Pith. sign in

REVIEW 4 major objections 5 minor 2 cited by

Generative Models for Parameter Space Reduction applied to Reduced Order Modelling

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read Generative models shrink the parameter space that slows PDE surrogates.

desk verdict Promising PPINN variant, but the ROM comparison leaks training data into the test latents, so the headline improvement is not cleanly demonstrated. read the letter →

arxiv 2506.09721 v1 pith:2DUFDOEO submitted 2025-06-11 math.NA cs.NA

Guglielmo Padula , Gianluigi Rozza This is my paper
classification math.NAcs.NA MSC 65N9968T07
keywords generativemodelsparameterspacereductionreducedordermodellingphysics-informedneuralnetworksgeometricallyparametrisedPDEslatentautoencodersshapedeformation
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

Solving PDEs on families of deformed 3D shapes is expensive, and surrogate models that map geometric parameters directly to solutions lose accuracy as the number of parameters grows. This paper proposes to replace the raw deformation parameters with the low-dimensional latent codes produced by a generative model, and reports that this substitution improves accuracy and speed for two kinds of surrogates: data-driven reduced order models and parametrised physics-informed neural networks. The evidence includes a Poisson equation on deformed Stanford Bunny meshes, a two-phase naval hull flow with a drag coefficient target, and a Laplace equation solved by PPINNs, with latent dimensions of 5–10 replacing 54–81 original parameters. A sympathetic reader would take the core insight to be that the effective dimensionality of a geometric design family, not the nominal parameter count, is what controls surrogate performance, and that generative models can extract that effective dimensionality from sample geometries.

What carries the argument

The load-bearing mechanism is the generative model's latent space used as a reduced parameterisation: each deformed geometry $a_i$ is encoded into $s_i \in \mathbb{R}^d$ with $d \ll l$, where $l$ is the number of free-form-deformation control parameters. For DROMs the encoder is a standard autoencoder-type model (VAE, AAE, or BEGAN) whose decoder output is passed through a multilinear constraint projection, Algorithm 2.1, so that generated shapes keep, for example, a constant barycentre. For PPINNs the mechanism is a point-conditioned autoencoder $g_\psi : \mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R}^d$ that maps each point of a reference cloud together with the corresponding point of the deformed cloud to a per-point code, averages these to one $s_i$ per geometry, and lets a decoder $f_\alpha(s_i, \cdot)$ rebuild the deformed cloud point by point; a normalising flow, diffusion, or energy-based model then learns the distribution of the $s_i$ so new geometries can be sampled. These codes are the coordinate system in which the map from geometry to PDE solution is learned, and their lower dimension is what the argument credits for the improved accuracy and convergence.

What would settle it

Retrain the autoencoder (or the full generative model) on the training folds only, excluding the held-out geometry from the encoder's training set, and recompute the leave-one-out errors of Eqs. (34)–(35); if the generative parameterisation no longer outperforms the CFFD baseline in that protocol, the reported gain is an evaluation leak rather than genuine dimension reduction.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central claim is that the latent space of a generative model trained on a family of deformed point clouds is a reduced parameterisation for that family, and that using it in place of the original geometric parameters improves both DROMs and PPINNs. In the DROM experiments, 54-parameter and 81-parameter CFFD deformations are compressed to latent codes, including a 10-dimensional code for the naval hull, and Gaussian process, radial-basis, and tree regressors trained on the codes show lower max and mean errors and lower CPU times than the same regressors trained on the original parameters. In the PPINN experiment, a point-conditioned autoencoder pools per-point features into a 5-dimensional code per geometry, a normalising flow, diffusion model, or energy-based model learns the distribution of those codes, and PPINNs trained on the codes converge to lower loss and lower L1/L2 error against finite-element solutions than PPINNs trained with the 54-parameter FFD parameterisation. The paper attributes the improvement to a softer curse of dimensionality, citing the Gaussian process generalisation-error scaling $m^{-c/\dim(\Gamma)}$, and notes that generation under multilinear constraints such as a fixed barycentre remains compatible with the whole pipeline.

Load-bearing premise

The paper's errors are computed leave-one-out for the surrogate model, but the generative model that defines each sample's latent code is trained on all samples, including the held-out one, so the comparison does not isolate out-of-sample dimension reduction.

Editorial extensions

If this is right

  • A geometry-to-solution pipeline can be prefixed by a generative compression step, replacing dozens of deformation parameters with a latent code of about one fifth to one tenth of the original dimension.
  • The same reduced parameterisation lets PPINNs be trained with fewer parameter and point samples, yielding lower loss and better agreement with finite-element solutions.
  • Because the point-conditioned decoder rebuilds point clouds pointwise, a PPINN solution trained on a coarse mesh can be transferred to a finer mesh without retraining.
  • Constrained shape generation survives the reduction: multilinear constraints such as a fixed barycentre are enforced on the decoder output, so the latent parameterisation still lives inside the admissible geometry family.
  • Sampling time for the generative approach is lower than for the full parameterisation; the paper reports about a twofold speed-up for PPINN sampling.

Reading between the lines

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

  • A stricter evaluation would retrain the autoencoder on each training fold and only then encode the held-out geometry; such a protocol would separate true out-of-sample dimension reduction from the current leave-one-out scheme, in which the encoder has already seen every sample.
  • The same latent-space reduction should extend from regression surrogates to optimisation and uncertainty quantification over shapes, where gradients or sampling over a 5–10 dimensional latent space are far cheaper than over the original 54–81 dimensional parameter space.
  • If the curse-of-dimensionality explanation is right, the accuracy gap between latent and original parameterisations should grow with the original parameter count $l$ and narrow as the latent dimension $d$ approaches $l$; that predicted scaling is directly testable.
  • The point-conditioned encoder is not tied to a fixed mesh, so the approach may generalise to non-matching geometries, adaptive meshes, and time-dependent or coupled GPPDEs where the map from shape to solution is even more expensive.
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 manuscript proposes using generative models (VAE, AAE, BEGAN, NF, DDPM, EBM) to learn low-dimensional latent encodings of deformed geometries, and then using those latent coordinates as reduced parameters for Data-Driven Reduced Order Models (DROMs) and Parametrised Physics-Informed Neural Networks (PPINNs). The central empirical claim is that replacing the original CFFD/FFD parameters (54 or 81 dimensions) with a generative-model latent code improves the accuracy of DROMs and the convergence of PPINNs, thereby mitigating the curse of dimensionality. Two DROM test cases are reported (Poisson on deformed Stanford Bunnies and two-phase flow around DTCHull), plus one PPINN test case (Laplace on deformed Stanford Bunnies). The paper also contains a self-contained review of divergences and generative-model losses, and an appendix with architectural details.

Significance. If the reported gains were shown to hold for genuinely out-of-sample geometries, the proposal would be practically valuable: a low-dimensional latent parameterization of geometry could make non-intrusive ROMs and PPINNs usable in shape optimization and digital twin settings where the original geometric parameter space is high-dimensional. The paper has concrete strengths: it gives explicit algorithms (Algorithm 2.1 for constrained generation, Algorithm 2.2 for point-conditioned encoding), describes the loss functions of the generative models, and tests on a nontrivial multiphase CFD case. However, the central experimental evidence is currently weakened by an asymmetric evaluation protocol: for the generative-model rows of the leave-one-out errors, the latent code of the held-out sample is produced by a model trained on all samples, whereas the CFFD baseline has no fitted encoder. This prevents the paper from isolating a generalizable out-of-sample benefit of dimensionality reduction. The reported inconsistencies in dataset size and missing latent dimensions further complicate interpretation.

major comments (4)
  1. [§2.1, Eqs. (34)–(35)] The leave-one-out comparison is asymmetric between the generative-model rows and the CFFD baseline. For a generative-model row, the test input s_i is the latent code produced by an encoder trained on all m geometries, including geometry i, so s_i is an in-sample encoding optimized to reconstruct geometry i; only the ROM M_{-i} is retrained. For CFFD, the test input is the raw deformation parameter that was known for every geometry and is not fitted to the data. Consequently, the reported error reduction does not isolate a generalizable benefit of dimension reduction; it may reflect the autoencoder's ability to reproduce training geometries. To support the abstract's claim about 'create new geometries', the authors need to train the encoder on a split that excludes the test geometry (or use a separate test set of geometries), and evaluate the full encoder+ROM pipeline on those held-out geometries.
  2. [§2.1, after Eq. (35)] The text states 'In both test cases considered, m=100', but the captions of Figs. 2–4 and 6–8 refer to '600 rabbits' and '600 ships'. The error metrics (34)–(35) depend on m through the max/min normalization and the average, so this is not a purely cosmetic discrepancy. Please clarify which value of m was used for the reported numbers and why the captions differ.
  3. [§2.1, first DROM test case] The reduced latent dimension d is never stated for the Stanford Bunny DROM experiment; only the DTCHull case explicitly reports 'The reduced parametrisation (obtained using Generative Models) is 10'. Since d controls the claimed reduction from 54 parameters and directly affects the curse-of-dimensionality argument, this value must be reported for each generative model and each test case.
  4. [§2.2, Algorithm 2.2 and Fig. 10] The PPINN experiment has the same leakage as the DROM experiment: the point-conditioned autoencoder is trained on the same geometries a_i that are later used to define the FEM reference solutions and the PPINN inputs, and no train/validation split or repeated-seed statistics are reported. This makes it impossible to distinguish the benefit of a lower-dimensional input from the benefit of having fitted the latent representation to the evaluation geometries. Please provide an out-of-sample evaluation and report the mean and standard deviation over multiple seeds.
minor comments (5)
  1. [Throughout] The manuscript alternates between 'DROMs' and 'DROMS' and between 'PPINNs' and 'PPINNS'; please unify the terminology.
  2. [§1.3.2, Eq. (17) and surrounding text] The Adversarial Autoencoder section says 'f_alpha : R^n -> R^k is decoder with weights alpha', but a decoder should map from the latent space R^k to the data space R^n; please correct the domain and codomain.
  3. [Eqs. (34)–(35)] The error definitions do not state explicitly what mu_i is for the generative-model rows. Please clarify that mu_i = s_i (the latent code) in those rows and mu_i = gamma_i for the CFFD/FFD baseline.
  4. [Fig. 10] The y-axis tick labels appear garbled in the PDF rendering (for example '3 x 10 1' instead of likely scientific notation); please check the typesetting.
  5. [Section 3] In the conclusion, 'Free From Deformation' should be 'Free Form Deformation'.

Circularity Check

1 steps flagged · score 5.0 of 10

Generative-model latent codes are fitted on all samples, so the leave-one-out DROM gain is partly an in-sample property rather than an out-of-sample prediction.

  1. fitted input called prediction [Section 2.1, Eqs. (34)-(35) and the paragraphs on CFFD training data and reduced parametrisation]
    "The training geometries a1,...,am were generated using Constrained Free Form Deformation [24] (CFFD)... These samples compose the training data for the Generative Models. ... As shown in Figs. 2, 3, 4 the reduced parametrisation obtained using Generative Models (each reduced parameter of a sample corresponds to the sample’s latent representation in the Generative Model’s latent space) increases the reduced order models performance in all the cases tested. ..."

    By construction, for the generative-model rows, μ_i is the latent code s_i returned by a generative model trained on every geometry a_1,...,a_m, including the held-out geometry i. The leave-one-out protocol of Eqs. (34)-(35) retrains only the ROM M_{-i}; it never retrains or re-encodes the generative model. Hence the test input for fold i is a fitted encoding of the test geometry, so the error is not an out-of-sample measure of the full generative-parameter pipeline, and the CFFD baseline (raw deformation parameters, no fitted encoder) is compared asymmetrically.

full rationale

The paper is an empirical/methodological chapter, not a proof-based derivation, so most of its theoretical sections are standard background and do not enter a circular chain. The DROM method is inherited from the authors' prior article [24], and the self-citation is heavy, but the experiments are new and are compared with an external baseline [52], so self-citation alone does not force the result. The load-bearing circularity is in the validation protocol: for the generative-model rows of the DROM comparison, the reduced parameter used in the leave-one-out error is the latent code s_i produced by an autoencoder trained on all m geometries, while Eqs. (34)-(35) retrain only the downstream ROM M_{-i}. The test input for fold i is therefore a fitted encoding of the held-out geometry, not an independently predicted parameter, so the reported accuracy gain does not measure the full encoder-plus-ROM pipeline on a new geometry; this is the fitted-input-called-prediction pattern. The PPINN experiment has a related lack of an out-of-sample split for the autoencoder codes, but because the reported comparison is framed as fitting/training convergence it is less clearly circular. The central outcome is not mathematically forced — the ROM output is not defined as the latent code — so the circularity is partial rather than total. The manuscript also contains internal inconsistencies (m=100 vs captions saying 600 samples) that affect reproducibility but are not circularity.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests mainly on an unstated fairness premise about the fitted latent space, plus the point-correspondence assumption for PPINNs. No new physical or mathematical entities are invented.

free parameters (3)
  • Latent dimension d (DROM Stanford Bunny) = not reported
    The generative model compression size is chosen by hand; it determines the input dimension of the DROM and is essential to the claimed improvement.
  • Latent dimension d (DROM DTCHull) = 10
    Explicitly stated as the reduced parametrisation dimension; chosen by hand.
  • Latent dimension d (PPINN Stanford Bunny) = 5
    Explicitly stated; chosen by hand.
assumptions (4)
  • domain assumption All deformed point clouds are in bijection with subsamples of the same reference point cloud bar a.
    Assumed in Sec. 2.2 before Algorithm 2.2; needed for the point-conditioned autoencoder and for transferring deformations to finer meshes.
  • standard math ReLU feedforward networks can approximate any target probability measure in Wasserstein distance.
    Used in Sec. 1.2, cited from [38]; grounds the generative model framework.
  • ad hoc to paper The latent representation of a generative model forms a valid reduced parameterization for the PDE solution map.
    This is the central modeling premise of the paper; it is not proven, only empirically suggested.
  • standard math Multilinear constraints can be enforced via Algorithm 2.1 by solving least-norm updates per coordinate.
    Algorithm 2.1 is taken from [24] and requires the constraint to be multilinear; the paper relies on it for barycenter and volume constraints.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Generative Models for Parameter Space Reduction applied to Reduced Order Modelling." pith.science (2026). https://pith.science/paper/2DUFDOEO

@misc{pith2026250609721,
  author       = {Pith},
  title        = {Pith review of: Generative Models for Parameter Space Reduction applied to Reduced Order Modelling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2DUFDOEO}},
  note         = {Machine review of arXiv:2506.09721}
}
read the original abstract

Solving and optimising Partial Differential Equations (PDEs) in geometrically parameterised domains often requires iterative methods, leading to high computational and time complexities. One potential solution is to learn a direct mapping from the parameters to the PDE solution. Two prominent methods for this are Data-driven Non-Intrusive Reduced Order Models (DROMs) and Parametrised Physics Informed Neural Networks (PPINNs). However, their accuracy tends to degrade as the number of geometric parameters increases. To address this, we propose adopting Generative Models to create new geometries, effectively reducing the number of parameters, and improving the performance of DROMs and PPINNs. The first section briefly reviews the general theory of Generative Models and provides some examples, whereas the second focusses on their application to geometries with fixed or variable points, emphasising their integration with DROMs and PPINNs. DROMs trained on geometries generated by these models demonstrate enhanced accuracy due to reduced parameter dimensionality. For PPINNs, we introduce a methodology that leverages Generative Models to reduce the parameter dimensions and improve convergence. This approach is tested on a Poisson equation defined over deformed Stanford Bunny domains.

Discussion (0). Sign in to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. A Generative Model-Free Form Deformation Approach for the Generation of Mesh Motions with Applications to PDE

    math.NA 2026-07 conditional novelty 5.0 of 10

    A control-point ODE flow with a Bernstein-polynomial drift can match genus-0 meshes of different topology, and a POD+autoencoder+normalizing-flow generative model on the control trajectories yields lower-dimensional s...

  2. A Structured Review of Reduced Order Modeling for Domain Decomposition Problems: State of the Art and Perspectives

    math.NA 2026-01 conditional novelty 2.0 of 10

    A structured survey classifying ROM–domain-decomposition coupling methods into intrusive and data-driven families, with engineering-oriented recommendations and open-challenge scoping.

Reference graph

Works this paper leans on

53 extracted references · 52 canonical work pages · cited by 2 Pith papers

  1. [1]

    VIII International Conference on Computational Methods in Marine Engineering (2019)

    Tezzele, M., Demo, N., Rozza, G.: Shape optimization through proper orthogonal decomposi- tion with interpolation and dynamic mode decomposition enhanced by active subspaces. VIII International Conference on Computational Methods in Marine Engineering (2019)

    work page 2019

  2. [2]

    Journal of Scientific Computing, 94, 1-30 (2023)

    Siena, P., Girfolgio, G., Ballarin, F., Rozza G.: Data-driven reduced order modelling for patient-specific hemodynamics of coronary artery bypass grafts with physical and geometrical parameters. Journal of Scientific Computing, 94, 1-30 (2023)

    work page 2023

  3. [3]

    Demo, N., Tezzele, M., Mola, A., Rozza, G.: Hull shape design optimization with parameter space and model reductions, and self-learning mesh morphing, 9, 1-22 (2021)

    work page 2021

  4. [4]

    Numerical Methods in Engineering, 125, e7426 (2024)

    Ivagnes, A., Demo, N., Rozza, G.: A Shape Optimization Pipeline for Marine Propellers by means of Reduced Order Modeling Techniques. Numerical Methods in Engineering, 125, e7426 (2024)

    work page 2024

  5. [5]

    Numerical Methods in Engineering, 124, 1193-1210 (2022)

    Tezzele, M., Fabris, L., Sidari, M., Siccherio, M., Rozza, G.: A multifidelity approach coupling parameter space reduction and nonintrusive POD with application to structural optimization of passenger ship hulls. Numerical Methods in Engineering, 124, 1193-1210 (2022)

    work page 2022

  6. [6]

    Computers & Mathematics with Applications, 151, 115-127 (2023)

    Gonnella, C.I., Hess, M.W., Stabile, G., Rozza, G.: A two-stage deep learning architecture for model reduction of parametric time-dependent problems. Computers & Mathematics with Applications, 151, 115-127 (2023)

    work page 2023

  7. [7]

    Enhancing non-intrusive Reduced Order Models with space-dependent aggregation methods

    Ivagnes, A., Tonicello, N., Cinnella, P., Rozza G.: Enhancing non-intrusive Reduced Order Models with space-dependent aggregation methods. ArXiv, 2403.05710 (2024)

    work page Pith review arXiv 2024

  8. [8]

    Optimal Transport-inspired Deep Learning Framework for Slow-Decaying Kolmogorov n-width Problems: Exploiting Sinkhorn Loss and Wasserstein Kernel

    Khamlich, M., Pichi, F., Rozza, G.: Optimal Transport-inspired Deep Learning Framework for Slow-Decaying Problems: Exploiting Sinkhorn Loss and Wasserstein Kernel. ArXiv, 2308.13840 (2023)

    work page Pith review arXiv 2023

Show all 53 references
  1. [9]

    Computer Methods in Applied Mechanics and Engineering,392, 114687 (2022)

    Papapicco, D., Demo, D., Girfoglio, M., Stabile, G., Rozza, G.: The Neural Network shifted- proper orthogonal decomposition: A machine learning approach for non-linear reduction of hyperbolic equations. Computer Methods in Applied Mechanics and Engineering,392, 114687 (2022)

    2022

  2. [10]

    Computers & Fluids, 254, 105813 (2023)

    Pichi, F., Ballarin, F., Rozza, G., Hesthaven, J.S.: An artificial neural network approach to bifurcating phenomena in computational fluid dynamics. Computers & Fluids, 254, 105813 (2023)

    2023

  3. [11]

    The Journal of Open Source Software, 3, 661 (2018)

    Demo, N., Tezzele, M., Mola, A., Rozza, G.: EZyRB: Easy Reduced Basis method. The Journal of Open Source Software, 3, 661 (2018)

    2018

  4. [12]

    International Journal for Numerical Methods in Fluids, 83, 291–306 (2016)

    Lin, Z., Xiao, D., Fang, F., Pain, C.C., Navon, I.M.: Non-intrusive reduced order modelling with least squares fitting on a sparse grid. International Journal for Numerical Methods in Fluids, 83, 291–306 (2016)

    2016

  5. [13]

    & Schwab, C

    Chen, P. & Schwab, C. Adaptive Sparse Grid Model Order Reduction for Fast Bayesian Estimation and Inversion. Sparse Grids And Applications - Stuttgart 2014, 1-27 (2016)

    2016

  6. [14]

    & Allen, M

    Park, K. & Allen, M. A Gaussian process regression reduced order model for geometrically nonlinear structures. Mechanical Systems And Signal Processing. 184, 109720 (2023)

    2023

  7. [15]

    Procacci, A., Donato, L., Amaduzzi, R. et al. Parameter Estimation Using a Gaussian Process Regression-Based Reduced-Order Model and Sparse Sensing: Application to a Methane/Air Lifted Jet Flame. Flow Turbulence Combust112, 879–895 (2024)

    2024

  8. [16]

    Acta Astronautica, 185, 25–36 (2021)

    Sun, F., Su, W.-Y., Wang, M.-Y., Wang, R.-J.: RBF-POD reduced-order modeling of flow field in the curved shock compression inlet. Acta Astronautica, 185, 25–36 (2021)

    2021

  9. [17]

    Journal of Computational Physics, 439, 110378 (2021)

    Dutta, S., Farthing, M.W., Perracchione, E., Savant, G., Putti, M.: A greedy non-intrusive reduced order model for shallow water equations. Journal of Computational Physics, 439, 110378 (2021)

    2021

  10. [18]

    Journal of Computational and Applied Mathematics, 390, 113372 (2021)

    Sun, X., Pan, X., Choi, J.-I.: Non-intrusive framework of reduced-order modeling based on proper orthogonal decomposition and polynomial chaos expansion. Journal of Computational and Applied Mathematics, 390, 113372 (2021)

    2021

  11. [19]

    On: D’Elia, M., Gunzburger, M., Rozza, G

    Hijazi, S., Stabile, G., Mola, A., Rozza, G.: Non-intrusive polynomial chaos method ap- plied to Full-Order and reduced problems in computational fluid dynamics: a comparison 24 Guglielmo Padula and Gianluigi Rozza and perspectives. On: D’Elia, M., Gunzburger, M., Rozza, G. (e...

    2020

  12. [20]

    Computers & Mathematics with Applications, 143, 383-396 (2023)

    Demo, N., Strazzullo, M., Rozza, G.: An extended physics informed neural network for preliminary analysis of parametric optimal control problems. Computers & Mathematics with Applications, 143, 383-396 (2023)

    2023

  13. [21]

    The Journal of Open Source Software, 8, 5352 (2023)

    Coscia, D., Ivagnes, A., Demo, N., Rozza, G.: Physics-Informed Neural networks for Advanced modeling. The Journal of Open Source Software, 8, 5352 (2023)

    2023

  14. [22]

    Scientific Reports, 14, 3826 (2024)

    Coscia, D., Demo, N., Rozza, G.: Generative adversarial reduced order modelling. Scientific Reports, 14, 3826 (2024)

    2024

  15. [23]

    SIAM, Philadelphia (2020)

    Rozza, G., Stabile, G., Ballarin, F.: Advanced Reduced Order Methods and Applications in Computational Fluid Dynamics. SIAM, Philadelphia (2020)

    2020

  16. [24]

    Computer Methods in Applied Mechanics and Engineering, 423, 116823 (2024)

    Padula, G., Romor, F., Stabile, G., Rozza, G.: Generative models for the deformation of indus- trial shapes with linear geometric constraints: Model order and parameter space reductions. Computer Methods in Applied Mechanics and Engineering, 423, 116823 (2024)

    2024

  17. [25]

    European Conference on Computer Vision (2018)

    Ranjan, A., Bolkart, T., Sanyal, S., Black, M.J.: Generating 3D faces using Convolutional Mesh Autoencoders. European Conference on Computer Vision (2018)

    2018

  18. [26]

    2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (2020)

    Yuan, Y.-J, Lai, Y.-K., Yang, J., Duan, Q., Fu, H., Gao L.: Mesh variational autoencoders with edge contraction pooling. 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (2020)

    2020

  19. [27]

    MIT Press, Cambridge (2016)

    Goodfellow, I., Bengio, Y., Courville, A.: Deep Learning. MIT Press, Cambridge (2016)

    2016

  20. [28]

    Noncompact uniform universal approximation.Neural Networks

    Van Nuland, T. Noncompact uniform universal approximation.Neural Networks. 173, 106181 (2024)

    2024

  21. [29]

    Simonnet, M.: Measures and probabilities. (1996)

    1996

  22. [30]

    International Conference on Learning Representations (2014)

    Kingma, D.P., Welling M.: Auto-Encoding Variational Bayes. International Conference on Learning Representations (2014)

    2014

  23. [31]

    Advances in Neural Information Processing Systems 33 (2020)

    Ho, J., Jain, A., Abbeel, P.:Denoising Diffusion Probabilistic Models. Advances in Neural Information Processing Systems 33 (2020)

    2020

  24. [32]

    and Courville, A

    Goodfellow, I., Pouget-Abadie, J., Mirza, M., Xu, B., Warde-Farley, D., Ozair, S. and Courville, A. and Bengio, Y.: Advances in Neural Information Processing Systems 27 (2014)

    2014

  25. [33]

    International Conference on Learning Representations (2016)

    Makhzani, A., Shlens J., Jaitly, N., Goodfellow I.: Adversarial Autoencoders. International Conference on Learning Representations (2016)

    2016

  26. [34]

    Pro- ceedings of the 34th International Conference on Machine Learning (2017)

    Arjovsky, M., Chintala, S., Bottou, L.: Wasserstein Generative Adversarial Networks. Pro- ceedings of the 34th International Conference on Machine Learning (2017)

    2017

  27. [35]

    Sason and S

    I. Sason and S. Verd´ u: ”𝑓 -Divergence Inequalities,” in IEEE Transactions on Information Theory, 62, 5973 (2016)

    2016

  28. [36]

    On Choosing and Bounding Probability Metrics

    Gibbs, Alison L., Francis Edward Su. “On Choosing and Bounding Probability Metrics.” International Statistical Review / Revue Internationale de Statistique,70, 419 (2002)

    2002

  29. [37]

    Villani, C.: Optimal transport. (2008)

    2008

  30. [38]

    Yang, Y., Li, Z., Wang Y.: On the capacity of deep generative networks for approximating distributions, Neural Networks, 145, 144 (2022)

    2022

  31. [39]

    & Cresswell, J

    Loaiza-Ganem, G., Ross, B., Hosseinzadeh, R., Caterini, A. & Cresswell, J. Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connections. Transactions On Machine Learning Research. (2024)

    2024

  32. [40]

    ArXiv, 1703.10717 (2017)

    Berthelot, D., Schumm, T., Metz, L.: BEGAN: Boundary Equilibrium Generative Adversarial Networks. ArXiv, 1703.10717 (2017)

    2017 arXiv

  33. [41]

    Proceedings of the 32nd International Conference on Machine Learning (2015)

    Rezende, D., Mohamed, S.: Variational Inference with Normalizing Flows. Proceedings of the 32nd International Conference on Machine Learning (2015)

    2015

  34. [42]

    Ad- vances in Neural Information Processing Systems 32 (2019)

    Du, Y., Mordatch, I.: Implicit Generation and Generalization in Energy-Based Models. Ad- vances in Neural Information Processing Systems 32 (2019)

    2019

  35. [43]

    Software Impacts, 7 (2021)

    Tezzele M., Demo N., Mola A., Rozza G.: PyGeM: Python Geometrical Morphing. Software Impacts, 7 (2021)

    2021

  36. [44]

    MIT Press, Cambridge (2006) Generative Models for Parameter Space Reduction applied to Reduced Order Modelling 25

    Rasmussen, C.E., Williams, C.K.I: Gaussian Processes for Machine Learning. MIT Press, Cambridge (2006) Generative Models for Parameter Space Reduction applied to Reduced Order Modelling 25

    2006

  37. [45]

    Cambridge University Press, Cambridge (2009)

    Buhamann., D.: Radial Basis Functions. Cambridge University Press, Cambridge (2009)

    2009

  38. [46]

    24, 303(2005)

    Rao, C.R., Data Mining And Data Visualization. 24, 303(2005)

    2005

  39. [47]

    & Duchesnay, E

    Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., Vanderplas, J., Passos, A., Cournapeau, D., Brucher, M., Perrot, M. & Duchesnay, E. Scikit-learn: Machine Learning in Python. Journal Of Mach...

    2011

  40. [48]

    Proceedings of the 21st Annual Conference on Computer Graphics and Interactive Techniques - SIGGRAPH ’94, ACM Press (1994)

    Turk G., Levoy M.: Zippered Polygon Meshes from Range Images. Proceedings of the 21st Annual Conference on Computer Graphics and Interactive Techniques - SIGGRAPH ’94, ACM Press (1994)

    1994

  41. [49]

    SNAME Maritime Convention, SNAME (2019)

    White, P.F., Knight, B.G., Filip, G.P., Maki, K.J.: Numerical Simulations of the Duisburg Test Case Hull Maneuvering in Waves. SNAME Maritime Convention, SNAME (2019)

    2019

  42. [50]

    Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2017)

    Qi, C.R., Su, H., Mo, K., Guibas, L.J.: PointNet: Deep Learning on Point Sets for 3D Clas- sification and Segmentation. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2017)

    2017

  43. [51]

    Proceedings of the IEEE/CVF International Conference on Computer Vision (2019)

    Yang, G., Huang, X., Hao, Z., and Liu, M., Belongie, S., and Hariharan, B.: PointFlow: 3D Point Cloud Generation With Continuous Normalizing Flows. Proceedings of the IEEE/CVF International Conference on Computer Vision (2019)

    2019

  44. [52]

    Advanced Modeling and Simulation in Engineering Sciences, 9, 8 (2022)

    Oldenburg, J., Borowski, F., ¨Oner, A., Schmitz, K., Stiehm, M.: Geometry aware physics informed neural network surrogate for solving Navier–Stokes equation (GAPINN). Advanced Modeling and Simulation in Engineering Sciences, 9, 8 (2022)

    2022

  45. [53]

    Communications in Applied and Industrial Mathematics (2013) 4 Appendix Here we describe the details of the architectures and hyperparameters adopted to train the Generative Models

    Koshakji, A., Quarteroni, A., Rozza, G.: Free Form Deformation Techniques Applied to 3D Shape Optimization Problems. Communications in Applied and Industrial Mathematics (2013) 4 Appendix Here we describe the details of the architectures and hyperparameters adopted to train th...

    2013

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.