pith. sign in

arxiv: 2604.25181 · v1 · submitted 2026-04-28 · 💻 cs.LG

Shearlet Neural Operators for Anisotropic-Shock-Dominated and Multi-scale parametric partial differential equations

Fabio Pereira dos Santos , Julio de Castro Vargas Fernandes , Adriano Mauricio de Almeida Cortes This is my paper

Pith reviewed 2026-05-07 17:01 UTC · model grok-4.3

classification 💻 cs.LG
keywords neural operatorsshearlet transformparametric PDEsanisotropic featuresshock-dominated flowsFourier neural operatorsmultiscale approximationdiscontinuities
0
0 comments X

The pith

Shearlet Neural Operators replace Fourier bases with directional multiscale atoms to improve accuracy on parametric PDEs that contain anisotropy and shocks.

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

The paper introduces the Shearlet Neural Operator to overcome the limitations of Fourier Neural Operators when solving parametric PDEs that feature directional anisotropy, sharp gradients, and localized discontinuities. It does this by learning solution operators directly in the shearlet domain, where the representation atoms are tuned for edges, fronts, and multiscale structures, then reconstructing via the inverse transform. Across seven families of benchmark PDEs—including anisotropic advection, diffusion, and nonlinear conservation laws with straight, curved, spiral, and polygonal shocks—SNO delivers higher predictive accuracy and better feature fidelity than FNO baselines, with the largest gains in the most anisotropic and discontinuity-heavy cases. The approach keeps the efficient spectral convolution structure of FNO while adding locality and directional selectivity through the shearlet basis.

Core claim

By substituting the global Fourier transform with a shearlet-based representation inside the neural operator architecture, the model acquires an inductive bias aligned with the sparse directional structure of anisotropic and shock-dominated PDE solutions; the resulting SNO learns and predicts in the shearlet coefficient domain and recovers the physical field via the inverse shearlet transform, yielding consistent accuracy and fidelity gains over FNO on the tested parametric families without altering the overall spectral computation pipeline.

What carries the argument

The shearlet transform, whose atoms are directional, multiscale, and spatially localized and provide near-optimal sparse approximation for anisotropic features and discontinuities.

If this is right

  • SNO improves predictive accuracy and feature fidelity most strongly on problems with strongly anisotropic advection or diffusion and on conservation laws containing straight, curved, interacting, spiral, or polygonal shocks.
  • The architecture retains efficient spectral convolution while gaining locality and directional selectivity from the shearlet atoms.
  • The same replacement of Fourier by shearlet bases can be applied inside other neural operator families that currently rely on global frequency representations.
  • Gains are expected to be largest precisely where FNO performance degrades, i.e., in discontinuity-dominated and directionally biased regimes.

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 time-dependent problems with moving shocks or on inverse problems where the operator must be learned from sparse observations.
  • Hybrid shearlet-Fourier layers might be explored to handle both global smooth components and localized anisotropic features within a single model.
  • Because shearlets are known to be near-optimal for cartoon-like functions, the approach may naturally extend to image-based or geometry-driven parametric PDEs without additional architectural redesign.

Load-bearing premise

That the shearlet representation supplies a sufficiently strong inductive bias for the targeted PDE classes without introducing reconstruction artifacts or requiring architecture changes that negate efficiency gains in practice.

What would settle it

A direct head-to-head experiment on one of the seven benchmark families in which SNO fails to improve test error or produces visible artifacts in reconstructed shock locations or anisotropic fronts relative to the FNO baseline.

Figures

Figures reproduced from arXiv: 2604.25181 by Adriano Mauricio de Almeida Cortes, Fabio Pereira dos Santos, Julio de Castro Vargas Fernandes.

Figure 1
Figure 1. Figure 1: Comparison of spectral parameter support in Fourier and Shearlet Neural Operators. view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of the Ground Truth, SNO Prediction, FNO Prediction, their respective Absolute Errors, and the view at source ↗
Figure 3
Figure 3. Figure 3: Training loss comparison between SNO and FNO. The loss (MSE) is plotted on a logarithmic scale against view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of the Ground Truth, SNO Prediction, FNO Prediction, their respective Absolute Errors, and the view at source ↗
Figure 5
Figure 5. Figure 5: Training loss comparison for bent ridge advect between SNO and FNO. The loss (MSE) is plotted on a view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of the Ground Truth, SNO Prediction, FNO Prediction, their respective Absolute Errors, and the view at source ↗
Figure 7
Figure 7. Figure 7: Training loss comparison for anisotropic ridge advect between SNO and FNO. The loss (MSE) is plotted on a view at source ↗
Figure 8
Figure 8. Figure 8: Comparison of the Ground Truth, SNO Prediction, FNO Prediction, their respective Absolute Errors, and the view at source ↗
Figure 9
Figure 9. Figure 9: Training loss comparison for sheared Kelvin-Helmholtz Stripes between SNO and FNO. The loss (MSE) is view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of the Ground Truth, SNO Prediction, FNO Prediction, their respective Absolute Errors, and the view at source ↗
Figure 11
Figure 11. Figure 11: Training loss comparison for polygonal shock case between SNO and FNO. The loss (MSE) is plotted on a view at source ↗
Figure 12
Figure 12. Figure 12: Comparison of the Ground Truth, SNO Prediction, FNO Prediction, their respective Absolute Errors, and the view at source ↗
Figure 13
Figure 13. Figure 13: Training loss comparison for multi angle shocks case between SNO and FNO. The loss (MSE) is plotted on view at source ↗
Figure 14
Figure 14. Figure 14: Comparison of the Ground Truth, SNO Prediction, FNO Prediction, their respective Absolute Errors, and the view at source ↗
Figure 15
Figure 15. Figure 15: Training loss comparison for spiral shock case between SNO and FNO. The loss (MSE) is plotted on a view at source ↗
read the original abstract

Neural operators have emerged as powerful data-driven surrogates for learning solution operators of parametric partial differential equations (PDEs). However, widely used Fourier Neural Operators (FNOs) rely on global Fourier representations, which can be inefficient for resolving anisotropic structures, sharp gradients, and spatially localized discontinuities that arise in shock-dominated and multiscale regimes. To address these limitations, we introduce the Shearlet Neural Operator (SNO), a neural operator architecture that replaces the Fourier transform with a shearlet-based representation. Shearlets offer directional, multiscale, and spatially localized atoms with near-optimal sparse approximation of anisotropic features, providing an inductive bias aligned with PDE solutions containing edges, fronts, and shocks. SNO learns in the shearlet domain and reconstructs predictions via the inverse transform, retaining efficient spectral computation while improving locality and directional selectivity. Across seven benchmark PDE families, including strongly anisotropic advection, anisotropic diffusion, and nonlinear conservation laws with straight, curved, interacting, spiral, and polygonal shock structures, SNO consistently improves predictive accuracy and feature fidelity over FNO baselines, with the largest gains observed in anisotropic and discontinuity-dominated settings.

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

0 major / 3 minor

Summary. The manuscript introduces the Shearlet Neural Operator (SNO), which replaces the Fourier transform in neural operators with a shearlet-based representation to better capture directional, multiscale, and localized features in solutions of parametric PDEs. The central claim is that SNO yields consistent gains in predictive accuracy and feature fidelity over FNO baselines across seven benchmark families (anisotropic advection, anisotropic diffusion, and nonlinear conservation laws with straight, curved, interacting, spiral, and polygonal shocks), with the largest improvements in anisotropic and discontinuity-dominated regimes.

Significance. If the empirical results hold under rigorous controls, the work is significant for extending neural operator architectures with an inductive bias aligned to the sparsity properties of anisotropic and shock-containing PDE solutions. It preserves the resolution-independent, spectral-efficiency structure of the operator-learning framework while addressing a known limitation of global Fourier bases, which could improve surrogate modeling in applications such as fluid dynamics and materials science.

minor comments (3)
  1. The abstract states that SNO 'consistently improves predictive accuracy' across seven families but provides no numerical values, error bars, or training-protocol details; adding at least one concrete metric (e.g., relative L2 error reduction on a representative anisotropic case) would strengthen the summary.
  2. The description of the architecture (forward shearlet transform, learned coefficient operations, inverse transform) should include an explicit statement confirming that the discrete shearlet implementation used is exactly invertible up to machine precision, to rule out reconstruction artifacts as a confounding factor.
  3. The experimental section should report parameter counts and FLOPs for SNO versus FNO on each benchmark to verify that efficiency gains are not offset by increased model size or transform overhead.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the accurate summary of our work and for recommending minor revision. The referee's description correctly captures the motivation for replacing the Fourier basis with shearlets and the reported gains on anisotropic and shock-dominated PDE benchmarks. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines SNO as an architectural replacement of the Fourier transform in FNO by the shearlet transform, with learned operations on shearlet coefficients followed by the inverse transform. This change is motivated by known properties of shearlets for sparse approximation of anisotropic features and is evaluated via direct empirical comparison on external benchmark PDE families. No equations, predictions, or central claims reduce by construction to fitted parameters, self-referential definitions, or unverified self-citations; the reported accuracy and feature-fidelity gains are independent measurements on held-out test cases. The derivation chain remains self-contained against the stated benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the mathematical properties of shearlets for sparse approximation of anisotropic features and on the assumption that benchmark PDEs are representative of target regimes; no free parameters or invented entities are named in the abstract.

axioms (1)
  • domain assumption Shearlets offer directional, multiscale, and spatially localized atoms with near-optimal sparse approximation of anisotropic features
    Invoked in the abstract to justify the inductive bias alignment with PDE solutions containing edges, fronts, and shocks.

pith-pipeline@v0.9.0 · 5505 in / 1261 out tokens · 70054 ms · 2026-05-07T17:01:50.536729+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages

  1. [1]

    Chapter 6 - the biological basis of diffusion anisotropy

    Christian Beaulieu. Chapter 6 - the biological basis of diffusion anisotropy. In Heidi Johansen-Berg and Timothy E.J. Behrens, editors,Diffusion MRI, pages 105–126. Academic Press, San Diego, 2009

    work page 2009

  2. [2]

    Towards understanding the spectral bias of deep learning, 2020

    Yuan Cao, Zhiying Fang, Yue Wu, Ding-Xuan Zhou, and Quanquan Gu. Towards understanding the spectral bias of deep learning, 2020

    work page 2020

  3. [3]

    Clarendon Press, Oxford, UK, 2nd edition, 1975

    John Crank.The Mathematics of Diffusion. Clarendon Press, Oxford, UK, 2nd edition, 1975

    work page 1975

  4. [4]

    Total variation diminishing runge–kutta schemes.Mathematics of Computation, 67(221):73–85, 1998

    Sigal Gottlieb and Chi-Wang Shu. Total variation diminishing runge–kutta schemes.Mathematics of Computation, 67(221):73–85, 1998

    work page 1998

  5. [5]

    Shearlets: Sparse multidimensional representations using anisotropic dilation and shear operators

    Kanghui Guo, Gitta Kutyniok, and Demetrio Labate. Shearlets: Sparse multidimensional representations using anisotropic dilation and shear operators. InWavelets and Splines, 05 2005

    work page 2005

  6. [6]

    Hmgno: Hybrid multigrid neural operator with low-order numerical solver for partial differential equations.Neural Networks, 190:107649, 2025

    Yifan Hu, Weimin Zhang, Fukang Yin, and Jianping Wu. Hmgno: Hybrid multigrid neural operator with low-order numerical solver for partial differential equations.Neural Networks, 190:107649, 2025

    work page 2025

  7. [7]

    Isakov.Inverse Problems for Partial Differential Equations

    V . Isakov.Inverse Problems for Partial Differential Equations. Applied Mathematical Sciences. Springer New York, 2013

    work page 2013

  8. [8]

    Frequency principle: Fourier analysis sheds light on deep neural networks.Communications in Computational Physics, 28(5):1746–1767, June 2020

    Zhi-Qin John Xu, Yaoyu Zhang, Tao Luo, Yanyang Xiao, and Zheng Ma. Frequency principle: Fourier analysis sheds light on deep neural networks.Communications in Computational Physics, 28(5):1746–1767, June 2020

    work page 2020

  9. [9]

    Neural operator: learning maps between function spaces with applications to pdes.J

    Nikola Kovachki, Zongyi Li, Burigede Liu, Kamyar Azizzadenesheli, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Neural operator: learning maps between function spaces with applications to pdes.J. Mach. Learn. Res., 24(1), January 2023

    work page 2023

  10. [10]

    Applied and Numerical Harmonic Analysis

    Gitta Kutyniok and Demetrio Labate, editors.Shearlets. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston, MA, 1 edition, 2012

    work page 2012

  11. [11]

    Shearlab 3d: Faithful digital shearlet transforms based on compactly supported shearlets.ACM Transactions on Mathematical Software, 42(1):1–42, January 2016

    Gitta Kutyniok, Wang-Q Lim, and Rafael Reisenhofer. Shearlab 3d: Faithful digital shearlet transforms based on compactly supported shearlets.ACM Transactions on Mathematical Software, 42(1):1–42, January 2016

    work page 2016

  12. [12]

    LeVeque.Finite Volume Methods for Hyperbolic Problems

    Randall J. LeVeque.Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, 2002

    work page 2002

  13. [13]

    Fourier neural operator for parametric partial differential equations, 2021

    Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Fourier neural operator for parametric partial differential equations, 2021

    work page 2021

  14. [14]

    Physics-informed neural operator for learning partial differential equations, 2023

    Zongyi Li, Hongkai Zheng, Nikola Kovachki, David Jin, Haoxuan Chen, Burigede Liu, Kamyar Azizzadenesheli, and Anima Anandkumar. Physics-informed neural operator for learning partial differential equations, 2023

    work page 2023

  15. [15]

    Multiscale deeponet for nonlinear operators in oscillatory function spaces for building seismic wave responses, 2021

    Lizuo Liu and Wei Cai. Multiscale deeponet for nonlinear operators in oscillatory function spaces for building seismic wave responses, 2021. 19 arXivTemplateA PREPRINT

    work page 2021

  16. [16]

    Learning nonlinear operators via deeponet based on the universal approximation theorem of operators.Nature Machine Intelligence, 3(3):218–229, March 2021

    Lu Lu, Pengzhan Jin, Guofei Pang, Zhongqiang Zhang, and George Em Karniadakis. Learning nonlinear operators via deeponet based on the universal approximation theorem of operators.Nature Machine Intelligence, 3(3):218–229, March 2021

    work page 2021

  17. [17]

    An upper limit of decaying rate with respect to frequency in linear frequency principle model

    Tao Luo, Zheng Ma, Zhiwei Wang, Zhiqin John Xu, and Yaoyu Zhang. An upper limit of decaying rate with respect to frequency in linear frequency principle model. In Bin Dong, Qianxiao Li, Lei Wang, and Zhi-Qin John Xu, editors,Proceedings of Mathematical and Scientific Machine Learning, volume 190 ofProceedings of Machine Learning Research, pages 205–214. P...

    work page 2022

  18. [18]

    Theory of the frequency principle for general deep neural networks, 2019

    Tao Luo, Zheng Ma, Zhi-Qin John Xu, and Yaoyu Zhang. Theory of the frequency principle for general deep neural networks, 2019

    work page 2019

  19. [19]

    On the exact computation of linear frequency principle dynamics and its generalization.SIAM Journal on Mathematics of Data Science, 4(4):1272–1292, 2022

    Tao Luo, Zheng Ma, Zhi-Qin John Xu, and Yaoyu Zhang. On the exact computation of linear frequency principle dynamics and its generalization.SIAM Journal on Mathematics of Data Science, 4(4):1272–1292, 2022

    work page 2022

  20. [20]

    McWilliams

    James C. McWilliams. Submesoscale currents in the ocean.Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 472(2189):20160117, 05 2016

    work page 2016

  21. [21]

    Robert M. Miura. Linear and nonlinear waves (g. b. whitham).SIAM Review, 18(4):776–781, 1976

    work page 1976

  22. [22]

    Hamprecht, Yoshua Bengio, and Aaron Courville

    Nasim Rahaman, Aristide Baratin, Devansh Arpit, Felix Draxler, Min Lin, Fred A. Hamprecht, Yoshua Bengio, and Aaron Courville. On the spectral bias of neural networks, 2019

    work page 2019

  23. [23]

    Raissi, P

    M. Raissi, P. Perdikaris, and G.E. Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations.Journal of Computational Physics, 378:686–707, 2019

    work page 2019

  24. [24]

    Influence of cross perturbations on turbulent kelvin–helmholtz instability.Fluids, 9(3), 2024

    Mae Sementilli, Rozie Zangeneh, and James Chen. Influence of cross perturbations on turbulent kelvin–helmholtz instability.Fluids, 9(3), 2024

    work page 2024

  25. [25]

    Smith.Uncertainty Quantification: Theory, Implementation, and Applications, Second Edition

    R.C. Smith.Uncertainty Quantification: Theory, Implementation, and Applications, Second Edition. Computa- tional science and engineering. Society for Industrial and Applied Mathematics, 2024

    work page 2024

  26. [26]

    T. J. Sullivan.Introduction to Uncertainty Quantification, volume 63 ofTexts in Applied Mathematics. Springer, 2015

    work page 2015

  27. [27]

    Energy propagation in deep convolutional neural networks, 2018

    Thomas Wiatowski, Philipp Grohs, and Helmut Bölcskei. Energy propagation in deep convolutional neural networks, 2018

    work page 2018

  28. [28]

    Overview frequency principle/spectral bias in deep learning, 2024

    Zhi-Qin John Xu, Yaoyu Zhang, and Tao Luo. Overview frequency principle/spectral bias in deep learning, 2024

    work page 2024

  29. [29]

    Training behavior of deep neural network in frequency domain, 2019

    Zhi-Qin John Xu, Yaoyu Zhang, and Yanyang Xiao. Training behavior of deep neural network in frequency domain, 2019

    work page 2019

  30. [30]

    Mscalefno: Multi-scale fourier neural operator learning for oscillatory function spaces, 2024

    Zhilin You, Zhenli Xu, and Wei Cai. Mscalefno: Multi-scale fourier neural operator learning for oscillatory function spaces, 2024

    work page 2024

  31. [31]

    Sh-cnn: Shearlet convolutional neural network for gender classification

    Chaymae Ziani and Abdelalim Sadiq. Sh-cnn: Shearlet convolutional neural network for gender classification. Advances in Science, Technology and Engineering Systems Journal, 5:1328–1334, 12 2020. 20

    work page 2020