Pith. sign in

REVIEW 4 major objections 5 minor 39 references

Trajectory-Aware Flow Matching for Topology Optimisation

T0 review · 4 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read By interpolating a Gaussian noise field toward the BESO reference topology, flow matching can generate compliant 2D and 3D topology candidates from design conditions alone, and encoding the BESO optimisation history into the probability pat

desk verdict Trajectory-aware flow matching is a genuinely new path-design idea, but the central empirical advantage over the linear baseline is not yet statistically supported. read the letter →

arxiv 2607.14652 v1 pith:QY4D44B5 submitted 2026-07-16 cs.LG cs.NAmath.NA

Shusheng Xiao , Jinshuai Bai , Hyogu Jeong , Yunfei Xi , Yilin Gui , YuanTong Gu This is my paper
classification cs.LGcs.NAmath.NA
keywords flowmatchingtopologyoptimisationBESOgenerativedesignprobabilitypathtrajectoryguidancevolumefractionconditionalgeneration
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper's central claim is that the choice of probability path, not just the generative model family, determines whether a learned topology generator produces structurally feasible designs. It first shows that a linear flow-matching path from noise to a converged BESO topology already beats a diffusion baseline on compliance ratio, volume-fraction error, IoU/Dice/boundary fidelity, and sampling cost (20 Euler steps vs 1000 reverse steps). The main contribution is a trajectory-aware path that steers generation through volume-fraction-indexed intermediate BESO states, weighted by a parameter lambda so the path is neither pure endpoint transport nor a forced replay of one optimiser's history. Under 1000 training examples, a moderate weight (lambda = 0.25) gives the best feasibility and topology fidelity, matching the paper's quadratic path-velocity mismatch analysis. A reader should care because it suggests that recorded optimisation history can be folded into training-time path supervision at zero inference cost, making generative design exploration faster and more reliable.

What carries the argument

The central object is the trajectory-aware conditional probability path. BESO intermediate topologies are recorded at descending volume fractions, mapped to flow time, and piecewise-linearly interpolated into a guide q_i(t). A lambda-weighted centreline m_i^lambda(t) = (1-lambda) rho_ref_i + lambda q_i(t) is used to define x_t^lambda = (1-t) x0 + t m_i^lambda(t) and the supervised target velocity u_t^lambda = -x0 + m_i^lambda(t) + t d/dt m_i^lambda(t). Interpolating between a Gaussian source and the reference topology keeps endpoint supervision, while the guide injects mechanics-driven intermediate material layouts into training. The same ODE solver is used at inference, so the trajectory in

What would settle it

Run a controlled suite where the same U-Net is trained as a diffusion model with identical data, architecture, and seed protocol, then evaluated across binarisation thresholds from 0.3 to 0.7. If a diffusion model with 1000 steps matches or beats FMTO's IoU, feasible-sample rate, and compliance ratio at the threshold that best suits each method — or if FMTO's advantage disappears away from 0.5 — the central claim that trajectory-aware flow matching is superior would be refuted.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes that flow matching with a trajectory-aware probability path is a viable conditional topology generator. In the 2D setting, linear FMTO reaches median compliance ratios close to the BESO reference and 20-step sampling that is roughly 50x faster than the 1000-step diffusion baseline, with higher IoU and Dice. Adding BESO trajectory guidance at lambda = 0.25 improves constrained best compliance, feasible-sample rate, and topology fidelity when training data are limited to 1000 instances; too much guidance (lambda approaching 1) degrades feasibility, and the non-monotonic trend is explained by a path-velocity mismatch quadratic G(lambda). The same trajecto

Load-bearing premise

The load-bearing premise is that the recorded BESO history, after interpolation and lambda-weighting, is closer than a straight noise-to-topology path to the ideal generative path a flow model would need to learn — and that the diffusion baseline is comparably tuned; if either fails, the reported gains shrink.

Editorial extensions

If this is right

  • Because trajectory guidance changes only the training path, generated topologies need no extra optimisation or surrogate guidance at inference — just 20 Euler steps.
  • In limited-data regimes, a moderate lambda improves both structural performance and volume-fraction satisfaction, suggesting flow matching can be data-efficient for design generation.
  • The 3D demonstration shows the path-construction idea transfers to voxel-based topology generation without algorithmic changes.
  • If the path-velocity mismatch analysis holds, lambda controls an explicit trade-off: too little guidance leaves noise-to-topology mixtures, and too much restricts transport flexibility.
  • Best-of-N sampling from FMTO can find candidates with recomputed compliance below the BESO reference, making generative sampling a legitimate design-exploration tool rather than just approximation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same trajectory-guiding recipe should apply to any TO method that emits intermediate states — SIMP density fields, level-set boundaries, or MMC parameters; a testable prediction is that the gain from guidance grows with the physical informativeness of those states.
  • Because lambda and anchor density are chosen empirically, a practical extension is to select lambda by a validation criterion such as feasible-sample rate or failure rate, or to anneal lambda during training.
  • The fixed 0.5 binarisation threshold sits inside all compliance and fidelity metrics; an obvious stress test is a threshold sweep, which would show whether the reported advantages are design-level or partly threshold artifacts.
  • The paper's ideal-centreline analysis is interpretive; a more direct test is to measure G(lambda) on a held-out set using the learned velocity error and check that the measured optimum matches the empirical best lambda.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes a flow-matching-based generative framework for topology optimisation (FMTO). Linear FMTO constructs an endpoint-based probability path between Gaussian noise and a BESO reference topology; trajectory-aware FMTO replaces the static endpoint centreline with a volume-fraction-indexed, piecewise-linear interpolation of intermediate BESO states, weighted by a scalar λ. The authors derive a path–velocity mismatch objective G(λ) and an error-propagation bound connecting mismatch to velocity-field error, terminal error, and failure probability. Numerical experiments compare linear FMTO, trajectory-aware FMTO with several λ, and a diffusion baseline (DMTO) on 2D datasets of BESO-generated topologies, including a limited-data setting, an anchor-density study, and a 3D extension. The central claims are that moderate trajectory guidance (λ=0.25) improves generation quality, volume-fraction satisfaction, and sampling stability under limited data, and that FMTO requires roughly 50× fewer sampling steps than DMTO while producing better or comparable compliance and fidelity metrics.

Significance. If the empirical claims are robust, the work is a useful contribution to generative topology optimisation: it replaces adversarial training and long reverse-diffusion sampling with a supervised flow-matching objective, and it introduces a principled way to inject physics-guided optimisation history into probability-path construction without inference-time optimisation. The manuscript has notable strengths: compliance is re-evaluated with external FEA rather than read from network outputs; the U-Net capacity is matched across methods; a step-sensitivity ablation supports the 20-step FMTO setting; and the 3D study tests voxel-level fidelity and volume feasibility. The theoretical path-velocity mismatch analysis is elegant as an interpretive framework. However, the headline quantitative claim—moderate trajectory weighting is best—rests on a single seed per configuration with small and partly contradictory differences, and the limited-data diffusion baseline appears to have collapsed; these issues need to be addressed before the conclusions can be accepted as stated.

major comments (4)
  1. [§4.2, Tables 4 and 5] The central claim that λ=0.25 is the best trajectory weight in the limited-data setting is supported by single runs, with no confidence intervals, multiple seeds, or paired significance tests. Several differences are small and directionally mixed: mean |Vf−V*f| is 0.0104 vs 0.0111, failure rate is 8.96% vs 9.80%, IoU is 0.8189 vs 0.7952, but median C/CBESO is slightly worse for λ=0.25 (1.0184 vs 1.0131). These differences could readily arise from training noise. I request at least 3–5 independent seeds per configuration, reported as mean±std or with paired bootstrap intervals, and a statistical test on the λ=0.25 vs λ=0 comparison.
  2. [§4.2, Table 4] The DMTO baseline in the limited-data setting is not a meaningful control: its median C/CBESO is reported as 1.14×10^7 and its failure rate is 82.88%. This indicates the diffusion model collapsed or is severely undertuned. The paper does not provide enough DMTO implementation detail (noise schedule, conditioning, sampling, hyperparameters, training epochs) to judge whether this is representative of diffusion-based generative TO. The claim that trajectory-aware FMTO outperforms DMTO under limited data is therefore not supported by this table. Please either provide a properly tuned diffusion baseline with comparable capacity and training budget, or explicitly restrict the comparison to the more successful Example 1 setting.
  3. [§3.4 and Appendices A–B] The theoretical rationale depends on an unobservable ideal generative centreline m*_i(t), and the error-propagation argument relies on the transfer inequality ελ ≤ A G(λ)+Rλ with the assumption Rλ≈R0. As written, G(λ) cannot be evaluated and Eq. (27) cannot be used to predict or test λ*; the paper later acknowledges that λ is selected empirically. This makes the statement in §3.4 that moderate guidance 'reduces the mismatch' essentially post hoc. I am not asking for a fully predictive theory, but the paper should either (i) provide a measurable proxy for G(λ) or ελ as a function of λ and test the predicted U-shape, or (ii) explicitly frame §3.4 as a qualitative intuition and remove the implication that it explains the empirical optimum.
  4. [§3.5, Eq. (33); §4.2 Tables 4–5] All compliance, volume-fraction, and topology-fidelity metrics depend on the fixed binarisation threshold of 0.5. No sensitivity analysis is reported. If trajectory-aware training produces sharper or more concentrated continuous density fields, the 0.5 threshold may bias the comparison in its favour. Please add a threshold sweep (e.g., 0.3, 0.4, 0.5, 0.6, 0.7) for at least the limited-data λ comparison, or report continuous-field metrics such as the volume fraction of the clamped field, to show the qualitative conclusions do not hinge on the threshold choice.
minor comments (5)
  1. [§4.3] The anchor-density parameter α is introduced in §4.3 without a formal definition. Please specify how α maps to the recording schedule of BESO states (e.g., save every α volume-fraction interval) and how the interpolation knots are constructed.
  2. [Table 4] The DMTO median of 1.14×10^7 looks like an overflow or failed FEA artifact. If this number is real, an explanation is needed; if it is an artifact of thresholding or FEA, the evaluation should exclude or flag such cases clearly.
  3. [Fig. 2 caption and §4.1] The qualitative selection protocol excludes 'clearly invalid samples, such as those with disconnected load-bearing paths, are excluded by inspection.' This is not reproducible. Please replace the visual inspection with an automated connectivity/validity filter or state explicitly that Figs. 2 and 5 are illustrative only.
  4. [§4.4] The 3D study does not include FEA-based compliance evaluation, so the claim of structural performance in 3D is limited. This is acknowledged in §5, but the results section should state up front that Table 8 measures topology fidelity and volume feasibility only.
  5. [Throughout] Minor notational points: (i) 'FMTO λ=0' is used to denote linear FMTO in Tables 4–5; please state this explicitly the first time it appears. (ii) In Eq. (20), the derivative ˙q_i(t) depends on the piecewise-linear interpolation but the notation does not make the segment index explicit; this is clear from context but could be clarified. (iii) The Data Availability statement says 'on request'; releasing code and trained models would substantially aid reproducibility.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the empirical comparisons are externally evaluated and the theory is explicitly interpretive.

full rationale

The paper's core derivation chain is not circular. Linear FMTO is a standard conditional straight-line flow-matching path (Eqs. 16-17), and trajectory-aware FMTO constructs its probability path and target velocity from recorded BESO states (Eqs. 19-25). The central experimental claims are evaluated against quantities that are not re-inserted as training targets in a way that forces the result: compliance is recomputed with external finite-element analysis on thresholded generated topologies, and fidelity metrics are measured against BESO references that are the explicit targets of the generative model rather than outputs derived from the model itself. The theoretical path-velocity mismatch analysis (Section 3.4, Appendix A) relies on a 'hypothetical ideal generative centreline' and unobserved quantities BG, NG, CG, which makes it interpretive rather than falsifiable, but the paper explicitly labels it as interpretation: 'This analysis is used as an interpretation rather than an unconditional guarantee.' The error-propagation argument in Appendix B similarly rests on stated assumptions and is not used to generate the numerical results. The only self-citation is the review [24] used for the general statement that engineering stability remains a central challenge; it is motivational and not load-bearing for the derivation. The selection of lambda = 0.25 is empirical, and the paper acknowledges that 'the trajectory weight and anchor density are selected empirically in the numerical studies.' The lack of multiple-seed uncertainty and baseline-tuning detail is a correctness/robustness concern, not a circularity concern.

Assumptions & free parameters 4 free parameters · 6 assumptions · 1 invented entities

The method itself has two central hyperparameters (lambda and anchor density) that are chosen empirically. The theoretical rationale depends on an invented ideal centreline and an assumed transfer inequality, neither of which is measured. No new physics entities are introduced.

free parameters (4)
  • Trajectory weight lambda = 0.25 (best in sweep; tested 0, 0.25, 0.5, 0.75, 1)
    Balances endpoint reference topology vs BESO trajectory guide in Eq. (21); selected empirically in Example 2, not predicted by theory.
  • Trajectory anchor density alpha = 0.10 (best overall fidelity; 0.25 best compliance; 0.05 worst)
    Spacing of recorded BESO states; selected empirically in Example 3.
  • Binarization threshold = 0.5
    Fixed threshold in Eq. (33); all volume, compliance and fidelity metrics depend on it, with no sensitivity analysis.
  • Mismatch weight eta in G(lambda) = unspecified (eta > 0)
    Introduced in Appendix A Eq. (A.5) to weight velocity vs position mismatch; never assigned a value and not used in experiments, so the theory is not a quantitative predictor.
assumptions (6)
  • ad hoc to paper There exists an ideal generative centreline m*_i(t) and the BESO guide is a useful approximation to it for topology evolution.
    Appendix A defines the mismatch using m*, which is unobservable; the usefulness of the BESO guide is assumed rather than measured.
  • ad hoc to paper Learning-error transfer inequality epsilon_lambda <= A G(lambda) + R_lambda and residual comparability R_lambda ~ R_0.
    Appendix B Eqs. (B.2)-(B.3): the key link between path mismatch and velocity-field error is assumed.
  • domain assumption Lipschitz condition on the learned velocity field and a Gronwall-type ODE stability bound with comparable numerical solver errors.
    Appendix B Eq. (B.5): standard but unverified for the U-Net velocity field.
  • domain assumption Bad-structure topologies are separated from the reference by distance delta in L2.
    Appendix B Eq. (B.7): used to convert terminal error to failure probability via Markov's inequality.
  • standard math Conditional flow matching with a Gaussian source distribution yields a valid generative ODE for binary topology fields, and 20 Euler steps are sufficient.
    Sections 2.2 and 3.5: standard flow matching theory applied to topology fields.
  • domain assumption BESO converges to a mechanically meaningful local optimum and volume-fraction-indexed states align across different optimisation instances.
    Sections 2.1 and 3.1: trajectory recording assumes consistent alignment of optimisation progress.
invented entities (1)
  • Hypothetical ideal generative centreline m*_i(t)
    purpose: Reference for defining path-velocity mismatch G(lambda) and the optimal lambda* formula (Eq. 27).
    No falsifiable prediction is attached to m*; it is an unobserved construct introduced to make the trade-off argument.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Trajectory-Aware Flow Matching for Topology Optimisation." pith.science (2026). https://pith.science/paper/QY4D44B5

@misc{pith2026260714652,
  author       = {Pith},
  title        = {Pith review of: Trajectory-Aware Flow Matching for Topology Optimisation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QY4D44B5}},
  note         = {Machine review of arXiv:2607.14652}
}
read the original abstract

Topology optimisation (TO) often requires repeated finite element analysis and sensitivity-based material updates, which can be costly when multiple candidate designs are needed under varying physical and design conditions. Generative TO offers a route to rapid design exploration, but existing models may rely on adversarial training, long reverse-diffusion sampling, or external guidance to maintain structural feasibility and physical consistency. This study develops a flow matching-based topology optimisation (FMTO) framework for conditional topology generation. Linear FMTO is first formulated as an endpoint-based baseline by interpolating between a Gaussian source field and the BESO reference topology. To introduce mechanically meaningful intermediate states, a trajectory-aware FMTO formulation is proposed, where volume-fraction-indexed BESO states are used to construct the probability path and target velocity field. This incorporates physics-guided optimisation history into generative flow learning without adding inference-time optimisation. A path--velocity mismatch analysis explains why moderate trajectory weighting can improve generation stability, whereas excessive guidance may over-constrain the learned transport. Numerical examples show that FMTO generates diverse topology candidates with improved compliance-related performance, volume-fraction satisfaction, topology fidelity, and substantially fewer sampling steps than a diffusion-based baseline. Under limited training data, trajectory-aware FMTO achieves the best overall performance with a moderate trajectory weight. Studies on trajectory-anchor density and three-dimensional topology generation further demonstrate the influence of path design and the applicability of the proposed framework beyond two-dimensional problems.

Figures

Figures reproduced from arXiv: 2607.14652 by the authors.

Figure 1
Figure 1. Schematic illustration of the proposed FMTO path construction and sampling workflow. (a) Endpoint [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Qualitative comparison of linear FMTO and DMTO under representative validation conditions. Each [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Compliance-ratio distributions of linear FMTO and DMTO over 100 generated samples for five [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Controlled step-quality analysis for linear FMTO and DMTO on five representative validation cases. [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]
Figure 5
Figure 5. Figure 5: Qualitative comparison of linear FMTO, trajectory-aware FMTO, and DMTO in the limited-data [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]
Figure 6
Figure 6. Figure 6: Qualitative comparison of trajectory-aware FMTO with different BESO trajectory anchor densities. [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]
Figure 7
Figure 7. Figure 7: Compliance-ratio distributions of trajectory-aware FMTO with different BESO trajectory anchor [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]
Figure 8
Figure 8. Figure 8: Prescribed 3D design domains and corresponding BESO reference topologies for two representative [PITH_FULL_IMAGE:figures/full_fig_p023_8.png]
Figure 9
Figure 9. Figure 9: Qualitative comparison of 3D topology samples generated by (a) linear FMTO, (b) trajectory-aware [PITH_FULL_IMAGE:figures/full_fig_p024_9.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

39 extracted references · 3 canonical work pages

  1. [1]

    G. I. N. Rozvany, Aims, scope, methods, history and unified terminology of computer- aided topology optimization in structural mechanics, Structural and Multidisciplinary Optimization 21 (2001) 90–108. doi:10.1007/s001580050174

    work page doi:10.1007/s001580050174 2001

  2. [2]

    M. P. Bendsøe, Optimal shape design as a material distribution problem, Structural optimization 1 (1989) 193–202

    1989

  3. [3]

    M. Zhou, G. I. N. Rozvany, The coc algorithm, part ii: Topological, geometrical and generalized shape optimization, Computer Methods in Applied Mechanics and Engineering 89 (1991) 309–336. doi:10.1016/0045-7825(91)90046-9

    doi:10.1016/0045-7825(91)90046-9 1991

  4. [4]

    Y. M. Xie, G. P. Steven, A simple evolutionary procedure for structural optimization, Computers & structures 49 (1993) 885–896

    1993

  5. [5]

    O. M. Querin, G. P. Steven, Y. M. Xie, Evolutionary structural optimisation using a bidirectional algorithm, Engineering Computations 15 (1998) 1031–1048. doi:10.1108/ 02644409810244129

    1998

  6. [6]

    M. Y. Wang, X. Wang, D. Guo, A level set method for structural topology optimization, Computer Methods in Applied Mechanics and Engineering 192 (2003) 227–246. doi:10. 1016/S0045-7825(02)00559-5

    2003

  7. [7]

    Zhang, J

    W.-H. Zhang, J. Yuan, J. Zhang, X. Guo, A new topology optimization approach based on moving morphable components (mmc) and the ersatz material model, Structural and Multidisciplinary Optimization 53 (2016) 1243–1260. doi:10.1007/s00158-015-1272-3

    work page doi:10.1007/s00158-015-1272-3 2016

  8. [8]

    Sigmund, K

    O. Sigmund, K. Maute, Topology optimization approaches, Structural and Multidisci- plinary Optimization 48 (2013) 1031–1055. doi:10.1007/s00158-013-0978-6

    doi:10.1007/s00158-013-0978-6 2013

Show all 39 references
  1. [9]

    G. I. N. Rozvany, A critical review of established methods of structural topology opti- mization, Structural and Multidisciplinary Optimization 37 (2009) 217–237. doi:10.1007/ s00158-007-0217-0. 28

    2009

  2. [10]

    Zhu, W.-H

    J.-H. Zhu, W.-H. Zhang, L.Xia, Topologyoptimizationinaircraftandaerospacestructures design, Archives of Computational Methods in Engineering 23 (2016) 595–622. doi:10. 1007/s11831-015-9151-2

    2016

  3. [11]

    Behzadi, H

    M. Behzadi, H. T. Ilies, Real-time topology optimization in 3d via deep transfer learning, arXiv preprint arXiv:2102.07657 (2021)

    2021 arXiv

  4. [12]

    E. Ulu, R. Zhang, L. B. Kara, A data-driven investigation and estimation of optimal topologies under variable loading configurations, Computer Methods in Biomechanics and Biomedical Engineering: Imaging & Visualization 4 (2016) 61–72

    2016

  5. [13]

    Sosnovik, I

    I. Sosnovik, I. Oseledets, Neural networks for topology optimization, Russian Journal of Numerical Analysis and Mathematical Modelling 34 (2019) 215–223

    2019

  6. [14]

    Banga, H

    S. Banga, H. Gehani, S. Bhilare, S. Patel, L. Kara, 3d topology optimization using convo- lutional neural networks, arXiv preprint arXiv:1808.07440, 2018

    2018 arXiv

  7. [15]

    D. W. Abueidda, S. Koric, N. A. Sobh, Topology optimization of 2d structures with nonlinearities using deep learning, Computers & Structures 237 (2020) 106283

    2020

  8. [16]

    Zheng, H

    S. Zheng, H. Fan, Z. Zhang, Z. Tian, K. Jia, Accurate and real-time structural topology prediction driven by deep learning under moving morphable component-based framework, Applied Mathematical Modelling 97 (2021) 522–535

    2021

  9. [17]

    Liang, D

    K. Liang, D. Zhu, F. Li, A fourier neural operator-based lightweight machine learning framework for topology optimization, Applied Mathematical Modelling 129 (2024) 714– 732

    2024

  10. [18]

    Lutheran, S

    A. Lutheran, S. Das, A. Tabarraei, Physics-informed transformer for real-time high-fidelity topology optimization, arXiv preprint arXiv:2604.03522, 2026

    2026 arXiv

  11. [19]

    Garayalde, M

    G. Garayalde, M. Torzoni, M. Bruggi, A. Corigliano, Real-time topology optimization via learnable mappings, International Journal for Numerical Methods in Engineering 125 (2024) e7502

    2024

  12. [20]

    Q. Ma, E. C. De Meter, S. Basu, To-node: topology optimization with neural ordinary differential equation, International Journal for Numerical Methods in Engineering 125 (2024) e7428

    2024

  13. [21]

    S. Oh, Y. Jung, S. Kim, I. Lee, N. Kang, Deep generative design: Integration of topology optimization and generative models, Journal of Mechanical Design 141 (2019) 111405

    2019

  14. [22]

    S. Jang, S. Yoo, N. Kang, Generative design by reinforcement learning: Enhancing the diversity of topology optimization designs, Computer-Aided Design 146 (2022) 103225

    2022

  15. [23]

    Radler, E

    A. Radler, E. Volkmann, J. Brandstetter, A. Berzins, Generative topology optimization: Exploring diverse solutions in structural design, arXiv preprint arXiv:2502.13174, 2025

    2025 arXiv

  16. [24]

    J. Bai, Y. Wang, H. Jeong, S. Chu, Q. Wang, L. Alzubaidi, X. Zhuang, T. Rabczuk, Y. M. Xie, X.-Q. Feng, Y. Gu, Towards the future of physics- and data-guided AI frameworks in computational mechanics, Acta Mechanica Sinica 41 (2025) 225340. doi:10.1007/s10409-025-25340-x

    2025 doi

  17. [25]

    Y.Yu, T.Hur, J.Jung, I.G.Jang, Deeplearningfordetermininganear-optimaltopological design without any iteration, arXiv preprint arXiv:1801.05463, 2018. 29

    2018 arXiv

  18. [26]

    Z. Nie, T. Lin, H. Jiang, L. B. Kara, Topologygan: Topology optimization using genera- tive adversarial networks based on physical fields over the initial domain, arXiv preprint arXiv:2003.04685, 2020

    2003 arXiv

  19. [27]

    F. Mazé, F. Ahmed, Diffusion models beat gans on topology optimization, arXiv preprint arXiv:2208.09591, 2022

    2022 arXiv

  20. [28]

    Lutheran, S

    A. Lutheran, S. Das, A. Tabarraei, Latent space diffusion for topology optimization, arXiv preprint arXiv:2508.05624, 2025

    2025 arXiv

  21. [29]

    Giannone, F

    G. Giannone, F. Ahmed, Diffusing the optimal topology: A generative optimization ap- proach, arXiv preprint arXiv:2303.09760, 2023

    2023 arXiv

  22. [30]

    J. Ho, A. Jain, P. Abbeel, Denoising diffusion probabilistic models, Advances in neural information processing systems 33 (2020) 6840–6851

    2020

  23. [31]

    Lipman, R

    Y. Lipman, R. T. Q. Chen, H. Ben-Hamu, M. Nickel, M. Le, Flow matching for generative modeling, in: International Conference on Learning Representations, 2023

    2023

  24. [32]

    A. Tong, K. Fatras, N. Malkin, G. Huguet, Y. Zhang, J. Rector-Brooks, G. Wolf, Y. Ben- gio, Improving and generalizing flow-based generative models with minibatch optimal transport, arXiv preprint arXiv:2302.00482 (2023)

    2023 arXiv

  25. [33]

    X. Liu, C. Gong, Q. Liu, Flow straight and fast: Learning to generate and transfer data with rectified flow, arXiv preprint arXiv:2209.03003 (2022)

    2022 arXiv

  26. [34]

    S. H. Lim, Y. Wang, A. Yu, E. Hart, M. W. Mahoney, X. S. Li, N. B. Erichson, Elucidating the design choice of probability paths in flow matching for forecasting, arXiv preprint arXiv:2410.03229 (2024)

    2024 arXiv

  27. [35]

    M. P. Bendsoe, O. Sigmund, Topology optimization: theory, methods, and applications, Springer Science & Business Media, 2013

    2013

  28. [36]

    Huang, Y

    X. Huang, Y. M. Xie, Convergent and mesh-independent solutions for the bi-directional evolutionary structural optimization method, Finite elements in analysis and design 43 (2007) 1039–1049

    2007

  29. [37]

    Lipman, M

    Y. Lipman, M. Havasi, P. Holderrieth, N. Shaul, M. Le, B. Karrer, R. T. Chen, D. Lopez- Paz, H.Ben-Hamu, I.Gat, Flowmatchingguideandcode, arXivpreprintarXiv:2412.06264 (2024)

    2024 arXiv

  30. [38]

    Albergo, N

    M. Albergo, N. M. Boffi, E. Vanden-Eijnden, Stochastic interpolants: A unifying frame- work for flows and diffusions, Journal of Machine Learning Research 26 (2025) 1–80

    2025

  31. [39]

    Rashed, D

    M. Rashed, D. F. V. Madeira, B. Gholami, C. Guerbuez, Y. Yang, N. Thuerey, On the generalization in topology optimization via sensitivity-conditioned bernoulli flow matching, arXiv preprint arXiv:2606.02179, 2026. 30

    2026 arXiv

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.