Learning Topology-Aware Representations for Mechanics-Relevant TPMS Metamaterials

Songlin Duan; Shuo Yang; Yujia Zheng; Tobin Filleter

cs.LGOct 14, 2026

Learning Topology-Aware Representations for Mechanics-Relevant TPMS Metamaterials

Authors: Songlin Duan, Shuo Yang, Yujia Zheng, Tobin Filleter

Organizations: University of Toronto, Canada · University of Illinois Urbana-Champaign, USA

Abstract

Learning informative representations of mechanical Triply Periodic Minimal Surface (TPMS) metamaterials is important for mechanics analysis and structural design, yet existing methods have rarely explicitly supervised learned representations with information about global structural connectivity that is relevant to mechanical response. We propose a topology-aware representation learning framework that integrates persistent homology into a DMAE backbone to explicitly encode global structural connectivity. A persistence-guided masking strategy prioritizes patches associated with persistent topological features, while an auxiliary persistent-homology decoder encourages the latent representation to retain information about topological feature counts across connected components, tunnels, and voids. We evaluate the learned representations on two complementary TPMS datasets using family classification, mechanical-property classification, unsupervised clustering, and regression. Across the two datasets, our method demonstrates strong performance in both structural family and mechanical-property classification, improving family classification by up to 3.9% while achieving the best linear-probe performance. More notably, unsupervised property-clustering performance improves by up to 26.7% and 51.5% across the two datasets, while strong regression performance is retained with $R^2=0.993$ . These results establish persistent topology as an effective inductive bias for representation learning of complex three-dimensional mechanical metamaterials.

Figures & tables

Figure 1: Localized topological changes can induce disproportionate changes in the mechanical response of TPMS structures. Starting from the reference TPMS (left), the Cut (middle) introduces a physical crack at an existing structural connection, whereas the Merge (right) joins two separated tunnels into a single connected passage. Despite only small changes in solid volume fraction ( $\Delta V=-0.421\%$ and $-1.240\%$ , respectively), both modifications produce clear changes in the persistence diagrams (PDs, summaries of topological features) and directional elastic responses. The Cut yields maximum reductions of $9.75\%$ in $E_{z}$ and $10.01\%$ in $G_{yz}$ , whereas the Merge reduces $E_{x}$ by $6.63\%$ and $G_{xy}$ by $6.85\%$ . Additional topological perturbations, full directional stiffness changes, and detailed analysis are provided in Appendix A.1 and Fig. S1 .

Representative work	Scope / focus					Topology-aware
	Meta.	TPMS	3D struct.	Mech. rel.	Rep. learn.	TDA / PH	Topo head	Topo mask
General metamaterial modeling
GraphMetaMat ( Maurizi et al., 2025 )	✓	–	✓	✓	–	–	–	–
Composable Energy Surrogates ( Beatson et al., 2020 )	✓	–	✓	✓	–	–	–	–
3D Metamaterial Force Control ( Wan et al., 2022 )	✓	–	✓	✓	–	–	–	–
Metamaterial representation & inverse design

Table 1: Comparison of representative work across metamaterial and TPMS modeling, mechanics-aware methods, representation learning model, and topology-aware structural learning.

Figure 2: Overview of the proposed topology-aware representation learning framework. Starting from voxelized TPMS signed-distance fields, the framework integrates persistence-guided masking and a persistent-homology prediction objective into masked representation learning to preserve topology-relevant information in the latent representation.

Model	MetaDB		Interpolation TPMS
	$\mathrm{F}_{\mathrm{Lin}}\uparrow$	$\mathrm{F}_{k\mathrm{NN}}\uparrow$	$\mathrm{Acc}_{\mathrm{Lin}}^{\mathrm{Str}}\uparrow$	$\mathrm{Acc}_{\mathrm{Lin}}^{\mathrm{Rel}}\uparrow$	$\mathrm{Acc}_{k\mathrm{NN}}^{\mathrm{Str}}\uparrow$	$\mathrm{Acc}_{k\mathrm{NN}}^{\mathrm{Rel}}\uparrow$
General representation learning
MVAE	0.009	0.581	0.038	0.051	0.446	0.559
SVAE	0.479	0.955	0.638	0.663	0.787	0.914
VICReg	0.913	0.853	0.263	0.299	0.370	0.456
I-JEPA	0.912	0.902	0.403	0.446	0.497	0.618

Table 2: Task 1 – Family classification.

\mathrm{F}_{\mathrm{Lin}}

and

\mathrm{F}_{k\mathrm{NN}}

denote macro-F1 scores on MetaDB using a linear probe and

k

NN classifier, respectively. For Interpolation TPMS,

\mathrm{Acc}_{\mathrm{Lin}}^{\mathrm{Str/Rel}}

and

\mathrm{Acc}_{k\mathrm{NN}}^{\mathrm{Str/Rel}}

denote strict and relaxed accuracies with respect to the dominant-family labels. Higher is better for all metrics.

Figure 3: Summary of mechanical-property downstream performance on (a) MetaDB and (b) Interpolation TPMS. The radar plots provide a compact comparison of frozen latent representations:

\mathrm{F1}_{k\mathrm{NN}}

for

C_{11}

property classification (Task 2),

\eta^{2}

\mathrm{Acc}_{\mathrm{Match}}

, and

\mathrm{F}_{\mathrm{Macro}}

for

C_{11}

property clustering (Task 3), and

R^{2}

for

C_{11}

property regression (Task 4). Higher is better for all metrics shown. Detailed numerical results for all metrics and baselines are provided in the Appendix (Tables S2 , S3 , and S4 ).

Figure 4: Ablation study of persistence-guided masking and the auxiliary PD prediction head. (a) reports representative metrics from Tasks 1, 2, and 4, while (b) reports the three Task 3 clustering metrics. We compare the DMAE baseline without either topology-aware mechanism ( w/o mask head ), variants removing either the persistence-guided mask ( w/o mask ) or PD head ( w/o head ), and the full model using both mechanisms ( full ). Complete numerical results across all evaluation metrics and both datasets are provided in Appendix Table S5 . Bars show the mean over five seeds, with error bars indicating the corresponding standard deviation. Higher is better for all metrics.

Figure 5: Direct latent-space alignment on (a) MetaDB and (b) Interpolation TPMS. PD-Score,

H_{1}

-Count,

H_{2}

-Count, and

C_{11}

-Property measure the correspondence between pairwise latent distances and the persistence-guided masking distribution, persistence-feature counts, and mechanical-property differences, respectively. Higher Spearman correlation indicates stronger alignment.

Appendix figures & tables12 assets

Supplementary material from the paper’s appendix.

Appendix

Figure S1: Topological perturbations and their mechanical responses. Five geometries are compared: the baseline structure and four locally modified variants (Cut, Single hole, Double separate, and Merged). The top row shows the corresponding geometries and localized modifications, the second row presents their persistence diagrams for $H_{0}$ , $H_{1}$ , and $H_{2}$ , and the third row shows the directional Young’s modulus surfaces. The table summarizes the relative changes in solid volume fraction and the six principal elastic moduli with respect to the baseline.

Figure S2: Persistence masking-guided PD-score distributions under localized topological perturbations. Normalized PD scores are projected onto the baseline structure and four locally modified variants. Warmer colors indicate regions with larger PD scores and therefore higher probabilities of being selected by the persistence-guided masking strategy.

Structure	Occupied voxels	Occupancy (%)	$\mathbf{H_{0}}$	$\mathbf{H_{1}}$	$\mathbf{H_{2}}$
Baseline	162,042	18.315	373	2190	1150
Cut	161,520	18.256	370 ( $-3$ )	2194 ( $+4$ )	1153 ( $+3$ )
Single hole	181,533	20.518	503 ( $+130$ )	2012 ( $-178$ )	1111 ( $-39$ )
Double separate	180,767	20.432	498 ( $+125$ )	1982 ( $-208$ )	1111 ( $-39$ )
Merged	180,465	20.398	498 ( $+125$ )	1987 ( $-203$ )	1115 ( $-35$ )

Appendix

Table S1: Persistent-homology feature counts for the controlled topological perturbations. The table reports the occupied voxel count, voxel occupancy, and the numbers of finite persistence features in

H_{0}

H_{1}

, and

H_{2}

. Values in parentheses denote the change relative to the baseline. All structures contain one infinite

H_{0}

feature and no infinite

H_{1}

H_{2}

features.

Figure S3: Overview of the geometric origin, mechanical characteristics, and representative applications of TPMS shell metamaterials. TPMS shell architectures originate from smooth periodic minimal surfaces and exhibit interconnected geometries that enable favorable mechanical performance and multifunctional applications, including lightweight structures, thermal management, and catalysis ( Wang et al., 2025 ; Wang et al., 2022 ; fly123, 2014 ; Basavalingappa et al., 2017 ) .

Figure S4: Representative TPMS structures with diverse persistent-homology signatures. Twenty TPMS samples were selected to span distinct persistence-diagram characteristics. Each voxelized geometry is shown alongside its corresponding persistence diagram, with the selected set exhibiting different combinations of $H_{0}$ , $H_{1}$ , and $H_{2}$ feature counts as well as substantial variation in their birth–death distributions. This selection provides a representative visualization of the broad topological diversity present in the TPMS dataset.

Figure S5: Representative finite-element homogenization example. A representative TPMS structure is shown together with its homogenized elastic response obtained from the voxel-based finite-element analysis. The corresponding $6\times 6$ effective stiffness matrix $\mathbf{C}$ summarizes the anisotropic elastic properties of the structure, while the three-dimensional directional Young’s modulus surface $E(\mathbf{n})$ , derived from the compliance matrix $\mathbf{S}=\mathbf{C}^{-1}$ , visualizes the variation in effective stiffness with loading direction.

Model	MetaDB				Interpolation TPMS
	$\mathrm{Acc}_{k\mathrm{NN}}\uparrow$	$\mathrm{F}_{k\mathrm{NN}}\uparrow$	$\mathrm{Acc}_{\mathrm{Lin}}\uparrow$	$\mathrm{F}_{\mathrm{Lin}}\uparrow$	$\mathrm{Acc}_{k\mathrm{NN}}\uparrow$	$\mathrm{F}_{k\mathrm{NN}}\uparrow$	$\mathrm{Acc}_{\mathrm{Lin}}\uparrow$	$\mathrm{F}_{\mathrm{Lin}}\uparrow$
General representation learning
MVAE	0.509	0.509	0.631	0.631	0.719	0.717	0.512	0.511
SVAE	0.766	0.765	0.793	0.793	0.863	0.863	0.698	0.697
VICReg	0.853	0.853	0.913	0.913	0.799	0.801	0.835	0.837
I-JEPA	0.902	0.902	0.911	0.912	0.831	0.833	0.863	0.864

Appendix

Table S2: Task 2 – Low/medium/high

C_{11}

elastic-property classification.

\mathrm{Acc}

denotes balanced accuracy and

\mathrm{F}

denotes macro-F1. The subscripts

k\mathrm{NN}

and

\mathrm{Lin}

correspond to the

k

NN and linear probes, respectively. Higher is better for all metrics.

Model	MetaDB			Interpolation TPMS
	$\eta^{2}\uparrow$	$\mathrm{Acc}_{\mathrm{Match}}\uparrow$	$\mathrm{F}_{\mathrm{Macro}}\uparrow$	$\eta^{2}\uparrow$	$\mathrm{Acc}_{\mathrm{Match}}\uparrow$	$\mathrm{F}_{\mathrm{Macro}}\uparrow$
General representation learning
MVAE	0.036	0.044	0.041	0.014	0.372	0.367
SVAE	0.127	0.519	0.436	0.049	0.399	0.394
VICReg	0.068	0.457	0.457	0.020	0.382	0.377
I-JEPA	0.014	0.393	0.291	0.081	0.370	0.366

Appendix

Table S3: Task 3 –

C_{11}

property clustering.

\eta^{2}

measures the fraction of continuous

C_{11}

variation explained by the latent clusters.

\mathrm{Acc}_{\mathrm{Match}}

and

\mathrm{F}_{\mathrm{Macro}}

evaluate agreement with the discrete property groups after optimal cluster-to-class matching. Higher is better for all metrics.

Model	MetaDB			Interpolation TPMS
	$\mathrm{MAE}\downarrow$	$\mathrm{RMSE}\downarrow$	$R^{2}\uparrow$	$\mathrm{MAE}\downarrow$	$\mathrm{RMSE}\downarrow$	$R^{2}\uparrow$
General representation learning
MVAE	0.049	0.078	0.246	0.039	0.050	0.110
SVAE	0.028	0.051	0.471	0.024	0.031	0.616
VICReg	0.011	0.049	0.903	0.010	0.014	0.935
I-JEPA	0.010	0.048	0.907	0.007	0.010	0.965

Appendix

Table S4: Task 4 –

C_{11}

property regression. MAE measures the average absolute prediction error, RMSE places greater emphasis on large errors, and

R^{2}

measures the fraction of

C_{11}

variation explained by the regression prediction. Lower is better for MAE and RMSE, while higher is better for

R^{2}

Interpolation TPMS
	Task 1		Task 2	Task 3			Task 4
Ablation	$\mathrm{Acc}^{\mathrm{Str}}_{k\mathrm{NN}}\uparrow$	$\mathrm{Acc}^{\mathrm{Rel}}_{k\mathrm{NN}}\uparrow$	$F_{k\mathrm{NN}}\uparrow$	$\eta^{2}\uparrow$	$\mathrm{ARI}\uparrow$	$\mathrm{NMI}\uparrow$	$R^{2}_{C_{11}}\uparrow$	$R^{2}_{C_{21}}\uparrow$
w/o mask & head	$0.8191\pm 0.0126$	$0.9324\pm 0.0061$	$0.8710\pm 0.0068$	$0.2721\pm 0.0060$	$0.138\pm 0.019$	$0.124\pm 0.020$	$\mathbf{0.9925\pm 0.0003}$	$0.9901\pm 0.0002$
w/o mask	$0.8218\pm 0.0077$	$0.9358\pm 0.0069$	$0.8845\pm 0.0062$	$0.4045\pm 0.0317$	$0.196\pm 0.041$	$0.191\pm 0.046$	$0.9925\pm 0.0005$	$\mathbf{0.9964\pm 0.0003}$
w/o head	$0.8232\pm 0.0155$	$0.9365\pm 0.0115$	$0.8682\pm 0.0062$	$0.2508\pm 0.0328$	$0.116\pm 0.040$	$0.116\pm 0.030$	$0.9917\pm 0.0002$	$0.9961\pm 0.0003$
Full	$\mathbf{0.8309\pm 0.0074}$	$\mathbf{0.9391\pm 0.0073}$	$\mathbf{0.8870\pm 0.0049}$	$\mathbf{0.4116\pm 0.0363}$	$\mathbf{0.203\pm 0.015}$	$\mathbf{0.213\pm 0.013}$	$0.9923\pm 0.0006$	$0.9962\pm 0.0003$

Appendix

Table S5: Ablation study on Interpolation TPMS and MetaDB. Results are reported as mean

\pm

standard deviation over five random seeds. For Interpolation TPMS, Task 1 reports strict and relaxed

k

NN family-classification accuracy; for MetaDB, Task 1 reports macro-F1 from the linear and

k

NN probes. Task 2 reports

F_{k\mathrm{NN}}

for

C_{11}

property classification; Task 3 reports

\eta^{2}

, ARI, and NMI for

C_{11}

property clustering; and Task 4 reports

R^{2}

for

C_{11}

and the full stiffness tensor

C_{21}

. Higher is better for all metrics.

		Task 1: Family classification				Task 2: $C_{11}$ classification		Task 3: $C_{11}$ clustering			Task 4: Regression
Parameter	Value	$\mathrm{Acc}^{\mathrm{Str}}_{\mathrm{Lin}}\uparrow$	$\mathrm{Acc}^{\mathrm{Rel}}_{\mathrm{Lin}}\uparrow$	$\mathrm{Acc}^{\mathrm{Str}}_{k\mathrm{NN}}\uparrow$	$\mathrm{Acc}^{\mathrm{Rel}}_{k\mathrm{NN}}\uparrow$	$F_{k\mathrm{NN}}\uparrow$	$F_{\mathrm{Lin}}\uparrow$	$\eta^{2}\uparrow$	$\mathrm{Acc}_{\mathrm{Match}}\uparrow$	$F_{\mathrm{Macro}}\uparrow$	$R^{2}_{C_{11}}\uparrow$	$R^{2}_{C_{21}}\uparrow$
Mask ratio $\rho$	0.55	0.8376	0.8484	0.8275	0.9317	0.8869	0.9218	0.3901	0.5459	0.5349	0.99225	0.99628
	0.65	0.8606	0.8750	0.8272	0.9360	0.8846	0.9157	0.3134	0.5493	0.5473	0.99185	0.99636
	$0.75^{\dagger}$	0.8764	0.8879	0.8347	0.9396	0.8793	0.9075	0.3166	0.5508	0.5465	0.99190	0.99633
	0.85	0.8908	0.9023	0.8491	0.9450	0.8728	0.9108	0.1500	0.4409	0.4412	0.99204	0.99613
$\lambda_{\mathrm{PD}}$	0.01	0.8872	0.8994	0.8279	0.9310	0.8778	0.9210	0.0972	0.4518	0.4529	0.99184	0.99610

Appendix

Table S6: Parameter sensitivity analysis on the Interpolation TPMS dataset. We evaluate the sensitivity to masking ratio

\rho

, PD-loss weight

\lambda_{\mathrm{PD}}

, and patch size

p

. For the masking-ratio sweep,

\lambda_{\mathrm{PD}}=0.02

and

p=4

are fixed; for the PD-loss-weight sweep,

\rho=0.75

and

p=4

are fixed; and for the patch-size sweep,

\rho=0.75

and

\lambda_{\mathrm{PD}}=0.05

are fixed. Bold values indicate the best performance within each sensitivity block.

\dagger

denotes the value adopted in the main configuration.

		Task 1: Family classification		Task 2: $C_{11}$ classification				Task 3: $C_{11}$ clustering			Task 4: Regression
Parameter	Value	$F_{\mathrm{Lin}}\uparrow$	$F_{k\mathrm{NN}}\uparrow$	$\mathrm{BA}_{\mathrm{Lin}}\uparrow$	$F_{\mathrm{Lin}}\uparrow$	$\mathrm{BA}_{k\mathrm{NN}}\uparrow$	$F_{k\mathrm{NN}}\uparrow$	$\eta^{2}\uparrow$	$\mathrm{Acc}_{\mathrm{Match}}\uparrow$	$F_{\mathrm{Macro}}\uparrow$	$R^{2}_{C_{11}}\uparrow$	$R^{2}_{C_{21}}\uparrow$
$\lambda_{\mathrm{PD}}$	0.005	0.9945	0.9844	0.9569	0.9569	0.9513	0.9512	0.3235	0.6593	0.6723	0.9026	0.9169
	$0.010^{\dagger}$	0.9944	0.9845	0.9641	0.9640	0.9513	0.9512	0.3228	0.6598	0.6728	0.8991	0.9360
	0.015	0.9930	0.9877	0.9600	0.9599	0.9539	0.9537	0.3217	0.6598	0.6728	0.8565	0.9033
	0.020	0.9931	0.9840	0.9635	0.9635	0.9539	0.9537	0.3202	0.6687	0.6809	0.9007	0.9192
	0.030	0.9937	0.9827	0.9560	0.9560	0.9503	0.9502	0.3073	0.5811	0.5603	0.8753	0.9122

Appendix

Table S7: Parameter sensitivity analysis on the MetaDB dataset. We evaluate the sensitivity to the PD-loss weight

\lambda_{\mathrm{PD}}

, while fixing the masking ratio to

\rho=0.75

and patch size to

p=4

. Bold values indicate the best performance within the sensitivity sweep.

\dagger

denotes the value adopted in the main configuration.

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