section describes MAW-ECM’s greedy pruning of an initial fixed ECM rule by testing integration-point removal under local exactness conditions and adaptive weight redistribution, using repeated candidate sweeps to mitigate path dependence and compact the rule. It then covers regression of sampled weights into continuous adaptive fields, which here does not enforce positivity or exact volume preservation.

5.1. Redistribution problem

  • Each tentative removal is accepted only if redistributed weights on the retained support satisfy local exactness constraints at every sampled latent state and meet admissibility conditions.A pruning trial removes one candidate point and seeks weights for the remaining support; if suitable weights exist, the removal is accepted.Source: §5.1 · p. 15

5.1.1. Regularized optimization problem

  • Redistribution minimizes changes from previously accepted weights while enforcing local equalities and positivity, with graph regularization coupling neighboring latent states and smoothing irregularities.The graph regularization becomes necessary when positivity is violated; its strength is controlled by αG.Source: §5.1.1 · pp. 15–16

5.1.2. Solution strategy

  • Algorithm 1 first solves uncoupled least-norm problems, then uses null-space active sets for positivity enforcement; a candidate is rejected if local rank conditions fail.Pruning cannot proceed below the maximum number of independent local constraints.Source: §5.1.2 · pp. 16–18
  • Most weight fields are removed without graph-coupled positivity enforcement, which typically becomes necessary only near the final sparse solution.The authors attribute this to volume preservation: after a point is removed, remaining weights tend to increase on average, favoring positivity.Source: §5.1.2 · p. 19

5.2. Candidate sweep and branch selection

  • Algorithm 2 explores up to ntry feasible pruning branches, testing candidates with smaller average weights first and selecting the feasible trial with the lowest graph Dirichlet energy when relevant.The energy ranking matters only when positivity enforcement modifies the redistribution; otherwise the returned energy is zero and no further branches are explored.Source: §5.2 · p. 19

5.3. Global pruning loop

  • Algorithm 3 repeatedly applies candidate sweeps to progressively transform the initial fixed ECM rule into a more compact adaptive cubature rule.The hierarchy proceeds from a tentative pruning step to candidate exploration and then the global pruning loop.Source: §5 · p. 14, §5.3 · pp. 19–20
  • The global loop repeatedly prunes the common Gauss-point support while preserving local exactness, stopping when no further feasible pruning step is found.Its retained-point weight matrix supplies samples of the adaptive fields over the latent space.Source: §5.3 · pp. 20–21

5.4. Regression of the adaptive weight fields

  • Generic regression does not guarantee positive weights or exact total-volume preservation, particularly outside the training data’s convex hull.The numerical examples use the unconstrained regression techniques employed for the nonlinear decoder; constraint-preserving regression is left for future study.Source: §5.4 · p. 21

5.5. Summary

  • Box 5.1 summarizes the complete offline construction of the MAW–ECM rule.Source: §5.5 · p. 21

6. Benchmark 1: homogenization of a metamaterial unit cell

The first benchmark applies MAW–ECM to a finite-strain metamaterial unit cell under first-order homogenization. The setup constructs a nonlinear decoder and adaptive integration rule from finite-element training trajectories.

6.1. Problem setting

  • The metamaterial cell has 1,315 bilinear quadrilateral elements and 5,260 integration points.The cell is deformed through imposed boundary displacements.Source: §6.1 · p. 21
  • The benchmark uses a large-strain, plane-strain homogenization model with an isotropic compressible Neo–Hookean solid.Its two input parameters represent macroscopic normal and shear deformation components.Source: §6.1 · pp. 21–22
  • Snap-through of curved ligaments produces negative incremental stiffness under compression and an asymmetric tensile–compressive response.These behaviors make the cell a challenging model-reduction benchmark.Source: §6.1 · p. 21
  • First-order homogenization is used as a practical illustration, but its macroscopic kinematics are too restrictive to represent the full complexity of bistable unit cells.The paper notes that richer nonlinear multiscale formulations may require additional microstructural fields.Source: §6.1 · p. 21
  • The workflow builds a decoder and latent coordinates from training trajectories, compresses the integrand matrix for a fixed-weight ECM rule, then prunes and regresses adaptive weights.The resulting online rule keeps a common support while its weights vary over the latent manifold.Source: §6.1 · p. 22

6.2. Training data

The metamaterial training set spans a symmetric two-parameter domain and uses two zig-zag loading trajectories. Meeting the stress-accuracy criterion with a standard HROM requires 52 displacement modes and an ECM rule retaining 1,340 integration points.

6.2. Training data

  • The training domain varies compression/stretching over [−0.42, 0.42] and shear over [−0.05, 0.05], with 4851 parameter states.The lower compression bound is the most compressed configuration before self-contact occurs.Source: §6.2 · p. 23
  • Two continuous zig-zag trajectories traverse the Cartesian training grid, covering nonnegative and nonpositive shear regions.The domain is symmetric even though opposite-sign parameter inputs can produce stress levels differing by up to a factor of 25.Source: §6.2 · p. 23

6.3. Standard HROM

  • A normalized maximum homogenized-stress error below 0.1% over the training set requires εD = 10−6, 52 displacement modes, and εA = 10−4.This was the least expensive configuration meeting the stated accuracy criterion.Source: §6.3 · p. 23
  • Small-singular-value modes are needed to capture internal buckling details governing the low-stress compressive branch, despite contributing little to snapshot energy.The tight displacement tolerance reflects the markedly different tensile and compressive response scales.Source: §6.3 · p. 24
  • The standard HROM retains 1,340 of 5,260 integration points and requires 52 displacement modes despite the benchmark having two input parameters.The retained points are approximately 25% of the original set.Source: §6.3 · p. 24

6.4. Manifold HROM

The manifold HROM identifies two latent coordinates from the input parameters, reconstructs 50 slave coordinates with a nonlinear closure, and builds a fixed-weight ECM rule. It reduces coordinates and integration points relative to the standard HROM, with an approximately fourfold increase in homogenized-stress error.

6.4.1. Identification of latent coordinates

  • For this static hyperelastic benchmark, the solution manifold has dimension two, and the latent coordinates associated with training states are numerically identified with the input parameters.The identification relies on a single-valued displacement parametrization by the prescribed input.Source: §6.4.1 · pp. 24–25
  • Because the latent state can be obtained directly from the prescribed input, the reduced equilibrium equations need not be solved online; adaptive cubature is needed only for homogenized stresses.The equilibrium-residual integrands are therefore excluded from the cubature training problem.Source: §6.4.1 · p. 25

6.4.2. Nonlinear closure

  • An anisotropic Gaussian RBF closure with quadratic polynomial terms maps two latent coordinates to 50 slave coordinates, using 1,540 centers from 4,851 snapshots.The relative reconstruction error is 4.30·10−6 at retained centers and 1.03·10−4 over the complete dataset.Source: §6.4.2 · p. 26

6.4.3. Fixed-weight ECM

  • The fixed-weight, output-only ECM rule uses four stress integration conditions per state and retains 54 Gauss points across 48 finite elements.It is constructed for homogenized stress because the latent state is prescribed directly by the macroscopic input.Source: §6.4.3 · p. 27
  • The manifold HROM reduces independent coordinates from 52 to 2 and integration points from 1,340 to 54, while its homogenized-stress error is approximately four times the standard HROM error.The manifold rule has 54 points; the standard HROM has 1,340.Source: Table 1 · p. 27, §6.4.3 · p. 28

6.5. Manifold-adaptive weight ECM

MAW–ECM prunes a feasible fixed-weight rule by redistributing positive weights over the sampled manifold. The resulting adaptive rules remain compact across initializations, while graph regularization yields smoother weight fields and better stress predictions.

6.5. Manifold-adaptive weight ECM

  • MAW–ECM starts from a feasible fixed-weight ECM support and greedily removes points while redistributing weights over sampled manifold states under local exactness and positivity conditions.The output-only rule has four PK1-stress conditions plus a volume constraint, giving a lower bound of five points.Source: §6.5 · p. 28

6.5.1. Influence of the fixed-weight initialization and pruning efficiency

  • Starting rules of 54–187 points are pruned to 9 or 10, nearly independent of initialization and about twice the lower bound of 5.The pruning process first reaches approximately 18–20 points before positivity violations appear.Source: Table 2 · p. 28, §6.5.1 · pp. 28–29
  • Increasing the initial support from 54 to 87 points leaves runtime below one minute, while always activating regularization raises runtimes from 50 seconds to about 20 minutes and from 44 seconds to over two hours, respectively.The two-stage strategy’s cost is governed mainly by pruning iterations that activate graph-regularized positivity enforcement.Source: §6.5.1 · p. 29

6.5.2. Adaptive cubature rule and weight-field regression

  • In the 10-point rule, the dominant weight rises from about 85% to 87% of total volume, while the remaining nine weights carry the nontrivial variation across latent coordinates.The reconstructed weight fields are smooth, and their RBF interpolation error is 3.89 × 10^-9 over the dataset.Source: §6.5.2 · pp. 29–30

6.5.3. Effect of graph regularization

  • Graph regularization reduces local fluctuations in sampled adaptive weights and improves the quality of the resulting stress predictions.The unregularized rule’s stress deviations are concentrated in compression, including the negative-stiffness response associated with progressive buckling of the internal beams.Source: §6.5.3 · pp. 30, 32, 33, Figure 10 · p. 32
  • With graph regularization, reducing support from 54 to 10 points increases homogenized-stress error by a factor of about 2.4; without it, the factor reaches about 18.The cited stress comparisons average errors over two training trajectories; unregularized deviations are concentrated in compression, where the regularized prediction stays close to the FE response.Source: §6.5.3 · pp. 30, 32, Table 3 · p. 31

7.1. Problem setting

The damage benchmark models a quarter plate with a circular hole under traction, using displacement as the output state. Models are trained on monotonic tension and tested on a history with unloading and reloading.

7.1. Problem setting

  • The benchmark is a small-strain, plane-strain plate with a circular hole governed by isotropic damage with linear hardening.The modeled quarter has symmetry on its left and lower boundaries, and traction amplitude µ is the input.Source: §7.1 · p. 33, Figure 11 · p. 33
  • The output is the unconstrained nodal displacement state, and constitutive memory means the response cannot be parameterized by current load µ alone.The reduced equilibrium equations must recover the structural response.Source: §7.1 · p. 33
  • Training uses a monotonic tensile path; validation tests generalization on a trajectory with unloading and reloading.The training path reaches µmax = 70 MPa over 1000 increments; the test trajectory has 1500 increments.Source: Figure 11 · p. 33, §7.1 · p. 34

7.2. Standard HROM

The standard HROM uses a linear displacement basis and a fixed-weight ECM rule for the damage problem. It retains 2370 Gauss points and achieves low displacement errors on both the monotonic training path and the cyclic test.

7.2. Standard HROM

  • The displacement snapshots are decomposed into an undamaged linear contribution and a nonlinear correction, with the latter represented by a truncated SVD basis.No unloading, reloading, or load-reversal trajectories are included among the training snapshots.Source: §7.2 · p. 34
  • The displacement basis separates a single linear reference mode from 15 orthogonal damage-correction modes, for a total dimension of 16.Snapshots from the undamaged regime form the linear basis; the nonlinear snapshots are projected onto its orthogonal complement before truncation.Source: §7.2 · p. 34
  • The standard HROM retains 2370 of 22050 Gauss points, about 10.7% of the original set, concentrated mainly where damage is most severe.The fixed ECM rule distributes these points over 1140 elements.Source: §7.2 · p. 35, Figure 13 · p. 35
  • Training and cyclic-test relative displacement errors are 1.38 · 10^-4 and 1.28 · 10^-4, respectively.The test includes unloading, reloading, and an unseen compression regime.Source: §7.2 · pp. 35–36, Figure 14 · p. 36

7.3. Manifold HROM

The manifold HROM compresses the damage response to two master coordinates by identifying a smooth scalar coordinate for the nonlinear trajectory and fitting a nonlinear closure. It reduces fixed-rule sampling from 2370 to 277 points, with errors increasing by about an order of magnitude relative to the standard HROM.

7.3.1. Identification of latent variables

  • A smoothest linear combination of normalized nonlinear amplitudes yields a monotone, injective scalar coordinate for the normalized nonlinear training trajectory.The method uses a graph Laplacian over ordered load samples; the identified coordinate succeeds where the individual components fail the monotonicity requirement.Source: §7.3.1 · p. 37, Figure 15 · p. 38

7.3.2. Nonlinear closure

  • The nonlinear closure regresses normalized slave amplitudes against the normalized nonlinear coordinate, whose constancy during elastic unloading and reloading leaves the tangent basis unchanged on those branches.Localized oscillations and kinks near the most damaged states motivate smoothing the regression with cubic least-squares B-splines.Source: §7.3.2 · p. 38, Figure 16 · p. 39

7.3.3. Fixed-weight ECM

  • The manifold HROM reduces independent coordinates from 16 to 2 and integration points from 2370 to 277, with compression factors of 8 and approximately 8.6, respectively.Its displacement errors increase by approximately one order of magnitude over the standard HROM but remain nearly identical between training and generalization; quadratic Newton convergence was observed at every increment.Source: §7.3.3 · p. 39, Table 4 · p. 39
  • The normalized nonlinear coordinate stays constant during elastic unloading and reloading, then increases as damage progresses, paralleling the local internal variable rϵ.The paper interprets it as a structural-scale counterpart of that local variable.Source: §7.3.3 · p. 39

7.4. Manifold-adaptive weight ECM

MAW–ECM prunes fixed-weight rules to seven or eight points while enforcing local integration conditions and positivity. In the damage benchmark, the seven-point adaptive rule essentially preserves the fixed-weight manifold HROM’s accuracy, despite small localized deviations.

7.4. Manifold-adaptive weight ECM

  • The pruning constraints combine state-dependent projected internal forces, invariant elastic-branch conditions, and a volume constraint, giving a minimum of 5 points.The invariant conditions preserve exact projected internal forces over the virgin elastic branch, where the adaptive rule assigns the same weight vector across states.Source: §7.4 · p. 40

7.4.1. Influence of the fixed-weight initialization and pruning efficiency

  • The final seven-point rules retain a similar spatial pattern across initializations: points occupy the undamaged region, the hole vicinity, and the damaged region.The points near the hole lie where damage nucleates and stress concentration is highest.Source: §7.4.1 · p. 41, Figure 18 · p. 42
  • Graph regularization produces smoother sampled weight variation, whereas unregularized pruning yields abrupt changes that are less suitable for regression.Between 80% and 96% of points were eliminated without activating positivity enforcement or graph regularization.Source: §7.4.1 · p. 41
  • As pruning removes points, the retained weights vary more strongly with the latent coordinate to compensate for the discarded integration points.The plotted feasible rules contain 14, 12, 10, and 8 points along the pruning path.Source: §7.4.1 · p. 42, §7.4.2 · p. 44

7.4.2. Online accuracy assessment

  • Pruning the 277-point fixed rule to 7 points essentially preserves the fixed-weight manifold HROM’s displacement accuracy.Across initial rules of 97, 277, and 534 points, the final adaptive rules contain 7–8 points, only two or three above the lower bound of 5.Source: §7.4 · p. 40, §7.4.1 · p. 41, §7.4.2 · pp. 42–43
  • The adaptive rule reproduces the macroscopic cyclic stress–strain response with no visible discrepancy, despite a small latent-coordinate departure near damage onset.The nonlinear coordinate rejoins the FE reference once damage evolution is established; the weight-regression deviation has practically no influence on global displacement error.Source: §7.4.2 · p. 43
  • At the seven sampled points, FE and adaptive-HROM stress histories practically coincide through the test, apart from a small discrepancy at point 6 during the initial damage transition.The agreement extends beyond the trained tensile branch to unloading and reverse loading.Source: §7.4.2 · pp. 43–44

8. Concluding remarks

Across both benchmarks, nonlinear-manifold models with MAW–ECM substantially reduce integration-point counts while essentially preserving fixed-weight manifold-HROM accuracy. The authors identify local constraint-space dimension as a key determinant of final support and note several scope and implementation limitations.

8. Concluding remarks

  • MAW–ECM removes approximately 80% of remaining metamaterial points and more than 97% in damage (277 to 7–8), essentially preserving fixed-weight manifold-HROM accuracy.In both benchmarks, final support depends weakly on the size of the initial fixed ECM support and lies only a few points above local-condition lower bounds.Source: §8 · p. 44
  • Metamaterial state coordinates fell from 52 to 2 (26×) and fixed cubature from 1340 to 54 points (approximately 25×); damage reductions were 8× and approximately 8.6×, respectively.The authors suggest that adaptive-rule support is governed primarily by local constraint-space dimension rather than the global span of sampled integrands.Source: §8 · p. 44
  • Graph regularization supports reliable online weight regression, while local redistribution with positivity preserved handles most pruning; in one metamaterial case pruning time fell from over two hours to under one minute.The graph-coupled stage is activated only after positivity enforcement succeeds.Source: §8 · pp. 44–45
  • Both benchmarks use only two independent latent coordinates, and graph-based history-coordinate identification was demonstrated only on an ordered one-dimensional damage trajectory.Extension to genuinely multidimensional loading histories remains to be investigated.Source: §8 · p. 45
  • After pruning, generic regression and extrapolation do not automatically preserve weight positivity or invariant integration conditions; dynamics require carefully identified invariant constraints.The effects of the adaptive-weight tangent contribution on robustness also need systematic investigation, despite no convergence difficulties in the reported experiments.Source: §8 · p. 45

Appendix A. Consistent linearization of the MAW–ECM equilibrium equations

The appendix derives the consistent Newton–Raphson linearization of the MAW–ECM reduced equilibrium equations. Its tangent includes standard constitutive, manifold-curvature, and adaptive-weight variation contributions.

Appendix A. Consistent linearization of the MAW–ECM equilibrium equations

  • The reduced equilibrium equations are first recast in residual form before the tangent is obtained by differentiating with respect to latent coordinates.The appendix identifies the residual form as a recasting of Eq. 54.Source: Appendix A · p. 46
  • The consistent tangent combines constitutive linearization, manifold curvature, and variation of the adaptive cubature weights.The weight-variation contribution arises because the cubature weights depend on latent coordinates.Source: Appendix A · p. 46
  • The adaptive-weight tangent term uses derivatives of each weight field with respect to latent coordinates, obtained by differentiating its regression model.Source: Appendix A · pp. 46–47
  • The manifold-curvature contribution vanishes when the decoder is affine because its Jacobian is constant.Source: Appendix A · p. 47
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