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arxiv: 2509.12344 · v4 · submitted 2025-09-15 · 💻 cs.LG

FEDONet : Fourier-Embedded DeepONet for Spectrally Accurate Operator Learning

Arth Sojitra , Mrigank Dhingra , Omer San This is my paper

Pith reviewed 2026-05-18 15:57 UTC · model grok-4.3

classification 💻 cs.LG
keywords DeepONetFourier featuresoperator learningPDE surrogate modelingneural networksBurgers equationKuramoto-SivashinskyAllen-Cahn
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The pith

Embedding random Fourier features into DeepONet trunks improves accuracy on PDE operator learning tasks.

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

The paper introduces FEDONet by replacing standard fully connected layers in the DeepONet trunk with a Fourier-embedded network that uses random Fourier features. This change is intended to better capture complex spatial structures in solutions to partial differential equations. The resulting model is tested on Burgers', 2D Poisson, Eikonal, Allen-Cahn, and Kuramoto-Sivashinsky equations under varying dataset sizes and noise levels. FEDONet produces lower reconstruction errors than the baseline DeepONet in every case, with the largest gains appearing in chaotic and stiff regimes. A reader would care because the modification is simple to implement yet appears to make data-driven PDE surrogates noticeably more reliable.

Core claim

By leveraging random Fourier features to enrich spatial representation capabilities in the trunk network, the Fourier-Embedded DeepONet (FEDONet) demonstrates superior performance compared to the traditional DeepONet across a comprehensive suite of PDE-driven datasets, including Burgers', 2D Poisson, Eikonal, Allen-Cahn, and the Kuramoto-Sivashinsky equation.

What carries the argument

Fourier-Embedded trunk network that incorporates random Fourier features to enrich spatial representations within the DeepONet architecture.

If this is right

  • Consistently superior reconstruction accuracy across all tested benchmark PDEs.
  • Particularly large relative L2 error reductions in chaotic and stiff systems.
  • Performance advantages hold across multiple training dataset sizes and input noise levels.
  • Provides a broadly applicable methodology for building PDE surrogate models.

Where Pith is reading between the lines

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

  • The same embedding trick might improve spatial fidelity in other branch-trunk operator architectures.
  • Smaller training sets could become viable for some problems if the enriched trunk reduces data hunger.
  • The spectral character of the embedding invites direct comparisons with traditional spectral numerical methods on the same tasks.

Load-bearing premise

Embedding random Fourier features into the DeepONet trunk network will reliably enrich spatial representations for a broad range of PDEs without requiring problem-specific hyperparameter tuning or introducing new instabilities.

What would settle it

A direct comparison on additional PDE problems or under different noise conditions in which FEDONet fails to reduce L2 error relative to standard DeepONet would challenge the claim of consistent superiority.

Figures

Figures reproduced from arXiv: 2509.12344 by Arth Sojitra, Mrigank Dhingra, Omer San.

Figure 1
Figure 1. Figure 1: FEDONet: Fourier Embedded Deep operator Network [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Solving a 2D Poisson equation: Comparison of predicted and reference solution fields of DeepONet and FEDONet (FEDONet). Both models were evaluated on 1000 randomly selected unseen test samples, yielding average relative L 2 errors of 1.41% for the DeepONet and 1.09% for the FEDONet. FEDONet reproduces the reference field with sharper gradients and more accurate spatial localization of extrema. The residual… view at source ↗
Figure 3
Figure 3. Figure 3: Energy spectrum comparison between the ground truth, vanilla and Fourier [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: presents prediction for a test sample, showing the ground truth so￾lution alongside predictions from both models. While DeepONet captures the coarse solution structure, it struggles with steep gradients and exhibits visible smoothing near shock regions. In contrast, FEDONet produces signif￾icantly sharper and more accurate reconstructions, faithfully capturing shock fronts and fine-scale dynamics [PITH_FU… view at source ↗
Figure 5
Figure 5. Figure 5: Energy spectrum and pointwise absolute error across space and time for a [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Training loss convergence for the 1D Burgers’ Equation - Comparison of Deep [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Reconstructed Lorenz-63 attractor for a randomnly sampled initial condition for [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Phase-plane projection of the x–z states. accumulates noticeable errors, particularly in the x(t) and z(t) components, which undergo rapid transitions between lobes. These deviations grow sharply after t > 1.5, illustrating compounding approximation errors. FE￾DONet exhibits substantially lower and smoother error profiles, with better robustness during nonlinear state transitions. In [PITH_FULL_IMAGE:figu… view at source ↗
Figure 9
Figure 9. Figure 9: Pointwise prediction error in x(t), y(t), and z(t) for a representative sample [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Cumulative relative error over time. phase alignment, lower pointwise and cumulative error, and superior spectral accuracy in chaotic dynamical systems. The introduction of Fourier embed￾dings into the trunk network acts as a spectral preconditioner, enabling the model to overcome inductive biases and improve fidelity under sensitivity to initial conditions. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Spectral energy density of predicted x(t) trajectories compared with the ground truth. 3.4. Eikonal Equation We consider the two-dimensional Eikonal equation, which governs the propagation of wavefronts at unit speed and arises in a wide range of ap￾plications including geometric optics, level-set methods, and computational geometry. The equation is given by: ∥∇s(x)∥2 = 1, s(x) = 0, x ∈ ∂Ω, (16) where x =… view at source ↗
Figure 12
Figure 12. Figure 12: Solving a parametric Eikonal equation (airfoils): Comparison of predicted signed distance functions (SDFs) for a representative airfoil geometry. Relative L 2 error: 2.728 × 10−2 for DeepONet, 0.912 × 10−2 for FEDONet. On this benchmark, the baseline DeepONet attains an average relative L 2 error of 2.15%, while the proposed FEDONet achieves an improved 1.12% error, demonstrating substantially higher reco… view at source ↗
Figure 13
Figure 13. Figure 13: Solving a parametric Allen Cahn equation: Comparison of predicted spatio￾temporal field for a random initial condition. Relative L 2 error: 4.85% for DeepONet, 2.27% for FEDONet [PITH_FULL_IMAGE:figures/full_fig_p021_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Energy spectrum and pointwise absolute error across space and time for a [PITH_FULL_IMAGE:figures/full_fig_p022_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Solving a parametric KS equation: Median performing test-sample. Relative L 2 error: FEDONet = 16.36%, DeepONet = 89.63% [PITH_FULL_IMAGE:figures/full_fig_p023_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Solving a parametric KS equation: Best performing test sample. FEDONet retains the amplitude and frequency content of the chaotic trajectory, yielding a low relative L 2 error of 10.77%. In comparison, the vanilla DeepONet fails entirely, producing numerical ar￾tifacts, amplitude distortion, and full trajectory collapse. This robustness under extreme conditions emphasizes the value of frequency-enhanced r… view at source ↗
Figure 17
Figure 17. Figure 17: Solving a parametric KS equation: Worst performing test sample. Even in failure cases, FEDONet remains qualitatively reasonable. Relative L 2 error: FEDONet = 58.02%, versus near-complete divergence for DeepONet. Finally, [PITH_FULL_IMAGE:figures/full_fig_p025_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: KS Training convergence comparison. FEDONet achieves faster and more [PITH_FULL_IMAGE:figures/full_fig_p026_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Distribution of relative L 2 errors across 512 unseen test samples for KS Equa￾tion. FEDONet maintains lower error in both mean and variance, demonstrating improved robustness and generalization in chaotic regimes. terized by sensitive dependence on initial conditions and multiscale feature interactions. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_19.png] view at source ↗
read the original abstract

Deep Operator Networks (DeepONets) have recently emerged as powerful data-driven frameworks for learning nonlinear operators, particularly suited for approximating solutions to partial differential equations. Despite their promising capabilities, the standard implementation of DeepONets, which typically employs fully connected linear layers in the trunk network, can encounter limitations in capturing complex spatial structures inherent to various PDEs. To address this limitation, we use Fourier-Embedded trunk networks within the DeepONet architecture, leveraging random Fourier features to enrich spatial representation capabilities. The Fourier-Embedded DeepONet (FEDONet) demonstrates superior performance compared to the traditional DeepONet across a comprehensive suite of PDE-driven datasets, including the Burgers', 2D Poisson, Eikonal, Allen-Cahn, and the Kuramoto-Sivashinsky equation. To systematically evaluate the effectiveness of the architectures, we perform comparisons across multiple training dataset sizes and input noise levels. FEDONet delivers consistently superior reconstruction accuracy across all benchmark PDEs, with particularly large relative $L^2$ error reductions observed in chaotic and stiff systems. This work demonstrates the effectiveness of Fourier embeddings in enhancing neural operator learning, offering a robust and broadly applicable methodology for PDE surrogate modeling.

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 manuscript introduces FEDONet, a DeepONet variant that replaces the standard fully-connected trunk with one augmented by random Fourier features to better capture spatial structures when learning nonlinear operators from PDE data. It reports consistent L² accuracy gains over baseline DeepONet on Burgers', 2D Poisson, Eikonal, Allen-Cahn, and Kuramoto-Sivashinsky benchmarks, with larger relative improvements on chaotic and stiff problems, and evaluates robustness across training-set sizes and input-noise levels.

Significance. If the empirical gains prove robust, the work would supply a lightweight, architecture-level improvement that leverages established Fourier embeddings to raise the effective spectral fidelity of operator networks, offering a practical route to more accurate surrogates for a range of PDEs without substantially increasing model complexity.

major comments (2)
  1. [§3.2] §3.2 (Fourier embedding description): the scale hyperparameter σ of the random Fourier features is not stated to be held fixed across all five PDEs or chosen once for the entire study. Because the optimal σ is known to be PDE-dependent (high-frequency content in KS versus smoother fields in Poisson), this detail is load-bearing for the central claim of broad applicability “without requiring problem-specific hyperparameter tuning.”
  2. [§4] §4 (Experimental results): no standard deviations, multiple random seeds, or statistical significance tests accompany the reported L² error reductions. Without these, it is impossible to determine whether the observed improvements, especially the large relative gains on chaotic/stiff systems, are reliable or could be explained by initialization variance.
minor comments (2)
  1. [Title] The title’s phrase “spectrally accurate” is not supported by any explicit spectral-norm or Fourier-coefficient error analysis in the results; either add such quantification or revise the title wording.
  2. [§2.1] §2.1: the precise concatenation of random Fourier features with the trunk input coordinates is described only at a high level; an explicit equation would remove ambiguity about the embedding dimension and activation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on hyperparameter specification and statistical reporting. These points strengthen the manuscript's clarity and rigor. We address each major comment below and outline the corresponding revisions.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (Fourier embedding description): the scale hyperparameter σ of the random Fourier features is not stated to be held fixed across all five PDEs or chosen once for the entire study. Because the optimal σ is known to be PDE-dependent (high-frequency content in KS versus smoother fields in Poisson), this detail is load-bearing for the central claim of broad applicability “without requiring problem-specific hyperparameter tuning.”

    Authors: We agree that explicit documentation of σ is necessary to substantiate the claim of broad applicability. The original experiments employed a single fixed value of σ across all five PDE benchmarks, selected to provide a practical balance for the range of frequency content encountered without per-problem retuning. We will revise §3.2 to state this choice explicitly, including the specific value and a brief justification, thereby directly addressing the referee's concern. revision: yes

  2. Referee: [§4] §4 (Experimental results): no standard deviations, multiple random seeds, or statistical significance tests accompany the reported L² error reductions. Without these, it is impossible to determine whether the observed improvements, especially the large relative gains on chaotic/stiff systems, are reliable or could be explained by initialization variance.

    Authors: We concur that reporting variability across runs is essential for assessing the reliability of the observed gains. We will revise §4 to include results aggregated over multiple independent random seeds (reporting means and standard deviations) and will add a short discussion of statistical significance for the key comparisons, particularly on the chaotic and stiff equations. revision: yes

Circularity Check

0 steps flagged

No circularity: FEDONet is an empirical architectural proposal validated on external benchmarks

full rationale

The paper introduces FEDONet by replacing the trunk network in DeepONet with a Fourier-embedded version using random Fourier features, then reports empirical L2 error reductions versus baseline DeepONet on Burgers', 2D Poisson, Eikonal, Allen-Cahn, and Kuramoto-Sivashinsky datasets across varying training sizes and noise levels. No derivation chain exists that reduces a claimed prediction or uniqueness result to a fitted parameter or self-citation by construction. Performance claims rest on direct, independent experimental comparisons rather than any self-referential definition or ansatz smuggled through prior work. The method is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach depends on the domain assumption that random Fourier features provide a general enrichment of spatial representations for PDE operators without introducing fitting artifacts or requiring extensive validation.

axioms (1)
  • domain assumption Random Fourier features can be directly embedded into the trunk network of DeepONet to improve capture of complex spatial structures in PDE solutions.
    Invoked as the key modification to address limitations of standard fully connected trunks.

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Forward citations

Cited by 1 Pith paper

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

  1. UFO: A Domain-Unification-Free Operator Framework for Generalized Operator Learning

    cs.LG 2026-05 unverdicted novelty 6.0

    UFO is a cross-domain neural operator framework that achieves discretization decoupling via adaptive jointly-conditioned interactions among distinct domain representations.

Reference graph

Works this paper leans on

56 extracted references · 56 canonical work pages · cited by 1 Pith paper · 4 internal anchors

  1. [1]

    J. P. Boyd, Chebyshev and Fourier spectral methods, Courier Corpora- tion, 2001

    work page 2001

  2. [2]

    Gottlieb, S

    D. Gottlieb, S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, CBMS–NSF Regional Conference Series in Applied Mathematics, SIAM, 1977. doi:10.1137/1.9781611970425

    work page doi:10.1137/1.9781611970425 1977

  3. [3]

    W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numer- ical Recipes: The Art of Scientific Computing, Cambridge University Press, 1986

    work page 1986

  4. [4]

    Cybenko, Approximation by superpositions of a sigmoidal function, Mathematics of control, signals and systems 2 (4) (1989) 303–314

    G. Cybenko, Approximation by superpositions of a sigmoidal function, Mathematics of control, signals and systems 2 (4) (1989) 303–314

    work page 1989

  5. [5]

    Hornik, M

    K. Hornik, M. Stinchcombe, H. White, Multilayer feedforward networks are universal approximators, Neural networks 2 (5) (1989) 359–366

    work page 1989

  6. [6]

    Poggio, F

    T. Poggio, F. Girosi, Networks for approximation and learning, Proceed- ings of the IEEE 78 (9) (1990) 1481–1497. doi:10.1109/5.58326

    work page doi:10.1109/5.58326 1990

  7. [7]

    T. Chen, H. Chen, Universal approximation to nonlinear operators by neuralnetworkswitharbitraryactivationfunctionsanditsapplicationto dynamical systems, IEEE transactions on neural networks 6 (4) (1995) 911–917

    work page 1995

  8. [8]

    Sirovich, Turbulence and the dynamics of coherent structures

    L. Sirovich, Turbulence and the dynamics of coherent structures. i. co- herent structures, Quarterly of applied mathematics 45 (3) (1987) 561– 571

    work page 1987

  9. [9]

    Berkooz, P

    G. Berkooz, P. Holmes, J. L. Lumley, The proper orthogonal decomposi- tion in the analysis of turbulent flows, Annual review of fluid mechanics 25 (1) (1993) 539–575. 29

    work page 1993

  10. [10]

    C. E. Rasmussen, C. K. I. Williams, Gaussian Processes for Machine Learning, The MIT Press, 2005. arXiv:https://direct.mit.edu/book- pdf/2514321/book_9780262256834.pdf, doi:10.7551/mitpress/3206.001.0001. URLhttps://doi.org/10.7551/mitpress/3206.001.0001

    work page doi:10.7551/mitpress/3206.001.0001 2005

  11. [11]

    E. Kansa, Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics—i surface approxima- tions and partial derivative estimates, Computers & Mathematics with Applications 19 (8) (1990) 127–145. doi:https://doi.org/10.1016/0898- 1221(90)90270-T. URLhttps://www.sciencedirect.com/science/article/pii/089812219090270T

    work page doi:10.1016/0898- 1990

  12. [12]

    D. Lowe, D. Broomhead, Multivariable functional interpolation and adaptive networks, Complex systems 2 (3) (1988) 321–355

    work page 1988

  13. [13]

    T. Qin, K. Wu, D. Xiu, Data driven governing equations approxima- tion using deep neural networks, Journal of Computational Physics 395 (2019) 620–635

    work page 2019

  14. [14]

    L. Lu, P. Jin, G. Pang, Z. Zhang, G. E. Karniadakis, Learning nonlin- ear operators via deeponet based on the universal approximation the- orem of operators, Nature Machine Intelligence 3 (3) (2021) 218–229. doi:10.1038/s42256-021-00302-5. URLhttps://doi.org/10.1038/s42256-021-00302-5

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

  15. [15]

    Kovachki, Z

    N. Kovachki, Z. Li, B. Liu, K. Azizzadenesheli, K. Bhattacharya, A. Stu- art, A. Anandkumar, Neural operator: learning maps between function spaceswithapplicationstopdes, J.Mach.Learn.Res.24(1)(Jan.2023)

    work page 2023

  16. [16]

    Z. Li, N. Kovachki, K. Azizzadenesheli, B. Liu, K. Bhattacharya, A. Stu- art, A. Anandkumar, Neural operator: Graph kernel network for partial differential equations (2020). arXiv:2003.03485. URLhttps://arxiv.org/abs/2003.03485

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  17. [17]

    Z. Li, N. Kovachki, K. Azizzadenesheli, B. Liu, K. Bhattacharya, A. Stu- art, A. Anandkumar, Multipole graph neural operator for parametric partial differential equations (2020). arXiv:2006.09535. URLhttps://arxiv.org/abs/2006.09535 30

    work page arXiv 2020

  18. [18]

    Z. Li, N. Kovachki, K. Azizzadenesheli, B. Liu, K. Bhattacharya, A. Stu- art, A. Anandkumar, Fourier neural operator for parametric partial dif- ferential equations (2021). arXiv:2010.08895. URLhttps://arxiv.org/abs/2010.08895

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  19. [19]

    K. Wu, D. Xiu, Data-driven deep learning of partial differential equa- tions in modal space, Journal of Computational Physics 408 (2020) 109307

    work page 2020

  20. [20]

    Tripura, S

    T. Tripura, S. Chakraborty, Wavelet neural operator for solving parametric partial differential equations in computational mechanics problems, Computer Methods in Applied Mechanics and Engineering 404 (2023) 115783. doi:https://doi.org/10.1016/j.cma.2022.115783. URLhttps://www.sciencedirect.com/science/article/pii/S0045782522007393

    work page doi:10.1016/j.cma.2022.115783 2023

  21. [21]

    Gupta, X

    G. Gupta, X. Xiao, P. Bogdan, Multiwavelet-based operator learning for differential equations, in: A. Beygelzimer, Y. Dauphin, P. Liang, J. W. Vaughan (Eds.), Advances in Neural Information Processing Systems, 2021. URLhttps://openreview.net/forum?id=LZDiWaC9CGL

    work page 2021

  22. [22]

    Raonic, R

    B. Raonic, R. Molinaro, T. Rohner, S. Mishra, E. de Bezenac, Con- volutional neural operators, in: ICLR 2023 Workshop on Physics for Machine Learning, 2023

    work page 2023

  23. [23]

    Multi-grid tensorized fourier neural operator for high-resolution PDEs.arXiv preprint arXiv:2310.00120, 2023

    J. Kossaifi, N. Kovachki, K. Azizzadenesheli, A. Anandkumar, Multi- grid tensorized fourier neural operator for high-resolution pdes, arXiv preprint arXiv:2310.00120 (2023)

    work page arXiv 2023

  24. [24]

    Guo, H.-B

    Z.-H. Guo, H.-B. Li, Mgfno: Multi-grid architecture fourier neu- ral operator for parametric partial differential equations (2024). arXiv:2407.08615. URLhttps://arxiv.org/abs/2407.08615

    work page arXiv 2024

  25. [25]

    Z. Li, D. Z. Huang, B. Liu, A. Anandkumar, Fourier neural operator with learned deformations for pdes on general geometries, Journal of Machine Learning Research 24 (388) (2023) 1–26

    work page 2023

  26. [26]

    Nonlinear re- construction for operator learning of pdes with discontinuities.International Conference on Learning Representations (ICLR), 2023

    S. Lanthaler, R. Molinaro, P. Hadorn, S. Mishra, Nonlinear recon- struction for operator learning of pdes with discontinuities (2022). 31 arXiv:2210.01074. URLhttps://arxiv.org/abs/2210.01074

    work page arXiv 2022

  27. [27]

    Y. Qiu, N. Bridges, P. Chen, Derivative-enhanced deep operator network (2024). arXiv:2402.19242. URLhttps://arxiv.org/abs/2402.19242

    work page arXiv 2024

  28. [28]

    S. W. Cho, J. Y. Lee, H. J. Hwang, Learning time-dependent pde via graph neural networks and deep operator network for robust accuracy on irregular grids (2024). arXiv:2402.08187. URLhttps://arxiv.org/abs/2402.08187

    work page arXiv 2024

  29. [29]

    S. Li, X. Yu, W. Xing, M. Kirby, A. Narayan, S. Zhe, Multi-resolution active learning of fourier neural operators (2024). arXiv:2309.16971. URLhttps://arxiv.org/abs/2309.16971

    work page arXiv 2024

  30. [30]

    Lei, H.-B

    W.-M. Lei, H.-B. Li, U-wno: U-net enhanced wavelet neural op- erator for solving parametric partial differential equations (2024). arXiv:2408.08190. URLhttps://arxiv.org/abs/2408.08190

    work page arXiv 2024

  31. [31]

    P. Hu, R. Wang, X. Zheng, T. Zhang, H. Feng, R. Feng, L. Wei, Y. Wang, Z.-M. Ma, T. Wu, Wavelet diffusion neural operator (2025). arXiv:2412.04833. URLhttps://arxiv.org/abs/2412.04833

    work page arXiv 2025

  32. [32]

    G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, L. Yang, Physics-informed machine learning, Nature Reviews Physics 3 (2021) 422–440. doi:10.1038/s42254-021-00314-5. URLhttps://doi.org/10.1038/s42254-021-00314-5

    work page doi:10.1038/s42254-021-00314-5 2021

  33. [33]

    S. Wang, Y. Teng, P. Perdikaris, Understanding and mitigating gradient pathologies in physics-informed neural networks (2020). arXiv:2001.04536. URLhttps://arxiv.org/abs/2001.04536

    work page arXiv 2020

  34. [34]

    Goswami, A

    S. Goswami, A. Bora, Y. Yu, G. E. Karniadakis, Physics-informed deep neural operator networks (2022). arXiv:2207.05748. URLhttps://arxiv.org/abs/2207.05748 32

    work page arXiv 2022

  35. [35]

    Z. Li, H. Zheng, N. Kovachki, D. Jin, H. Chen, B. Liu, K. Azizzade- nesheli, A. Anandkumar, Physics-informed neural operator for learning partial differential equations (2023). arXiv:2111.03794. URLhttps://arxiv.org/abs/2111.03794

    work page arXiv 2023

  36. [36]

    M. S. Eshaghi, C. Anitescu, M. Thombre, Y. Wang, X. Zhuang, T. Rabczuk, Variational physics-informed neural operator (vino) for solving partial differential equations (2024). arXiv:2411.06587. URLhttps://arxiv.org/abs/2411.06587

    work page arXiv 2024

  37. [37]

    K. Chen, Y. Li, D. Long, W. W. XING, J. Hochhalter, S. Zhe, Pseudo physics-informed neural operators (2025). URLhttps://openreview.net/forum?id=CrmUKllBKs

    work page 2025

  38. [38]

    T. Wang, C. Wang, Latent neural operator for solving forward and in- verse pde problems (2024). arXiv:2406.03923. URLhttps://arxiv.org/abs/2406.03923

    work page arXiv 2024

  39. [39]

    T. Wang, C. Wang, Latent neural operator pretraining for solving time- dependent pdes (2024). arXiv:2410.20100. URLhttps://arxiv.org/abs/2410.20100

    work page arXiv 2024

  40. [40]

    Ahmad, S

    Z. Ahmad, S. Chen, M. Yin, A. Kumar, N. Charon, N. Trayanova, M. Maggioni, Diffeomorphic latent neural operators for data- efficient learning of solutions to partial differential equations (2024). arXiv:2411.18014. URLhttps://arxiv.org/abs/2411.18014

    work page arXiv 2024

  41. [41]

    D. Long, Z. Xu, Q. Yuan, Y. Yang, S. Zhe, Invertible fourier neu- ral operators for tackling both forward and inverse problems (2025). arXiv:2402.11722. URLhttps://arxiv.org/abs/2402.11722

    work page arXiv 2025

  42. [42]

    Z. Hao, Z. Wang, H. Su, C. Ying, Y. Dong, S. Liu, Z. Cheng, J. Song, J. Zhu, Gnot: A general neural operator transformer for operator learn- ing, in: International Conference on Machine Learning, PMLR, 2023, pp. 12556–12569

    work page 2023

  43. [43]

    Z. Li, K. Meidani, A. B. Farimani, Transformer for partial differential equations’ operator learning (2023). arXiv:2205.13671. URLhttps://arxiv.org/abs/2205.13671 33

    work page arXiv 2023

  44. [44]

    Z.Li, D.Shu, A.BaratiFarimani, Scalabletransformerforpdesurrogate modeling, Advances in Neural Information Processing Systems 36 (2023) 28010–28039

    work page 2023

  45. [45]

    S. K. Boya, D. Subramani, A physics-informed transformer neural oper- ator for learning generalized solutions of initial boundary value problems (2025). arXiv:2412.09009. URLhttps://arxiv.org/abs/2412.09009

    work page arXiv 2025

  46. [46]

    Bryutkin, J

    A. Bryutkin, J. Huang, Z. Deng, G. Yang, C.-B. Schönlieb, A. Aviles- Rivero, Hamlet: Graph transformer neural operator for partial differen- tial equations, arXiv preprint arXiv:2402.03541 (2024)

    work page arXiv 2024

  47. [47]

    Z. Li, N. Kovachki, C. Choy, B. Li, J. Kossaifi, S. Otta, M. A. Nabian, M. Stadler, C. Hundt, K. Azizzadenesheli, et al., Geometry-informed neural operator for large-scale 3d pdes, Advances in Neural Information Processing Systems 36 (2023) 35836–35854

    work page 2023

  48. [48]

    FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators

    J. Pathak, S. Subramanian, P. Harrington, S. Raja, A. Chattopadhyay, M. Mardani, T. Kurth, D. Hall, Z. Li, K. Azizzadenesheli, et al., Four- castnet: A global data-driven high-resolution weather model using adap- tive fourier neural operators, arXiv preprint arXiv:2202.11214 (2022)

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  49. [49]

    Z. Xiao, S. Kou, H. Zhongkai, B. Lin, Z. Deng, Amortized fourier neural operators, AdvancesinNeuralInformationProcessingSystems37(2024) 115001–115020

    work page 2024

  50. [50]

    Srinivasan, Ben Mildenhall, Sara Fridovich-Keil, Nithin Raghavan, Utkarsh Singhal, Ravi Ramamoorthi, Jonathan T

    M. Tancik, P. P. Srinivasan, B. Mildenhall, S. Fridovich-Keil, N. Ragha- van, U. Singhal, R. Ramamoorthi, J. T. Barron, R. Ng, Fourier features let networks learn high frequency functions in low dimensional domains (2020). arXiv:2006.10739. URLhttps://arxiv.org/abs/2006.10739

    work page arXiv 2020

  51. [51]

    On the Spectral Bias of Neural Networks

    N.Rahaman, A.Baratin, D.Arpit, F.Draxler, M.Lin, F.A.Hamprecht, Y. Bengio, A. Courville, On the spectral bias of neural networks (2019). arXiv:1806.08734. URLhttps://arxiv.org/abs/1806.08734

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  52. [52]

    Rahimi, B

    A. Rahimi, B. Recht, Random features for large-scale kernel machines, in: J. Platt, D. Koller, Y. Singer, S. Roweis (Eds.), Advances in Neural Information Processing Systems, Vol. 20, Curran Associates, Inc., 2007. 34

    work page 2007

  53. [53]

    S. Wang, H. Wang, P. Perdikaris, Learning the solution operator of para- metric partial differential equations with physics-informed deeponets, Science advances 7 (40) (2021) eabi8605. Appendix A. Whitening Effect of Fourier Feature Embeddings Letϕ(ζ) = √ 2 [sin(2πZζ),cos(2πZζ)]∈R 2M be the Fourier feature em- bedding of an inputζ∈R d, where each row of t...

    work page 2021

  54. [54]

    Fourier-Embedded DeepONets naturally align with the dominant spec- tral modes of the target operator, facilitating more efficient approxima- tion of high-frequency content

    work page

  55. [55]

    The learned basis can adapt to complex patterns in the data, unlike fixed basis expansions

    work page

  56. [56]

    In summary, Fourier Embeddings serve as a principled mechanism for spec- tral lifting, enabling DeepONet to perform a learned Galerkin-style decom- position of operators

    The operator learning framework parallels well-established numerical discretization techniques, such as Petrov-Galerkin and spectral meth- ods, but operates in a fully data-driven regime. In summary, Fourier Embeddings serve as a principled mechanism for spec- tral lifting, enabling DeepONet to perform a learned Galerkin-style decom- position of operators...

    work page