How the Classical Baseline Shapes Machine-Learning Speed Claims in Topology Optimization

Shaoliang Yang; Jun Wang

cs.CEOct 12, 2026

How the Classical Baseline Shapes Machine-Learning Speed Claims in Topology Optimization

Authors: Shaoliang Yang, Jun Wang

Organizations: Santa Clara University Santa Clara, CA 95053, USA

Abstract

Topology optimization places material in a design domain to optimize a structure's performance, such as its stiffness or heat conduction. Machine-learning methods often report large speed-ups over classical, non-learned optimization, but a slow classical code or a lower-quality learned design can exaggerate them. We re-ran released code for eight methods on one workstation and studied two in depth: Optimize Any Topology (OAT), a diffusion model that generates designs, and a graph-based neural optimizer. Replacing the classical code with an efficient implementation of the same algorithm cut these two methods' speed-ups by up to 20-fold. We then timed the classical code until its design was at least as stiff as the learned one; for OAT, we picked this point after the runs, which gives the classical code its best case. Published timings imply a 134x speed-up for OAT on a 256 x 256 grid; our matched single-design comparisons gave 1.5x to 3.5x. At matched stiffness, the graph-based optimizer was slower than the classical code in all six two-dimensional problems. Of the six other methods, two kept their advantage, one shrank, one reversed, and two were not comparable. In an exploratory audit, language models judged that 8 of 28 speed comparisons in randomly sampled, openly accessible articles lacked evidence of equal design quality; our own review counted 11, so the count depends on who judges. We provide a comparison procedure and a minimum report: classical code, hardware, design quality, what each timing includes, and data-generation and training costs.

Figures & tables

Study	Domain	Design	Main finding or recommendation
Woldseth et al. [13]	Neural topology optimization	Critical review	Asks for comparisons at equal volume and quality, and for training cost in any break-even claim
Sigmund [14]	Topology optimization	Recommendations	Recommends fair evaluation and benchmarking practice
Shin et al. [11]	Topology optimization	Review	Surveys machine-learning-based topology optimization
Regenwetter et al. [12]	Engineering design	Review	Surveys deep generative models in design, including structural optimization
Herrmann et al. [15]	Topology optimization	Controlled study	Neural-network discretizations can find better local optima under certain conditions
Sanu et al. [16]	Topology optimization	Controlled study	Neural networks add non-convexity, which can delay convergence in convex problems

Table 1: Related reviews and evaluations, and the contribution of this study.

Part	What is tested	Quality criterion	Ladder levels	Results
OAT re-test (Sect. 3.3.1 )	A diffusion model against our efficient classical code on five test problems, fixed in advance, at $64^{2}$ and $256^{2}$ ; classical runs of 5 to 150 iterations	Compliance error from OAT’s evaluator; a volume condition, added after the runs, also limits material	$S_{\rm pub}$ , $S_{1}$ , $S_{2}$ , $S_{3}$	Sects. 4.1 and 4.2
HGTO re-test (Sect. 3.3.2 )	A graph-based method against the authors’ classical code and our GPU port of it on six linear two-dimensional problems	Time to convergence, and time to reach HGTO’s final compliance from its independent evaluator	$S_{\rm pub}$ , $S_{1}$ , $S_{2}$ , $S_{3}$	Sects. 4.1 and 4.3
Census (Sect. 3.3.3 )	The six sampled speed claims whose public code ran, each in the article’s own comparison	Each article’s own quality measure (Table 3 )	L0 (code runs), L1 (as $S_{1}$ ), L2 (as $S_{2}$ – $S_{3}$ )	Sect. 4.4
Article audit (Sects. 3.4 – 3.6 )	57 sampled articles, 28 with speed claims, coded against rules R1–R4 by two language-model passes with model adjudication, then reviewed by the authors	Rule R1: matched quality or time	–	Sects. 4.6 – 4.8

Table 2: Parts of the study. The re-tests use the comparison ladder of Fig. 1 :

S_{\rm pub}

is implied by an article’s published times,

S_{1}

re-runs the released codes on our workstation,

S_{2}

gives the classical code an efficient state solve, and

S_{3}

times it until its design reaches the learned design’s quality. The census levels L0–L2 apply the same steps to each article’s own comparison. R1–R4 are the four audit rules (Sect. 3.5 ): R1 matched quality or time, R2 efficient baseline, R3 full cost, and R4 verified output. GPU, graphics processing unit.

Figure 1: Comparison levels and re-test procedure. Each row is one factor of the comparison; a highlighted cell marks the level at which it changes, and “same” repeats the value to its left.

S_{\rm pub}

is the ratio of the times an article reports on its own hardware;

S_{1}

re-runs the article’s classical code and the learning-based method on our workstation;

S_{2}

keeps the classical algorithm and budget with an efficient state solve;

S_{3}

compares the time needed for that implementation to reach the learned design’s evaluated quality, with matches for OAT (Optimize Any Topology, a diffusion model) identified after the runs. GPU/CPU, graphics/central processing unit; PARDISO and SuperLU are sparse direct equation solvers. The level-to-level ratios

H

I

, and

Q

form the identity in Eq. ( 5 ). Correctness gates precede timing, quality is scored with the learning-based method’s own evaluator, and census levels L0–L2 adapt these steps to the article-specific quality criteria of Sect. 3.3.3

Figure 2: Flow of records through language-model screening, seeded sampling, access assessment, two full-text codings, and adjudication. (a) Each stage lists the records it received, removed (

-

), and kept (

=

), and ends in its output; tagged stages include language-model steps. (b) Where all 4,484 identified records left the flow, with the 709 that passed screening shown at full width; the dashed segment holds eligible records outside the analysis sample. Blind re-screen referrals arose from unclear abstracts, not confirmed screening errors

Figure 3: Speed-up at the comparison levels of Fig. 1 : published, authors’ code, efficient solve, and matched quality. (a) Optimize Any Topology (OAT), a diffusion model, and HGTO, a graph-based method; the dashed segment is the value implied by the published timings on the article’s hardware. At matched quality, the filled

256^{2}

OAT marker uses the post hoc volume-condition rule and the open one the pre-registered rule. (b) Census comparisons with article-specific quality checks; open one-shot markers are L1 against its own code. Phase-field CNN compares convergence times, with lower learned compliance and an unchanged classical solver. Values below one favor classical optimization. OAT matched-quality points use the first budget of our graphics processing unit (GPU) baseline that meets the rule, identified after the runs; evaluator time is not counted. The axis is logarithmic. CNN, convolutional neural network

Method	Reported	Same-machine outcome	Quality criterion
In-depth re-tests
OAT diffusion generator, $256^{2}$ [ 8 ]	134× (implied by published timings)	78.6 $\times$ against OAT’s released SIMP code; 18.6× with an efficient GPU solve; at matched quality, single samples: 1.72× under the pre-registered rule, 3.47× with the volume condition	Compliance error from OAT’s evaluator; the volume condition also limits material
HGTO graph-based reparameterization [ 9 ]	4.2× and 9.6× (from the article’s reported times)	4.9× and 10.8× against the authors’ SIMP–OC; the same algorithm with our GPU solve was faster on all six configurations at matched quality (HGTO took 2.0× to 6.5× as long)	Time to convergence; final compliance from HGTO’s independent evaluator
Halle et al. predictor [ 10 ]	259 $\times$	Not compared: stopped at the pre-registered reconstruction check (57 of 100 within 1 %, 95 required)	Reconstruction check before timing
Census
Conductivity CNN [ 2 ]	2,280–24,747 $\times$	320–1,112 $\times$ at L1; 3.6 $\times$ at L2 with more accurate solve	Effective conductivity against the published values

Table 3: Eight released-code tests and one stopped reconstruction check. Optimize Any Topology (OAT), a diffusion model, and HGTO, a graph-based method, were fixed before any timed run for in-depth re-tests; a third in-depth re-test stopped at our reconstruction check. The frozen census covers every speed claim in the analysis sample that links public code (seven of 28); code ran for six, including one classical-only run, and the seventh is that stopped reconstruction, excluded from the eight-method total. Factors are classical time divided by the learning-based method’s time. Reported factors are derived from the times each article reports on its own hardware; L1 repeats own-code comparisons here, and L2 changes only classical solve or device, with unstated parameters set by rule. For the one-shot generator, the tested threading change did not speed the classical solve, so L2 retains L1 at matched quality. The last column gives each quality criterion; phase-field CNN compares convergence times at lower learned compliance. CNN, convolutional neural network; GPU/CPU, graphics/central processing unit; SIMP–OC, solid isotropic material with penalization and an optimality-criteria update.

Figure 4: Optimize Any Topology (OAT), a diffusion model: same-machine timing and quality. GPU/CPU, graphics/central processing unit; SIMP, solid isotropic material with penalization; CG, conjugate gradient; PARDISO, a sparse direct equation solver. (a) One-sample inference, setup-inclusive GPU and CPU classical runs, and authors’ loop-only classical timing (OAT’s direct-solver path was run at

64^{2}

only); dots are problems and bars are medians. (b) Compliance error (CE) under OAT’s evaluator. (c) Error of our GPU baseline minus OAT’s error across fresh 5–150-iteration budgets, one curve per problem, numbered as in Online Resource 2 (three examples appear in Fig. 5 ); filled marks at or below zero match and the dashed line is median OAT time. Error axes are signed linear-log, linear between

-1

and

1

; values above 100 points in (c) are pinned to the edge

Figure 5: Three $256^{2}$ test problems from the benchmark split of Optimize Any Topology (OAT), a diffusion model (rows), labeled by matching outcome. First column: the dataset’s reference design (gray), against which compliance error is measured, with supports (filled squares fully fixed, open circles fixed in one direction) and the point load (arrow). Other columns: OAT’s sample, the first classical budget whose compliance error is at most OAT’s, and the classical design after 150 iterations. Designs are thresholded at 0.5 as OAT’s evaluator scores them (black is solid); labels give compliance error (CE) and thresholded volume. A compliance-error match may not meet the volume-condition rule

Figure 6: HGTO, a graph-based neural design representation, and classical SIMP–OC on the same workstation. SIMP–OC combines solid isotropic material with penalization and an optimality-criteria update; GPU/CPU, graphics/central processing unit. (a) Wall time to convergence against mesh size for four cantilevers: HGTO on GPU, authors’ SIMP–OC with two or six CPU threads, and our GPU port, as keyed by the plotted lines. (b) Classical time divided by HGTO time for six configurations (half MBB is the symmetric half of the Messerschmitt–Bölkow–Blohm beam benchmark); values below one favor the classical baseline, and open diamonds show the GPU port at matched quality. (c) Final $960\times 320$ designs and compliance under the authors’ independent evaluator. Timing includes construction, optimization, and final evaluation; the inclined-load classical runs hit the update cap

Figure 7: Optimize Any Topology (OAT), a diffusion model: offline cost and graphics processing unit (GPU) board energy. (a) Upper bounds on the uses needed to repay data-generation and training cost, calculated from the mean saving across the five-problem workload at the first tested matching iteration budgets. Arrows indicate upper bounds because unmatched classical runs are censored at 150 iterations. Filled marks use the post hoc volume-condition rule; the open mark uses the pre-registered compliance-only rule. H100 offline time is divided by RTX 5090 per-design savings; these are GPU-time bounds, not financial or energy break-even estimates. (b) RTX 5090 board energy per design; dots are individual problems and bars are medians. Central processing unit (CPU) package and matched-quality energy were not measured

Stratum	Rule	Fail	Fail or partial
Speed	R1 matched quality/time	8/28 (29 %)	22/28 (79 %)
Speed	R2 efficient baseline	0/28 (0 %)	11/28 (39 %)
Speed	R3 full cost	7/25 (28 %)	20/25 (80 %)
Speed	R4 verified output	4/28 (14 %)	20/28 (71 %)
Speed	R1 or R2	8/28 (29 %)	24/28 (86 %)
Speed	Any rule	13/28 (46 %)	27/28 (96 %)

Table 4: Adjudicated fail and fail-or-partial rates. Fail requires evidence of a violation; partial marks incomplete evidence or a likely concern (Sect. 3.5 ). R3 is inapplicable to three speed claims; confidence intervals for these descriptive rates appear in Online Resource 4.

Figure 8: Model-coded availability and reporting indicators, with exact Clopper–Pearson 95 % confidence intervals for proportions [ 27 ] . Code, data, seeds, failures, and out-of-distribution testing cover the speed-claim and quality-only groups; hardware and timing indicators cover speed claims. Shares describe reporting in this coded set

Item	Report
Classical method	Formulation, update, filter, linear solver, and stopping rule
Hardware	CPU and GPU model, threads, and precision for each method
Timing	Setup, post-processing, warm-up, repeats, and statistic
Quality	Common finite-element objective, constraints, failures, and compared point
Offline cost	Data generation, training, hardware, and break-even use count
Artifacts	Code, weights, timing harness, and pinned environment

Table 5: Minimum information for a reconstructable speed comparison. CPU/GPU, central/graphics processing unit.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Article	Method	Physics	Dim.	Factor	R1	R2	R3	R4
Adam et al. [2]	Convolutional network	Thermal	2D	2,280 $\times$	Pass	Partial	Pass	Pass
Al Ali [56]	Language-model agent	Compliance	2D, 3D	3.33 $\times$	Fail	Partial	–	Partial
Banga et al. [39]	Convolutional network	Compliance	3D	1.67 $\times$	Partial	Pass	Fail	Fail
Black and Najafi [3]	Surrogate	Stress	2D	10,000 $\times$	Partial	Pass	Partial	Pass
Bujny et al. [46]	Other	Compliance	2D, 3D	12 $\times$	Partial	Pass	Partial	Partial
Cang et al. [29]	Other	Compliance	2D	10,000 $\times$	Fail	Partial	Partial	Pass

Appendix

Table A1: The 28 speed claims. Factor: classical time divided by the learning-based method’s time in the article’s primary comparison, as coded, with factors above 100 rounded to integers; a dash means that the article gave no single factor. A dash under R3 means that the rule was inapplicable. Verdicts are the protocol-fixed adjudicated verdicts of Table 4 under the rules of Table B1 ; they score the reported comparison, not the article’s contribution. Online Resource 3 gives the evidence and locators for each verdict. Dim., dimension; 2D and 3D, two- and three-dimensional. Homogenization, architected-material homogenization; GAN, generative adversarial network; VAE, variational autoencoder.

Article	Method	Physics	Dim.	R1	R4
Abueidda et al. [57]	Convolutional network	Compliance	2D	Partial	Fail
Ait Ouchaoui et al. [58]	GAN	Compliance	2D	Partial	Partial
Behzadi and Ilies [59]	GAN	Compliance	2D	Partial	Partial
Bhagat et al. [60]	Transformer	Multiphysics	2D, 3D	Partial	Fail
Chadha and Kumar [61]	Convolutional network	Compliance	2D	Partial	Partial
Chandrasekhar and Suresh [62]	Neural reparameterization	Compliance	2D	Partial	Partial

Appendix

Table A2: The 29 quality-only articles. R2 and R3 are shown for speed claims only; Online Resource 4 gives R2 for three quality-only articles. Verdicts are the protocol-fixed adjudicated verdicts of Table 4 under the rules of Table B1 ; they score the reported comparison, not the article’s contribution. Online Resource 3 gives the evidence and locators for each verdict. Dim., dimension; 2D and 3D, two- and three-dimensional. Homogenization, architected-material homogenization; GAN, generative adversarial network; VAE, variational autoencoder.

Rule	Fail	Partial
R1: matched quality or time	Inference against a converged classical run with worse learned output and no match; speed claim without quality comparison; or post-processing counted in quality but not in time	Loose matching or unstated stopping
R2: efficient baseline	Our re-implementation at least three times faster on the same hardware class and size; an abstract reports a weaker baseline than the results; unjustified GPU–CPU comparison at $\geq 10^{5}$ degrees of freedom; excessive iterations; or classical iteration at least ten times slower than a strong baseline	Suspected inefficiency without a comparable measure
R3: full cost	Neither data nor training cost given	Training cost without data-generation cost, or no amortization or per-query scope
R4: verified output	Only image-similarity metrics or surrogate predictions, or examples chosen without a stated rule	Finite-element verification without violations or failure rates

Appendix

Table B1: Fixed rules. R1 checks matched quality or time, R2 classical efficiency, R3 full cost, and R4 output verification. The full protocol and examples are in Online Resource 1. An unknown condition scores pass under the conservative fail-only headline. GPU/CPU, graphics/central processing unit.

Characteristic	Speed	Quality	All
Convolutional network	13	4	17
Neural reparameterization	1	7	8
Surrogate	5	2	7
Compliance physics	18	20	38
Architected-material homogenization	6	2	8
Two-dimensional	17	19	36

Appendix

Table B2: Selected categories of learning-based method and physics, with the complete dimension split, in the coded set; the method and physics rows do not sum to the total. Complete categories appear in Online Resource 3.

Item	$n$	Raw agreement	Claude $\kappa$	OpenAI $\kappa$	OpenAI vs Claude $\kappa$
Inclusion	110	88 %	0.62 (0.42–0.80)	0.61 (0.39–0.79)	0.48–0.56
Stratum	110	70 %	0.53 (0.40–0.65)	0.59 (0.47–0.71)	0.46–0.60
R1	82	55 %	0.35 (0.21–0.49)	0.23 (0.11–0.36)	0.16–0.25
R2	82	56 %	0.28 (0.12–0.43)	0.26 (0.11–0.41)	0.23–0.39
R3	82	83 %	0.66 (0.49–0.80)	0.47 (0.30–0.63)	0.42–0.56
R4	82	70 %	0.56 (0.42–0.70)	0.71 (0.58–0.83)	0.44–0.56

Appendix

Table B3: Agreement between coders, as Cohen’s

\kappa

with a 95 % bootstrap confidence interval (CI);

\kappa

corrects agreement for chance [ 34 ] . Claude: first pass vs blind second pass, before model adjudication; these models were blind to one another but from one provider. OpenAI: GPT-6-Sol vs GPT-6-Luna; OpenAI vs Claude: range over the four pairs of one OpenAI model and one Claude pass; both from the post hoc cross-provider check (Online Resource 4). Authors: the later unblinded review against the adjudicated verdicts.

n

and raw agreement refer to the Claude pair and the authors’ review.

Topology Optimization

Topology optimization finds material layouts that meet performance and manufacturing constraints across engineering systems.

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Sep 23, 2026J/K move · Enter open · S save

KATOsuper: Surrogate-accelerated neural topology optimization with sensitivity-consistent Fourier neural operators

Shengyu Yan, Jasmin Jelovica

Department of Mechanical Engineering, The University of British Columbia, 6250 Applied Science Ln, Vancouver, V6T 1Z4, Canada · Department of Civil Engineering, The University of British Columbia, 6250 Applied Science Ln, Vancouver, V6T 1Z4, Canada

Neural Surrogate Modeling Topology Optimization Surrogate Modeling Fourier Neural Operator Surrogate-Assisted Optimization

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Topology optimization (TO) remains computationally intensive due to repeated finite element analysis (FEA) evaluations required at each iteration. While neural network-based surrogates offer potential acceleration, existing approaches often suffer from gradient inconsistency between predicted objectives and sensitivities, leading to optimization instability. This work presents KATOsuper, an objective-agnostic framework that couples neural-reparameterized topology optimization with a Sensitivity-Consistent Fourier Neural Operator (SC-FNO). The framework employs the forward_split architecture, which derives deployed sensitivities via automatic differentiation through the predicted objective field and thereby preserves consistency between the predicted objective and the gradient used for optimization. The case studies include three 2D benchmark problems and three 3D structures considering compliance or stress minimization. A physics-informed multi-channel input encoding with Fourier position embedding enables resolution-invariant learning, supporting zero-shot extrapolation beyond the training resolution, with useful performance at moderate scaling factors and topology-preserving exploration at up to 64x without retraining. The framework extends to 3D through KATO3D, featuring novel KANConv3D blocks with learnable B-spline activations. KATOsuper demonstrates 15--110x deployment-time speedup over MATLAB baselines while maintaining competitive optimality, with the clearest gains observed in complex 3D and stress-optimization cases. The insight that sensitivity direction matters more than magnitude enables robust optimization even with approximate physics evaluation, extensible to other differentiable physics-driven design objectives.

Abstract

Topology Optimization

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