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arxiv: 2605.19959 · v1 · pith:QYGBGZ3Inew · submitted 2026-05-19 · 💻 cs.LG · math.FA

Learning Orthonormal Bases for Function Spaces

Hamidreza Kamkari , Mohammad Sina Nabizadeh , Justin Solomon This is my paper

Pith reviewed 2026-05-20 07:45 UTC · model grok-4.3

classification 💻 cs.LG math.FA
keywords orthonormal basesfunction spacesneural networksorthogonal groupODE flowsLie manifoldsadaptive basesfunctional data
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The pith

Any target orthonormal basis in function space can be approximated arbitrarily closely by integrating rank-2 skew-adjoint operators generated by a neural network from a reference basis.

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

The paper treats infinite-dimensional orthonormal bases as points on the manifold of the orthogonal group and represents them as endpoints of continuous paths connecting a fixed reference basis such as Fourier to a target basis. These paths obey ODEs whose generators are skew-adjoint integral operators; neural networks supply finite-rank approximations to those generators. A universality result establishes that the integrated flows remain dense in the orthogonal group even when the generator rank is restricted to two, under the relevant operator topology. The construction therefore lets a practitioner optimize a basis directly for a given dataset or operator while preserving orthonormality at every step.

Core claim

Even with a rank-2 generator, the integrated solutions of the ODE are dense in the orthogonal group under the appropriate operator topology, so any target orthonormal basis can be approximated arbitrarily closely from a reference basis.

What carries the argument

Finite-rank skew-adjoint generators of ODEs on the Lie manifold of the orthogonal group that produce continuous paths from a reference orthonormal basis to a target basis.

If this is right

  • A Fourier basis can be continuously deformed into the principal components of a given functional dataset while remaining orthonormal throughout.
  • Eigenfunctions of a linear operator on the function space can be obtained by optimizing the endpoint of such a neural-driven path.
  • Dynamic modes of energy-preserving physical simulations can be recovered as the learned basis without explicit mode decomposition.
  • The same parameterization supports end-to-end training of basis coefficients together with the generator network for downstream tasks.

Where Pith is reading between the lines

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

  • The density result may allow similar finite-rank control on other infinite-dimensional Lie groups that arise in constrained optimization over function spaces.
  • Numerical experiments could test whether low-rank generators require more training steps than higher-rank ones to reach comparable approximation quality.
  • The framework could be combined with existing functional data analysis pipelines to replace hand-chosen bases in kernel methods or spectral methods.

Load-bearing premise

Finite-rank skew-adjoint generators produced by neural networks can produce paths whose limits cover the full space of orthonormal bases in the relevant operator topology without requiring infinite-dimensional controls.

What would settle it

An explicit orthonormal basis together with a concrete operator-topology distance such that no sequence of rank-2 neural-generated paths approaches it within that distance.

Figures

Figures reproduced from arXiv: 2605.19959 by Hamidreza Kamkari, Justin Solomon, Mohammad Sina Nabizadeh.

Figure 1
Figure 1. Figure 1: Method and Applications Overview. (a) Parameterizing a change-of-basis map Qθ allows us to adapt a reference ONB to various applications: eigenfaces that describe the variance of the data, visualizing the training dynamics of a non-linear neural function by diagonalizing its neural tangent kernel, or dynamic modes of fluid-flow. (b) The learned bases inherit properties of the reference such as being discre… view at source ↗
Figure 2
Figure 2. Figure 2: Numerical Pipeline. (a) A Fourier basis is gradually transformed into principal functions of CelebA dataset. We select quadrature points Ω = b {ωj}D j=1 ∼ µ and time-discretization {t0 = 0, . . . , tL = T} and perform L Cayley steps. Each step takes O(r 3 + r 2D) that exploits the rank r structure of the generator. (b) Using off-the-shelf ODE solvers result in loss or explosion of the function norm, by con… view at source ↗
Figure 3
Figure 3. Figure 3: 1D Synthetic PCA on Function Space. (a) Samples from a distribution of signals described by (15), exhibiting a jump at x = 0.5 on a 1D domain. (b) The Fourier basis is transformed to diagonalize the covariance operator, allowing the basis to better capture the variance of signals. Our results improve upon the traditional finite-dimensional PCA construction, which is limited by the Nyquist cutoff inherent t… view at source ↗
Figure 4
Figure 4. Figure 4: 2D PCA in Function Space. (a) Fourier basis transformed to explain the INR zoo of SIREN networks representing the MNIST dataset. Note that the resulting basis represents a dataset that is itself discretization-free. (b) Transformed bases for the CelebA dataset. (c) Reconstruction of several datapoints using the Fourier basis and our transformed, optimized basis. With only a few basis elements, our basis re… view at source ↗
Figure 5
Figure 5. Figure 5: Two-moon NTK. Diagonalizing the NTK over the course of training a two-moon classifier (steps 1, 250, 500, and 5000). (a) The evolving logit landscape and training data. (b) Top eigenfunctions recovered by a discrete grid-solver; the grid’s fixed resolution leads to noticeable pixelation and a loss of high-frequency information. (c) Eigenfunctions obtained using our proposed discretization-free method. Our … view at source ↗
Figure 6
Figure 6. Figure 6: Learning the Koopman Operator. (a) Visual comparison of fluid-flow trajectories generated by our learned Koopman operator vs. the composition operator integrated via RK methods; our method is noisier but conserves energy. (b) L2 energy tracking over time, comparing standard solvers (1st, 2nd, and 4th-order RK) against iterative rollouts from Qθ. 8 Conclusions, Limitations, & Future Work In this work, we in… view at source ↗
Figure 7
Figure 7. Figure 7: Full Cayley Ablation. Comparison of the Cayley method against backward and forward Euler. (a) Example evolution of a basis function ϕ θ i (t) over time. The Cayley solver perfectly maintains signal energy, while forward Euler rapidly explodes and backward Euler dissipates. (b) Beyond norm preservation, the inner products of various basis functions, visualized as a gram matrix, also remain preserved through… view at source ↗
Figure 8
Figure 8. Figure 8: PCA for CIFAR10 [32] SIREN Zoo. (a) Visualizes the reference and learned basis after training; the learned bases resemble smooth principal components of the CIFAR10 dataset. (b) Reconstruction with the learned basis results in higher PSNRs. (c) Quantitatively, the captured energy is higher for our learned basis compared to the Fourier basis. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: PCA for CelebA. (a) Visualizes the reference and learned basis; the learned bases resemble smooth “eigenfaces” of CelebA. (b) Reconstruction with the learned basis obtains higher PSNRs and captures the structure of the face better with fewer elements. (c) Quantitatively, the cumulative captured energy is higher for our learned basis compared to Fourier basis. 23 [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Initial and Learned 2D and Multichannel Bases. Visualization a truncated set of initial and learned bases across applications using our index notation (see subsection D.1); the center of the mosaic corresponds to the direct current (DC) Fourier functions. (a) 2D Fourier and a learned bases fit to the principal components of the MNIST SIREN Zoo; the bases are visualized as a 2D mosaic indexed as φi1,i2 for… view at source ↗
Figure 11
Figure 11. Figure 11: Internal Network Dynamics. This figure illustrates the learned neural network outputs that transform the Fourier basis into the dynamic modes of the Taylor-Green vortices. Specifically, we visualize the first five neural fields, U 0...4 θ , across various time steps (a), alongside the spatial rotations exp(Mθ − M⊤ θ ) (b) that jointly characterize the generator Kθ(t). Passing the Fourier basis through the… view at source ↗
read the original abstract

Infinite-dimensional orthonormal basis expansions play a central role in representing and computing with function spaces due to their favorable linear algebraic properties. However, common bases such as Fourier or wavelets are fixed and do not adapt to the structure of a given problem or dataset. In this paper, we aim to represent these bases with neural networks and optimize them. Our key idea is that any target infinite-dimensional orthonormal basis can be viewed either as a point on the Lie manifold of the orthogonal group, or equivalently, as the endpoint of a continuous path on that manifold that connects a reference basis, e.g. Fourier, to that target. Paths on the Lie manifold satisfy ordinary differential equations (ODEs) governed by skew-adjoint integral operators. Using neural networks to define finite-rank generators of such ODEs allows us to parameterize and optimize orthonormal bases in function space. While relying on finite-rank generators to model infinite operators might seem restrictive, we prove a universality result: even with a rank-2 generator, the integrated solutions of the ODE are dense in the orthogonal group under the appropriate operator topology. In other words, for any target orthonormal basis, there exists a path originating from a reference basis and driven by finite-rank generators that gets arbitrarily close to that target basis. We demonstrate the flexibility of our framework by transforming the Fourier basis into the principal components of a functional dataset, eigenfunctions of linear operators, or dynamic modes of energy-preserving physical simulations.

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

1 major / 2 minor

Summary. The manuscript proposes parameterizing adaptive orthonormal bases in infinite-dimensional function spaces as endpoints of paths on the orthogonal group, where the paths are solutions to ODEs driven by skew-adjoint generators that are themselves parameterized by neural networks. A central theoretical claim is a universality result: even when the generators are restricted to rank-2 finite-rank operators, the reachable set of integrated flows is dense in the orthogonal group under the strong operator topology. The framework is illustrated by transforming a reference Fourier basis into principal components of functional data, eigenfunctions of linear operators, and dynamic modes of energy-preserving simulations.

Significance. If the density result holds, the approach supplies a principled, optimizable alternative to fixed bases such as Fourier or wavelets, with direct relevance to functional data analysis, operator learning, and structure-preserving scientific machine learning. The grounding in Lie-group controllability theory and the explicit statement that finite-rank controls suffice are strengths that distinguish the work from purely empirical basis adaptation methods.

major comments (1)
  1. [§4] §4 (Universality result): the proof sketch invokes controllability of the infinite-dimensional orthogonal group by plane rotations in the strong operator topology, but the manuscript does not explicitly address how the neural-network approximation error on the generator accumulates along the flow and whether the density statement remains uniform with respect to that approximation; a quantitative bound linking generator approximation to basis approximation error would strengthen the claim.
minor comments (2)
  1. [§2] The notation for the skew-adjoint integral operator and its finite-rank truncation should be introduced with a single consistent symbol rather than switching between K(t) and A_N(t) across sections.
  2. [Figure 2] Figure 2 (basis transformation examples): the caption should state the precise operator topology in which the reported approximation error is measured.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation and constructive feedback. We respond to the major comment below.

read point-by-point responses

  1. Referee: [§4] §4 (Universality result): the proof sketch invokes controllability of the infinite-dimensional orthogonal group by plane rotations in the strong operator topology, but the manuscript does not explicitly address how the neural-network approximation error on the generator accumulates along the flow and whether the density statement remains uniform with respect to that approximation; a quantitative bound linking generator approximation to basis approximation error would strengthen the claim.

    Authors: We appreciate the referee's suggestion to strengthen the universality claim. The proof in §4 establishes that the reachable set using exact rank-2 generators is dense in the orthogonal group under the strong operator topology, based on controllability results for the infinite-dimensional case. Regarding neural network approximation, we note that the generators are finite-rank and can be approximated arbitrarily well by neural networks due to their universal approximation properties for continuous functions on compact sets (as the finite-rank operators can be represented via finite-dimensional parameters). To address accumulation along the flow, we will revise the manuscript to include a brief argument that the solution map of the ODE is continuous with respect to the generator in the topology that induces the strong operator topology on the group. This continuity ensures that the density result carries over to approximately realized generators. A fully quantitative error bound would require additional estimates on the Lipschitz constants of the flow, which we believe is possible but may be technical; we will provide a qualitative statement and leave a quantitative version for future work if space permits. revision: partial

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper introduces a neural parameterization of paths on the infinite-dimensional orthogonal group via finite-rank skew-adjoint generators of ODEs, with the central universality claim—that rank-2 generators produce dense flows in the strong operator topology—presented as a proved controllability result from Lie-group theory rather than a fitted or self-referential construction. No equation or step reduces the density statement to data-driven inputs by construction, no self-citation chain bears the load of the existence result, and the framework does not rename empirical patterns or smuggle ansatzes. The derivation remains self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard results from Lie group theory and functional analysis for the existence and uniqueness of ODE flows on the orthogonal group; the neural network parameterization introduces trainable parameters whose effect is bounded by the universality theorem.

axioms (2)
  • standard math Solutions to the skew-adjoint operator ODE exist and remain on the orthogonal group for finite-rank generators.
    Invoked to justify that the integrated path stays orthonormal.
  • domain assumption The operator topology used for density is the appropriate one for function-space bases.
    Required for the universality statement to imply practical approximation of any target basis.

pith-pipeline@v0.9.0 · 5787 in / 1259 out tokens · 46953 ms · 2026-05-20T07:45:47.331263+00:00 · methodology

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Reference graph

Works this paper leans on

58 extracted references · 58 canonical work pages · 3 internal anchors

  1. [1]

    Principal component analysis.Wiley interdisciplinary reviews: computational statistics, 2(4):433–459, 2010

    Hervé Abdi and Lynne J Williams. Principal component analysis.Wiley interdisciplinary reviews: computational statistics, 2(4):433–459, 2010

    work page 2010

  2. [2]

    Futon: Fourier tensor network for implicit neural representations.arXiv preprint arXiv: 2602.13414, 2026

    Pooya Ashtari, Pourya Behmandpoor, Nikos Deligiannis, and Aleksandra Pizurica. Futon: Fourier tensor network for implicit neural representations.arXiv preprint arXiv: 2602.13414, 2026

    work page arXiv 2026

  3. [3]

    Continuous-time signal de- composition: An implicit neural generalization of pca and ica

    Shayan K Azmoodeh, Krishna Subramani, and Paris Smaragdis. Continuous-time signal de- composition: An implicit neural generalization of pca and ica. In2025 IEEE 35th International Workshop on Machine Learning for Signal Processing (MLSP), pages 1–6. IEEE, 2025

    work page 2025

  4. [4]

    Deep learning solution of the eigenvalue problem for differential operators.Neural Computation, 35(6):1100–1134, 2023

    Ido Ben-Shaul, Leah Bar, Dalia Fishelov, and Nir Sochen. Deep learning solution of the eigenvalue problem for differential operators.Neural Computation, 35(6):1100–1134, 2023

    work page 2023

  5. [5]

    Out-of-sample extensions for lle, isomap, mds, eigenmaps, and spectral clustering.Advances in neural information processing systems, 16, 2003

    Yoshua Bengio, Jean-françcois Paiement, Pascal Vincent, Olivier Delalleau, Nicolas Roux, and Marie Ouimet. Out-of-sample extensions for lle, isomap, mds, eigenmaps, and spectral clustering.Advances in neural information processing systems, 16, 2003

    work page 2003

  6. [6]

    Springer, 2002

    Pierre Brémaud.Mathematical principles of signal processing: Fourier and wavelet analysis. Springer, 2002

    work page 2002

  7. [7]

    An introduction to lie group integrators–basics, new developments and applications.Journal of Computational Physics, 257: 1040–1061, 2014

    Elena Celledoni, Håkon Marthinsen, and Brynjulf Owren. An introduction to lie group integrators–basics, new developments and applications.Journal of Computational Physics, 257: 1040–1061, 2014

    work page 2014

  8. [8]

    and Chen, Peter Yichen and Grinspun, Eitan , title =

    Yue Chang, Otman Benchekroun, Maurizio M. Chiaramonte, Peter Yichen Chen, and Eitan Grinspun. Shape space spectra.ACM Trans. Graph., 44(4), July 2025. ISSN 0730-0301. doi: 10.1145/3731148. URLhttps://doi.org/10.1145/3731148

    work page doi:10.1145/3731148 2025

  9. [9]

    Neural ordinary differential equations.Advances in neural information processing systems, 31, 2018

    Ricky TQ Chen, Yulia Rubanova, Jesse Bettencourt, and David K Duvenaud. Neural ordinary differential equations.Advances in neural information processing systems, 31, 2018

    work page 2018

  10. [10]

    Methods of mathematical physics, vol

    Richard Courant and David Hilbert. Methods of mathematical physics, vol. i.Phys. Today, 7 (5):17–17, 1954

    work page 1954

  11. [11]

    Deep learning on implicit neural representations of shapes.The Eleventh International Conference on Learning Representations, 2023

    Luca De Luigi, Adriano Cardace, Riccardo Spezialetti, Pierluigi Zama Ramirez, Samuele Salti, and Luigi Di Stefano. Deep learning on implicit neural representations of shapes.The Eleventh International Conference on Learning Representations, 2023

    work page 2023

  12. [12]

    Neuralef: Deconstructing kernels by deep neural networks

    Zhijie Deng, Jiaxin Shi, and Jun Zhu. Neuralef: Deconstructing kernels by deep neural networks. InInternational Conference on Machine Learning, pages 4976–4992. PMLR, 2022

    work page 2022

  13. [13]

    The cayley transform in the numerical solution of unitary differential systems.Advances in computational mathematics, 8(4):317–334, 1998

    Fasma Diele, Luciano Lopez, and R Peluso. The cayley transform in the numerical solution of unitary differential systems.Advances in computational mathematics, 8(4):317–334, 1998. 10

    work page 1998

  14. [14]

    Ziya Erkoc ¸, Fangchang Ma, Qi Shan, Matthias Nießner, and Angela Dai

    Emilien Dupont, Hyunjik Kim, SM Eslami, Danilo Rezende, and Dan Rosenbaum. From data to functa: Your data point is a function and you can treat it like one.arXiv preprint arXiv:2201.12204, 2022

    work page arXiv 2022

  15. [15]

    Sigmoid-weighted linear units for neural network function approximation in reinforcement learning.Neural networks, 107:3–11, 2018

    Stefan Elfwing, Eiji Uchibe, and Kenji Doya. Sigmoid-weighted linear units for neural network function approximation in reinforcement learning.Neural networks, 107:3–11, 2018

    work page 2018

  16. [16]

    How to train your neural ode: the world of jacobian and kinetic regularization

    Chris Finlay, Jörn-Henrik Jacobsen, Levon Nurbekyan, and Adam Oberman. How to train your neural ode: the world of jacobian and kinetic regularization. InInternational conference on machine learning, pages 3154–3164. PMLR, 2020

    work page 2020

  17. [17]

    Remarks on the Cayley Representation of Orthogonal Matrices and on Perturbing the Diagonal of a Matrix to Make it Invertible

    Jean Gallier. Remarks on the cayley representation of orthogonal matrices and on perturbing the diagonal of a matrix to make it invertible.arXiv preprint math/0606320, 2006

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  18. [18]

    Eigengame: Pca as a nash equilibrium.International Conference on Learning Representations, 2021

    Ian Gemp, Brian McWilliams, Claire Vernade, and Thore Graepel. Eigengame: Pca as a nash equilibrium.International Conference on Learning Representations, 2021

    work page 2021

  19. [19]

    On the relationships between svd, klt and pca.Pattern recognition, 14(1-6): 375–381, 1981

    Jan J Gerbrands. On the relationships between svd, klt and pca.Pattern recognition, 14(1-6): 375–381, 1981

    work page 1981

  20. [20]

    Courier Corporation, 2003

    Roger G Ghanem and Pol D Spanos.Stochastic finite elements: a spectral approach. Courier Corporation, 2003

    work page 2003

  21. [21]

    Anode: Unconditionally accurate memory- efficient gradients for neural odes

    Amir Gholaminejad, Kurt Keutzer, and George Biros. Anode: Unconditionally accurate memory- efficient gradients for neural odes. InProceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence, IJCAI-19, pages 730–736. International Joint Conferences on Artificial Intelligence Organization, 7 2019. doi: 10.24963/ijcai.2019/103....

    work page doi:10.24963/ijcai.2019/103 2019

  22. [22]

    JHU press, 2013

    Gene H Golub and Charles F Van Loan.Matrix computations. JHU press, 2013

    work page 2013

  23. [23]

    SIAM, 1977

    David Gottlieb and Steven A Orszag.Numerical analysis of spectral methods: theory and applications. SIAM, 1977

    work page 1977

  24. [24]

    Lie groups, lie algebras, and representations

    Brian C Hall. Lie groups, lie algebras, and representations. InQuantum Theory for Mathemati- cians, pages 333–366. Springer, 2013

    work page 2013

  25. [25]

    SIAM, 2002

    Nicholas J Higham.Accuracy and stability of numerical algorithms. SIAM, 2002

    work page 2002

  26. [26]

    Multilayer feedforward networks are universal approximators.Neural networks, 2(5):359–366, 1989

    Kurt Hornik, Maxwell Stinchcombe, and Halbert White. Multilayer feedforward networks are universal approximators.Neural networks, 2(5):359–366, 1989

    work page 1989

  27. [27]

    On cayley-transform methods for the discretization of lie-group equations

    Arieh Iserles. On cayley-transform methods for the discretization of lie-group equations. Foundations of Computational Mathematics, 1(2):129–160, 2001

    work page 2001

  28. [28]

    Adam: A Method for Stochastic Optimization

    Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization.arXiv preprint arXiv:1412.6980, 2014

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  29. [29]

    CelebA Dataset cropped with Haar-Cascade face detector, 2021

    Andreas M Kist. CelebA Dataset cropped with Haar-Cascade face detector, 2021. URL https://doi.org/10.5281/zenodo.5561092

    work page doi:10.5281/zenodo.5561092 2021

  30. [30]

    Hamiltonian systems and transformation in hilbert space.Proceedings of the National Academy of Sciences, 17(5):315–318, 1931

    Bernard O Koopman. Hamiltonian systems and transformation in hilbert space.Proceedings of the National Academy of Sciences, 17(5):315–318, 1931

    work page 1931

  31. [31]

    Neural operator: Learning maps between function spaces with applications to pdes.Journal of Machine Learning Research, 24(89):1–97, 2023

    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.Journal of Machine Learning Research, 24(89):1–97, 2023

    work page 2023

  32. [32]

    Learning multiple layers of features from tiny images

    Alex Krizhevsky and Geoffrey Hinton. Learning multiple layers of features from tiny images. Technical Report, 2009

    work page 2009

  33. [33]

    Gradient-based learning applied to document recognition.Proceedings of the IEEE, 86(11):2278–2324, 1998

    Yann LeCun, Léon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document recognition.Proceedings of the IEEE, 86(11):2278–2324, 1998. 11

    work page 1998

  34. [34]

    understands

    Bruno Lévy. Laplace-beltrami eigenfunctions towards an algorithm that" understands" geometry. InIEEE International Conference on Shape Modeling and Applications 2006 (SMI’06), pages 13–13. IEEE, 2006

    work page 2006

  35. [35]

    Neural Operator: Graph Kernel Network for Partial Differential Equations

    Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Neural operator: Graph kernel network for partial differential equations.arXiv preprint arXiv:2003.03485, 2020

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  36. [36]

    Fourier neural operator for parametric partial differen- tial equations.International Conference on Learning Representations, 2021

    Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Fourier neural operator for parametric partial differen- tial equations.International Conference on Learning Representations, 2021

    work page 2021

  37. [37]

    Physics-informed neural operator for learning partial differential equations.ACM/JMS Journal of Data Science, 1(3):1–27, 2024

    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.ACM/JMS Journal of Data Science, 1(3):1–27, 2024

    work page 2024

  38. [38]

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

    Lu Lu, Pengzhan Jin, Guofeng 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, 2021

    work page 2021

  39. [39]

    Elsevier, 1999

    Stéphane Mallat.A wavelet tour of signal processing. Elsevier, 1999

    work page 1999

  40. [40]

    Nerf: Representing scenes as neural radiance fields for view synthesis

    Ben Mildenhall, Pratul P Srinivasan, Matthew Tancik, Jonathan T Barron, Ravi Ramamoor- thi, and Ren Ng. Nerf: Representing scenes as neural radiance fields for view synthesis. Communications of the ACM, 65(1):99–106, 2021

    work page 2021

  41. [41]

    Simplicits: Mesh-free, geometry-agnostic elastic simulation.ACM Transactions on Graphics (TOG), 43(4): 1–11, 2024

    Vismay Modi, Nicholas Sharp, Or Perel, Shinjiro Sueda, and David IW Levin. Simplicits: Mesh-free, geometry-agnostic elastic simulation.ACM Transactions on Graphics (TOG), 43(4): 1–11, 2024

    work page 2024

  42. [42]

    Equivariant architectures for learning in deep weight spaces

    Aviv Navon, Aviv Shamsian, Idan Achituve, Ethan Fetaya, Gal Chechik, and Haggai Maron. Equivariant architectures for learning in deep weight spaces. In Andreas Krause, Emma Brunskill, Kyunghyun Cho, Barbara Engelhardt, Sivan Sabato, and Jonathan Scarlett, editors, Proceedings of the 40th International Conference on Machine Learning, volume 202 ofPro- ceed...

    work page 2023

  43. [43]

    Spectral inference networks: Unifying deep and spectral learning.International Conference on Learning Representations, 2019

    David Pfau, Stig Petersen, Ashish Agarwal, David GT Barrett, and Kimberly L Stachenfeld. Spectral inference networks: Unifying deep and spectral learning.International Conference on Learning Representations, 2019

    work page 2019

  44. [44]

    Maziar Raissi, Paris Perdikaris, and George 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

  45. [45]

    SIAM, 2011

    Yousef Saad.Numerical methods for large eigenvalue problems: revised edition. SIAM, 2011

    work page 2011

  46. [46]

    Dynamic mode decomposition of numerical and experimental data.Journal of fluid mechanics, 656:5–28, 2010

    Peter J Schmid. Dynamic mode decomposition of numerical and experimental data.Journal of fluid mechanics, 656:5–28, 2010

    work page 2010

  47. [47]

    Data-free learning of reduced-order kinematics

    Nicholas Sharp, Cristian Romero, Alec Jacobson, Etienne V ouga, Paul Kry, David IW Levin, and Justin Solomon. Data-free learning of reduced-order kinematics. InACM SIGGRAPH 2023 Conference Proceedings, pages 1–9, 2023

    work page 2023

  48. [48]

    Im- plicit neural representations with periodic activation functions.Advances in neural information processing systems, 33:7462–7473, 2020

    Vincent Sitzmann, Julien Martel, Alexander Bergman, David Lindell, and Gordon Wetzstein. Im- plicit neural representations with periodic activation functions.Advances in neural information processing systems, 33:7462–7473, 2020

    work page 2020

  49. [49]

    Prentice-Hall, Inc., 1986

    Henry Stark and John W Woods.Probability, random processes, and estimation theory for engineers. Prentice-Hall, Inc., 1986

    work page 1986

  50. [50]

    Pearson Prentice Hall Upper Saddle River, NJ, 2005

    Petre Stoica, Randolph L Moses, et al.Spectral analysis of signals, volume 452. Pearson Prentice Hall Upper Saddle River, NJ, 2005. 12

    work page 2005

  51. [51]

    Fourier features let networks learn high frequency functions in low dimensional domains.Advances in neural information processing systems, 33:7537–7547, 2020

    Matthew Tancik, Pratul Srinivasan, Ben Mildenhall, Sara Fridovich-Keil, Nithin Raghavan, Utkarsh Singhal, Ravi Ramamoorthi, Jonathan Barron, and Ren Ng. Fourier features let networks learn high frequency functions in low dimensional domains.Advances in neural information processing systems, 33:7537–7547, 2020

    work page 2020

  52. [52]

    Mechanism of the production of small eddies from large ones.Proceedings of the Royal Society of London

    Geoffrey Ingram Taylor and Albert Edward Green. Mechanism of the production of small eddies from large ones.Proceedings of the Royal Society of London. Series A-Mathematical and Physical Sciences, 158(895):499–521, 1937

    work page 1937

  53. [53]

    SIAM, 2000

    Lloyd N Trefethen.Spectral methods in MATLAB. SIAM, 2000

    work page 2000

  54. [54]

    Functional data analysis.Annual Review of Statistics and its application, 3:257–295, 2016

    Jane-Ling Wang, Jeng-Min Chiou, and Hans-Georg Müller. Functional data analysis.Annual Review of Statistics and its application, 3:257–295, 2016

    work page 2016

  55. [55]

    Neural fields in visual computing and beyond

    Yiheng Xie, Towaki Takikawa, Shunsuke Saito, Or Litany, Shiqin Yan, Numair Khan, Federico Tombari, James Tompkin, Vincent Sitzmann, and Srinath Sridhar. Neural fields in visual computing and beyond. InComputer graphics forum, volume 41, pages 641–676. Wiley Online Library, 2022

    work page 2022

  56. [56]

    Signal processing for implicit neural representations.Advances in Neural Information Processing Systems, 35: 13404–13418, 2022

    Dejia Xu, Peihao Wang, Yifan Jiang, Zhiwen Fan, and Zhangyang Wang. Signal processing for implicit neural representations.Advances in Neural Information Processing Systems, 35: 13404–13418, 2022

    work page 2022

  57. [57]

    A structured dictionary perspective on implicit neural representations

    Gizem Yüce, Guillermo Ortiz-Jiménez, Beril Besbinar, and Pascal Frossard. A structured dictionary perspective on implicit neural representations. InProceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 19228–19238, 2022

    work page 2022

  58. [58]

    finite-like

    Biao Zhang and Rico Sennrich. Root mean square layer normalization.Advances in neural information processing systems, 32, 2019. 13 A Proof of Theorem 1 We first restate the theorem for completeness: Theorem 1(Approximating the Orthogonal Group via Rank-2 Generators).Let QT denote the set of all operators Q(T) obtained by solving the initial value problem ...

    work page 2019