pith. sign in

arxiv: 2605.21499 · v1 · pith:GXL467GWnew · submitted 2026-05-05 · ⚛️ physics.flu-dyn · cs.LG

Conditional Neural Field based Reduced Order Model for Dynamic Ditching Load Prediction

Henning Schwarz , Pyei Phyo Lin , Jens-Peter M. Zemke , Thomas Rung This is my paper

Pith reviewed 2026-05-22 00:51 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn cs.LG
keywords conditional neural fieldsreduced order modelingditching loadsaircraft fuselagespatio-temporal predictionLSTM networkscomputational fluid dynamicssurrogate models
0
0 comments X

The pith

Conditional neural fields paired with LSTMs predict aircraft ditching loads accurately across different spatial grids using fewer parameters than grid-based methods.

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

The paper establishes that coordinate-based conditional neural fields can serve as discretization-independent reduced-order models for the spatio-temporal evolution of dynamic pressure loads on an aircraft fuselage during ditching. This approach matters because standard grid-dependent methods such as convolutional autoencoders tie the surrogate to one fixed mesh, making it hard to combine data from varied geometries or resolutions that arise in aerospace practice. By representing the load field as a function of spatial coordinates conditioned on a latent code and advancing that code with a long short-term memory network, the model reaches prediction accuracy comparable to convolutional autoencoders on single-grid data while using far fewer parameters. On the second dataset containing heterogeneous discretizations, the same architecture reconstructs loads accurately without grid-specific retraining or major fidelity loss. A sympathetic reader cares because the resulting surrogate can therefore draw on mixed training sources and be applied to new configurations without rebuilding the entire model.

Core claim

The central claim is that a conditional neural field, when paired with an LSTM network in the latent space, delivers spatio-temporal predictions of ditching loads whose accuracy on a fixed discretization approaches that of convolutional autoencoder models yet requires significantly fewer parameters, while the same architecture additionally reconstructs loads accurately when trained on data from heterogeneous spatial discretizations.

What carries the argument

The conditional neural field, which encodes the load distribution as a coordinate-based function of a latent vector so that the representation does not depend on any particular mesh topology or resolution.

If this is right

  • Training data collected on multiple meshes or geometries can be pooled directly without interpolation to a common grid.
  • The trained surrogate can be queried for load predictions at any desired spatial locations rather than only at nodes of the original mesh.
  • The reduced parameter count lowers memory and training cost for repeated evaluations in design studies.
  • The same latent-space LSTM evolution can be reused when the underlying geometry changes, provided the neural field is conditioned appropriately.
  • Predictions remain available even when the test discretization differs from every discretization seen during training.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method could be tested on problems with moving contact lines or deforming surfaces where grid topology changes at every time step, extending the heterogeneity already handled here.
  • Combining simulation data from one mesh family with experimental pressure measurements from irregular sensor layouts becomes feasible without explicit remapping.
  • If the latent space proves sufficiently smooth, the model might support gradient-based optimization of fuselage shape for reduced ditching loads by differentiating through the neural field.
  • The approach invites direct comparison with other coordinate-based representations such as implicit neural representations or Fourier feature mappings on the same ditching datasets.

Load-bearing premise

The coordinate-based formulation can learn load patterns that generalize across mesh resolutions and topologies without introducing significant artifacts or fidelity loss.

What would settle it

Training the model on one set of discretizations and then measuring large localized errors or grid-dependent artifacts when it is asked to predict loads on a new, unseen discretization of the same geometry would falsify the generalization claim.

Figures

Figures reproduced from arXiv: 2605.21499 by Henning Schwarz, Jens-Peter M. Zemke, Pyei Phyo Lin, Thomas Rung.

Figure 1
Figure 1. Figure 1: FIG. 1. Scheme of the conditional neural field architecture. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Exemplary normalized pressure loads on the D150 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Velocity pairs for the training set of dataset A. [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Reconstruction loss on the training and validation sets of dataset A for different dimensions [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Reconstruction loss obtained for the CNF with di [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Velocity pairs for the training set of dataset B. Blue [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Velocity pairs for the test sets. Red square refers to [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Reconstruction loss obtained for the CNF with di [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Simulated loads with the different unseen discretizations at three time steps of the test case and the corresponding [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
read the original abstract

Grid-based neural networks such as convolutional autoencoders are widely used in dimension reduction-based surrogate models for computational fluid dynamics. In recent years, the use of coordinate-based approaches like conditional neural fields has emerged. Their independence of the spatial discretization is a beneficial feature for various applications in computational fluid dynamics. This paper discusses the spatio-temporal prediction of aircraft ditching loads using a conditional neural field approach. The model is evaluated using two datasets for the dynamic loads of the fuselage of a DLR-D150 aircraft, one of which relates to a single fixed spatial discretization and the other that includes data from different discretizations. When paired with a long short-term memory (LSTM) network in the latent space, the neural field-based model achieves a spatio-temporal prediction accuracy for the first data set that is close to that of grid-dependent convolutional autoencoder-based models, and with significantly less parameters. Results for the second data set demonstrate the ability of the neural field-based approach to reconstruct ditching loads accurately for heterogeneous spatial discretizations. This allows for flexible use of training datasets generated for different geometries and/or discretizations, as well as the use of the surrogate model to predict loads for different configurations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes a conditional neural field (CNF) reduced-order model paired with an LSTM in the latent space for spatio-temporal prediction of dynamic ditching loads on a DLR-D150 aircraft fuselage. It evaluates the approach on two CFD datasets: one with a single fixed spatial discretization and a second with heterogeneous discretizations. The central claims are that the CNF-LSTM model achieves prediction accuracy close to grid-dependent convolutional autoencoder baselines while using significantly fewer parameters on the first dataset, and that it accurately reconstructs loads across varying spatial discretizations on the second dataset, enabling flexible use of mixed training data.

Significance. If the empirical claims are substantiated with quantitative metrics, this approach could offer a useful discretization-independent alternative to convolutional reduced-order models in fluid dynamics surrogate modeling. The potential for training on heterogeneous meshes and the reported parameter efficiency would be relevant for applications involving complex geometries or variable discretizations, such as aircraft ditching or other transient load predictions.

major comments (2)
  1. [Results] Results section (description of first dataset): The claim that the CNF-LSTM model achieves 'spatio-temporal prediction accuracy ... close to that of grid-dependent convolutional autoencoder-based models' is not supported by any reported quantitative error metrics (e.g., RMSE, relative L2 norm, or time-averaged errors), validation split details, or tabulated baseline comparisons, leaving the central accuracy and parameter-efficiency assertions only qualitatively described.
  2. [Results] Results section (second dataset): The assertion that the model 'reconstructs ditching loads accurately for heterogeneous spatial discretizations' rests on the untested assumption that coordinate conditioning alone compensates for differences in mesh resolution and topology; no ablation isolating mesh variation effects, no analysis of potential resolution-dependent bias or boundary artifacts, and no quantitative reconstruction errors on held-out heterogeneous meshes are provided.
minor comments (2)
  1. [Abstract] Abstract: The statement 'with significantly less parameters' should be accompanied by explicit parameter counts for the CNF model versus the convolutional baseline to allow direct comparison.
  2. [Methods] Methods: The precise formulation of the conditional neural field (e.g., how spatial coordinates and the latent code are concatenated or modulated) and the training procedure for the LSTM in latent space should be clarified with equations or pseudocode.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed comments. We agree that strengthening the quantitative support for our claims will improve the manuscript and address each major comment below with proposed revisions.

read point-by-point responses

  1. Referee: [Results] Results section (description of first dataset): The claim that the CNF-LSTM model achieves 'spatio-temporal prediction accuracy ... close to that of grid-dependent convolutional autoencoder-based models' is not supported by any reported quantitative error metrics (e.g., RMSE, relative L2 norm, or time-averaged errors), validation split details, or tabulated baseline comparisons, leaving the central accuracy and parameter-efficiency assertions only qualitatively described.

    Authors: We agree that the manuscript would benefit from explicit quantitative metrics to support the accuracy and parameter-efficiency claims. In the revised version, we will add a dedicated table in the Results section reporting RMSE, relative L2 norms, and time-averaged errors for the CNF-LSTM model against the convolutional autoencoder baselines on the first dataset. We will also specify the validation split details and include a direct comparison of parameter counts. These additions will provide the quantitative foundation for the stated claims without altering the core findings. revision: yes

  2. Referee: [Results] Results section (second dataset): The assertion that the model 'reconstructs ditching loads accurately for heterogeneous spatial discretizations' rests on the untested assumption that coordinate conditioning alone compensates for differences in mesh resolution and topology; no ablation isolating mesh variation effects, no analysis of potential resolution-dependent bias or boundary artifacts, and no quantitative reconstruction errors on held-out heterogeneous meshes are provided.

    Authors: We acknowledge the value of quantitative evidence and targeted analysis for the heterogeneous discretization case. We will revise the Results section to report quantitative reconstruction errors (RMSE and relative L2) on held-out heterogeneous meshes. We will also add an ablation study isolating mesh variation effects and a brief analysis of resolution-dependent bias and boundary artifacts to demonstrate that coordinate conditioning handles these variations effectively. This will directly address the concerns while preserving the manuscript's emphasis on flexibility across discretizations. revision: yes

Circularity Check

0 steps flagged

No significant circularity; empirical training on external CFD data with held-out validation.

full rationale

The paper trains a conditional neural field + LSTM surrogate end-to-end on CFD-generated ditching load data. Spatial predictions are produced by querying the coordinate-conditioned network at arbitrary points; temporal evolution is handled by the LSTM in latent space. Accuracy is measured against held-out CFD snapshots and compared to an independent convolutional autoencoder baseline. The heterogeneous-discretization claim is supported by training on mixed-mesh data and testing on unseen meshes, which is a standard empirical generalization test rather than a definitional reduction or self-referential fit. No equations, uniqueness theorems, or self-citations are invoked to force the reported performance by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard neural approximation assumptions plus the domain premise that fluid load fields admit continuous coordinate-based representations; no new physical entities are postulated and the only free parameters are the usual neural-network weights fitted to CFD data.

free parameters (1)
  • Neural network weights and biases
    Parameters of the conditional neural field and LSTM are optimized on the provided ditching-load datasets.
axioms (1)
  • domain assumption Fluid load distributions admit a continuous functional representation conditioned on latent variables and time that is independent of any particular spatial discretization.
    This premise is required for the coordinate-based neural field to reconstruct fields from heterogeneous meshes.

pith-pipeline@v0.9.0 · 5751 in / 1397 out tokens · 92106 ms · 2026-05-22T00:51:10.447445+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

45 extracted references · 45 canonical work pages

  1. [1]

    author author J. L. \ Lumley ,\ title title The structure of inhomogeneous turbulent flows , \ @noop journal journal Atmospheric turbulence and radio wave propagation \ ,\ pages 166--178 ( year 1967 ) NoStop

    work page 1967

  2. [2]

    author author L. Agostini ,\ title title Exploration and prediction of fluid dynamical systems using auto-encoder technology , \ https://doi.org/10.1063/5.0012906 journal journal Physics of Fluids \ volume 32 ,\ pages 067103 ( year 2020 ) NoStop

    work page doi:10.1063/5.0012906 2020

  3. [3]

    Eivazi , author H

    author author H. Eivazi , author H. Veisi , author M. H. \ Naderi ,\ and\ author V. Esfahanian ,\ title title Deep neural networks for nonlinear model order reduction of unsteady flows , \ https://doi.org/10.1063/5.0020526 journal journal Physics of Fluids \ volume 32 ,\ pages 105104 ( year 2020 ) NoStop

    work page doi:10.1063/5.0020526 2020

  4. [4]

    Pache \ and\ author T

    author author R. Pache \ and\ author T. Rung ,\ title title Data-driven surrogate modeling of aerodynamic forces on the superstructure of container vessels , \ https://doi.org/10.1080/19942060.2022.2044383 journal journal Engineering Applications of Computational Fluid Mechanics \ volume 16 ,\ pages 746--763 ( year 2022 ) NoStop

    work page doi:10.1080/19942060.2022.2044383 2022

  5. [5]

    Eivazi , author S

    author author H. Eivazi , author S. L. \ Clainche , author S. Hoyas ,\ and\ author R. Vinuesa ,\ title title Towards extraction of orthogonal and parsimonious non-linear modes from turbulent flows , \ https://doi.org/https://doi.org/10.1016/j.eswa.2022.117038 journal journal Expert Systems with Applications \ volume 202 ,\ pages 117038 ( year 2022 ) NoStop

    work page doi:10.1016/j.eswa.2022.117038 2022

  6. [6]

    \ Kang , author S

    author author Y.-E. \ Kang , author S. Yang ,\ and\ author K. Yee ,\ title title Physics-aware reduced-order modeling of transonic flow via -variational autoencoder , \ https://doi.org/10.1063/5.0097740 journal journal Physics of Fluids \ volume 34 ,\ pages 076103 ( year 2022 ) NoStop

    work page doi:10.1063/5.0097740 2022

  7. [7]

    Solera-Rico, C

    author author A. Solera-Rico, C. Sanmiguel Vila, M. Gómez-López, Y. Wang, A. Almashjary, S. Dawson, and R. Vinuesa ,\ title title -variational autoencoders and transformers for reduced-order modelling of fluid flows , \ https://doi.org/10.1038/s41467-024-45578-4 journal journal Nature Communications \ volume 15 ( year 2024 ),\ 10.1038/s41467-024-45578-4 NoStop

    work page doi:10.1038/s41467-024-45578-4 2024

  8. [8]

    U berr\

    author author H. Schwarz , author M. \" U berr\" u ck , author J.-P. M. \ Zemke ,\ and\ author T. Rung ,\ title title Machine learning based prediction of ditching loads , \ https://doi.org/10.2514/1.J064086 journal journal AIAA Journal \ volume 63 ,\ pages 1835--1854 ( year 2025 ) NoStop

    work page doi:10.2514/1.j064086 2025

  9. [9]

    Loft , author H

    author author M. Loft , author H. Schwarz ,\ and\ author T. Rung ,\ title title Data-driven pressure field prediction for ships in regular sea states , \ https://doi.org/10.1080/19942060.2025.2553337 journal journal Engineering Applications of Computational Fluid Mechanics \ volume 19 ,\ pages 2553337 ( year 2025 ) NoStop

    work page doi:10.1080/19942060.2025.2553337 2025

  10. [10]

    Long short -term memory,

    author author S. Hochreiter \ and\ author J. Schmidhuber ,\ title title Long short-term memory , \ https://doi.org/10.1162/neco.1997.9.8.1735 journal journal Neural Computation \ volume 9.8 ,\ pages 1735–1780 ( year 1997 ) NoStop

    work page doi:10.1162/neco.1997.9.8.1735 1997

  11. [11]

    Vaswani, N

    author author A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin ,\ title title Attention is all you need , \ in\ https://proceedings.neurips.cc/paper/2017/file/3f5ee243547dee91fbd053c1c4a845aa-Paper.pdf booktitle Advances in Neural Information Processing Systems \ ( year 2017 ) NoStop

    work page 2017

  12. [12]

    Wu , author S

    author author P. Wu , author S. Gong , author K. Pan , author F. Qiu , author W. Feng ,\ and\ author C. Pain ,\ title title Reduced order model using convolutional auto-encoder with self-attention , \ https://doi.org/10.1063/5.0051155 journal journal Physics of Fluids \ volume 33 ,\ pages 077107 ( year 2021 ) NoStop

    work page doi:10.1063/5.0051155 2021

  13. [13]

    Hemmasian \ and\ author A

    author author A. Hemmasian \ and\ author A. Barati Farimani ,\ title title Reduced-order modeling of fluid flows with transformers , \ https://doi.org/10.1063/5.0151515 journal journal Physics of Fluids \ volume 35 ,\ pages 057126 ( year 2023 ) NoStop

    work page doi:10.1063/5.0151515 2023

  14. [14]

    author author M. A. \ Maram , author G. Bletsos , author T. T. \ Nguyen , author A. Hassan , author M. Palm ,\ and\ author T. Rung ,\ https://arxiv.org/abs/2602.14907 title Adjoint-based shape optimization of a ship hull using a conditional variational autoencoder (cvae) assisted propulsion surrogate model , \ ( year 2026 ),\ https://arxiv.org/abs/2602.14...

    work page arXiv 2026

  15. [15]

    Pfaff , author M

    author author T. Pfaff , author M. Fortunato , author A. Sanchez-Gonzalez ,\ and\ author P. Battaglia ,\ title title Learning mesh-based simulation with graph networks , \ in\ @noop booktitle International Conference on Learning Representations \ ( year 2021 ) NoStop

    work page 2021

  16. [16]

    Brandstetter , author D

    author author J. Brandstetter , author D. E. \ Worrall ,\ and\ author M. Welling ,\ title title Message passing neural PDE solvers , \ in\ @noop booktitle International Conference on Learning Representations \ ( year 2022 ) NoStop

    work page 2022

  17. [17]

    Barwey , author V

    author author S. Barwey , author V. Shankar , author V. Viswanathan ,\ and\ author R. Maulik ,\ title title Multiscale graph neural network autoencoders for interpretable scientific machine learning , \ https://doi.org/https://doi.org/10.1016/j.jcp.2023.112537 journal journal Journal of Computational Physics \ volume 495 ,\ pages 112537 ( year 2023 ) NoStop

    work page doi:10.1016/j.jcp.2023.112537 2023

  18. [18]

    author author N. R. \ Franco , author S. Fresca , author F. Tombari ,\ and\ author A. Manzoni ,\ title title Deep learning-based surrogate models for parametrized pdes: Handling geometric variability through graph neural networks , \ @noop journal journal Chaos: An Interdisciplinary Journal of Nonlinear Science \ volume 33 ,\ pages 123121 ( year 2023 ) NoStop

    work page 2023

  19. [19]

    Serrano , author L

    author author L. Serrano , author L. Le Boudec , author A. Kassa\" Koupa\" , author T. X. \ Wang , author Y. Yin , author J.-N. \ Vittaut ,\ and\ author P. Gallinari ,\ title title Operator learning with neural fields: Tackling pdes on general geometries , \ in\ https://proceedings.neurips.cc/paper_files/paper/2023/file/df54302388bbc145aacaa1a54a4a5933-Pa...

    work page 2023

  20. [20]

    Xie , author T

    author author Y. Xie , author T. Takikawa , author S. Saito , author O. Litany , author S. Yan , author N. Khan , author F. Tombari , author J. Tompkin , author V. Sitzmann ,\ and\ author S. Sridhar ,\ title title Neural fields in visual computing and beyond , \ https://doi.org/https://doi.org/10.1111/cgf.14505 journal journal Computer Graphics Forum \ vo...

    work page doi:10.1111/cgf.14505

  21. [21]

    author author J. J. \ Park , author P. Florence , author J. Straub , author R. Newcombe ,\ and\ author S. Lovegrove ,\ title title Deepsdf: Learning continuous signed distance functions for shape representation , \ in\ @noop booktitle Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) \ ( year 2019 ) NoStop

    work page 2019

  22. [22]

    Yin , author M

    author author Y. Yin , author M. Kirchmeyer , author J.-Y. \ Franceschi , author A. Rakotomamonjy ,\ and\ author P. Gallinari ,\ title title Continuous PDE dynamics forecasting with implicit neural representations , \ in\ @noop booktitle The Eleventh International Conference on Learning Representations \ ( year 2023 ) NoStop

    work page 2023

  23. [23]

    Du , author M

    author author P. Du , author M. H. \ Parikh , author X. Fan , author X.-Y. \ Liu ,\ and\ author J.-X. \ Wang ,\ title title Conditional neural field latent diffusion model for generating spatiotemporal turbulence , \ https://doi.org/10.1038/s41467-024-54712-1 journal journal Nature Communications \ volume 15 ( year 2024 ),\ 10.1038/s41467-024-54712-1 NoStop

    work page doi:10.1038/s41467-024-54712-1 2024

  24. [24]

    Guo , author P

    author author J. Guo , author P. Du , author X. Fan , author Y. Li ,\ and\ author J.-X. \ Wang ,\ title title Conditional neural field for spatial dimension reduction of turbulence data: A comparison study , \ https://doi.org/10.1063/5.0310238 journal journal Physics of Fluids \ volume 38 ,\ pages 025153 ( year 2026 ) NoStop

    work page doi:10.1063/5.0310238 2026

  25. [25]

    Catalani , author J

    author author G. Catalani , author J. Fesquet , author X. Bertrand , author F. Tost , author M. Bauerheim ,\ and\ author J. Morlier ,\ title title Towards scalable surrogate models based on neural fields for large scale aerodynamic simulations , \ https://doi.org/https://doi.org/10.1016/j.compfluid.2025.106929 journal journal Computers & Fluids \ volume 3...

    work page doi:10.1016/j.compfluid.2025.106929 2025

  26. [26]

    Li , author N

    author author Z. Li , author N. B. \ Kovachki , author K. Azizzadenesheli , author B. Liu , author K. Bhattacharya , author A. Stuart ,\ and\ author A. Anandkumar ,\ title title Fourier neural operator for parametric partial differential equations , \ in\ https://openreview.net/forum?id=c8P9NQVtmnO booktitle International Conference on Learning Representa...

    work page 2021

  27. [27]

    Wang , author J

    author author S. Wang , author J. H. \ Seidman , author S. Sankaran , author H. Wang , author G. J. \ Pappas ,\ and\ author P. Perdikaris ,\ https://arxiv.org/abs/2405.13998 title Cvit: Continuous vision transformer for operator learning , \ ( year 2025 ),\ https://arxiv.org/abs/2405.13998 arXiv:2405.13998 [cs.LG] NoStop

    work page arXiv 2025

  28. [28]

    Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators

    author author L. Lu , author P. Jin , author G. Pang , author Z. Zhang ,\ and\ author G. E. \ Karniadakis ,\ title title Learning nonlinear operators via deeponet based on the universal approximation theorem of operators , \ https://doi.org/10.1038/s42256-021-00302-5 journal journal Nature Machine Intelligence \ volume 3 ,\ pages 218–229 ( year 2021 ) NoStop

    work page doi:10.1038/s42256-021-00302-5 2021

  29. [29]

    Pan , author S

    author author S. Pan , author S. L. \ Brunton ,\ and\ author J. N. \ Kutz ,\ title title Neural implicit flow: a mesh-agnostic dimensionality reduction paradigm of spatio-temporal data , \ http://jmlr.org/papers/v24/22-0365.html journal journal Journal of Machine Learning Research \ volume 24 ,\ pages 1--60 ( year 2023 ) NoStop

    work page 2023

  30. [30]

    Raissi, P

    author author M. Raissi , author P. Perdikaris ,\ and\ author G. Karniadakis ,\ title title Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations , \ https://doi.org/https://doi.org/10.1016/j.jcp.2018.10.045 journal journal Journal of Computational Physics \ ...

    work page doi:10.1016/j.jcp.2018.10.045 2018

  31. [31]

    Streckwall , author O

    author author H. Streckwall , author O. Lindenau ,\ and\ author L. Bensch ,\ title title Aircraft ditching: a free surface/free motion problem , \ https://doi.org/https://doi.org/10.1016/S1644-9665(12)60025-9 journal journal Archives of Civil and Mechanical Engineering \ volume 7 ,\ pages 177--190 ( year 2007 ) NoStop

    work page doi:10.1016/s1644-9665(12)60025-9 2007

  32. [32]

    Schwarz, P

    author author H. Schwarz, P. P. Lin, J.-P. M. Zemke and T. Rung ,\ title title Disentangled latent spaces for reduced order models using deterministic autoencoders , \ https://doi.org/https://doi.org/10.1016/j.compfluid.2025.106837 journal journal Computers & Fluids \ volume 302 ,\ pages 106837 ( year 2025 ) NoStop

    work page doi:10.1016/j.compfluid.2025.106837 2025

  33. [33]

    Ha , author A

    author author D. Ha , author A. M. \ Dai ,\ and\ author Q. V. \ Le ,\ title title Hypernetworks , \ in\ @noop booktitle International Conference on Learning Representations \ ( year 2017 ) NoStop

    work page 2017

  34. [34]

    Sitzmann , author M

    author author V. Sitzmann , author M. Zollh\" o fer ,\ and\ author G. Wetzstein ,\ title Scene representation networks: Continuous 3d-structure-aware neural scene representations , \ in\ @noop booktitle Proceedings of the 33rd International Conference on Neural Information Processing Systems \ ( publisher Curran Associates Inc. ,\ address Red Hook, NY, US...

    work page 2019

  35. [35]

    Sitzmann , author J

    author author V. Sitzmann , author J. Martel , author A. Bergman , author D. Lindell ,\ and\ author G. Wetzstein ,\ title title Implicit neural representations with periodic activation functions , \ in\ https://proceedings.neurips.cc/paper_files/paper/2020/file/53c04118df112c13a8c34b38343b9c10-Paper.pdf booktitle Advances in Neural Information Processing ...

    work page 2020

  36. [36]

    Perez , author F

    author author E. Perez , author F. Strub , author H. de Vries , author V. Dumoulin ,\ and\ author A. Courville ,\ title title Film: Visual reasoning with a general conditioning layer , \ https://doi.org/10.1609/aaai.v32i1.11671 journal journal Proceedings of the AAAI Conference on Artificial Intelligence \ volume 32 ( year 2018 ),\ 10.1609/aaai.v32i1.11671 NoStop

    work page doi:10.1609/aaai.v32i1.11671 2018

  37. [37]

    Dumoulin , author E

    author author V. Dumoulin , author E. Perez , author N. Schucher , author F. Strub , author H. d. \ Vries , author A. Courville ,\ and\ author Y. Bengio ,\ title title Feature-wise transformations , \ https://doi.org/10.23915/distill.00011 journal journal Distill \ ( year 2018 ),\ 10.23915/distill.00011 ,\ note https://distill.pub/2018/feature-wise-transf...

    work page doi:10.23915/distill.00011 2018

  38. [38]

    Rahaman , author A

    author author N. Rahaman , author A. Baratin , author D. Arpit , author F. Draxler , author M. Lin , author F. Hamprecht , author Y. Bengio ,\ and\ author A. Courville ,\ title title On the spectral bias of neural networks , \ in\ @noop booktitle Proceedings of the 36th International Conference on Machine Learning ,\ series Proceedings of Machine Learning...

    work page 2019

  39. [39]

    Jacot , author F

    author author A. Jacot , author F. Gabriel ,\ and\ author C. Hongler ,\ title title Neural tangent kernel: Convergence and generalization in neural networks , \ in\ https://proceedings.neurips.cc/paper_files/paper/2018/file/5a4be1fa34e62bb8a6ec6b91d2462f5a-Paper.pdf booktitle Advances in Neural Information Processing Systems ,\ Vol. volume 31 ,\ editor ed...

    work page 2018

  40. [40]

    Tancik , author P

    author author M. Tancik , author P. P. \ Srinivasan , author B. Mildenhall , author S. Fridovich-Keil , author N. Raghavan , author U. Singhal , author R. Ramamoorthi , author J. T. \ Barron ,\ and\ author R. Ng ,\ title title Fourier features let networks learn high frequency functions in low dimensional domains , \ in\ @noop booktitle Proceedings of the...

    work page 2020

  41. [41]

    Salimans \ and\ author D

    author author T. Salimans \ and\ author D. P. \ Kingma ,\ title title Weight normalization: A simple reparameterization to accelerate training of deep neural networks , \ in\ https://proceedings.neurips.cc/paper_files/paper/2016/file/ed265bc903a5a097f61d3ec064d96d2e-Paper.pdf booktitle Advances in Neural Information Processing Systems ,\ Vol. volume 29 ,\...

    work page 2016

  42. [42]

    author author D. P. \ Kingma \ and\ author J. Ba ,\ title title Adam: A method for stochastic optimization , \ in\ @noop booktitle Proceedings of the 3rd International Conference on Learning Representations (ICLR) \ ( year 2015 ) NoStop

    work page 2015

  43. [43]

    Zintgraf , author K

    author author L. Zintgraf , author K. Shiarli , author V. Kurin , author K. Hofmann ,\ and\ author S. Whiteson ,\ title title Fast context adaptation via meta-learning , \ in\ https://proceedings.mlr.press/v97/zintgraf19a.html booktitle Proceedings of the 36th International Conference on Machine Learning ,\ series Proceedings of Machine Learning Research ...

    work page 2019

  44. [44]

    Paszke , author S

    author author A. Paszke , author S. Gross , author F. Massa , author A. Lerer , author J. Bradbury , author G. Chanan , author T. Killeen , author Z. Lin , author N. Gimelshein , author L. Antiga , author A. Desmaison , author A. Kopf , author E. Yang , author Z. DeVito , author M. Raison , author A. Tejani , author S. Chilamkurthy , author B. Steiner , a...

    work page 2019

  45. [45]

    Bensch , author V

    author author L. Bensch , author V. Shigunov ,\ and\ author H. Söding ,\ title title Computational method to simulate planned ditching of a transport airplane , \ in\ https://doi.org/https://doi.org/10.1016/B978-008044046-0.50307-9 booktitle Computational Fluid and Solid Mechanics 2003 ,\ editor edited by\ editor K. Bathe \ ( publisher Elsevier Science Lt...

    work page doi:10.1016/b978-008044046-0.50307-9 2003