Pith. sign in

REVIEW 3 major objections 6 minor 59 references

Physics-informed reduced-order modelling with equivariant spectral submanifolds

T0 review · 3 major / 6 minor · reviewed 2026-08-08 · deepseek-v4-flash

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

Pith's one-line read This paper establishes that spectral submanifolds of equivariant systems inherit the full symmetry group, and turns that structure into a data-driven reduced-order algorithm whose fitted manifold and dynamics are exactly symmetric…

desk verdict The equivariance of SSMs is proven cleanly, but the algorithm's exact symmetry constraint hinges on a data-driven subspace selection that can fail without a singular-value gap; still deserving of serious review. read the letter →

arxiv 2608.04239 v1 pith:3IKSU2OX submitted 2026-08-04 cs.LG cs.NAmath.DSmath.NA

Georg Maierhofer This is my paper · ORCID
classification cs.LGcs.NAmath.DSmath.NA MSC 37D1037C8137M21
keywords equivariancespectralsubmanifoldsmodelreductionnonlineardynamicsdata-drivenmodellingextendednormalformdelayembedding
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Spectral-submanifold (SSM) reduction builds low-dimensional nonlinear models of high-dimensional systems by fitting the unique smoothest invariant manifold tangent to a chosen spectral subspace of the linearised dynamics. This paper seeks to establish that when the full system has a linear symmetry group, the SSM, its coordinate chart, and the reduced dynamics are automatically equivariant: the symmetry passes through every stage of the reduction. The author converts that fact into an algorithm, eSSM, which restricts every fitted coefficient to a symmetry-adapted space, so the learned model is exactly symmetric by construction. The practical payoff is a large reduction in free parameters and fit time at comparable or better prediction accuracy. This matters because symmetries are common in physical and engineering models, and this gives a principled way to build them into nonlinear reduced-order modelling.

What carries the argument

The load-bearing object is the spectral submanifold (SSM): the unique smoothest invariant manifold tangent to a chosen spectral subspace $E$ of the linearised dynamics. The argument runs through three equivariance-preserving choices: a $G$-invariant complement $E^c$ (e.g. the orthogonal complement in the $G$-averaged inner product), a restricted representation $R_0$ frozen from an orbit-augmented singular value decomposition of the data, and coefficient parametrisations through the nullspace of the equivariance operator $M_k$ (equivalently, the image of the group-averaging projection $\mathcal{R}_k$). The same treatment carries into the extended normal form, where the group action preserves the near-resonant support because the diagonalised action commutes with the linear part $\Lambda$. These linear constraints are what convert a symmetry of the full system into a reduced parameter count.

What would settle it

Take a genuinely $G$-equivariant system, fit an eSSM model, and compare the prediction started at a transformed initial condition $S x_0$ with the group transform of the prediction at $x_0$; agreement within numerical tolerance confirms the equivariance of the fitted model. A sharper test is to repeat the fit when the leading $d$ singular values of the orbit-augmented snapshot matrix are almost degenerate, so that the frozen representation is unstable: the fitted manifold should then visibly break equivariance and the reduced predictions should diverge under symmetry transformations.

Watch

Extended reading notes

Core claim

The central discovery is that equivariance is not an extra constraint bolted onto spectral-submanifold reduction but a structural property of the manifolds. For a system $\dot{x}=Ax+f(x)$ equivariant under a compact linear group $G$, the unique smoothest spectral submanifold $W(E)$ of an invariant spectral subspace $E$ is itself $G$-invariant; the graph parametrisation $h$ obeys $h(S|_E\eta)=S h(\eta)$, the reduced vector field $r$ obeys $r(S|_E\eta)=S|_E r(\eta)$, and the extended normal form is equivariant in diagonalising coordinates. The admissible Taylor coefficients of these maps are exactly the fixed points of the group-averaging projection $\mathcal{R}_k$, so the symmetry constraints are linear equations on the coefficients. The eSSM algorithm imposes those constraints exactly at every fitted polynomial order, producing a reduced model that is symmetric by construction.

Load-bearing premise

The construction depends on the data actually possessing the symmetry supplied by the user and on the leading $d$ singular values of the orbit-augmented snapshot matrix being well separated, because the restricted representation is frozen once from that SVD; if either assumption fails, the equivariance constraint imposed on the fit is the wrong one.

Editorial extensions

If this is right

  • The fitted eSSM model is exactly $G$-equivariant by construction, so it cannot drift into symmetry-broken dynamics even when trained on finite or noisy data.
  • Parameter counts fall sharply with the symmetry group: the parity group prunes all even-degree monomials, and a fourth-order cyclic group cuts free parameters by about 75 percent at every fitted manifold order.
  • Because a $G$-covariant observable leads to a $G$-equivariant delay embedding, the method applies to partial-state measurements rather than requiring full-state data.
  • The discrete-time variant uses a multiplicative resonance condition and inherits the same equivariance guarantees, covering iterated maps as well as continuous flows.
  • On a chaotic benchmark outside the strict theoretical assumptions, the equivariant fit attains accuracy comparable to the unconstrained SSM fit in roughly 40 percent of the runtime.

Reading between the lines

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

  • The same coefficient-space projection could be applied to other data-driven reduced-order methods whose unknowns are polynomial coefficients, so the equivariance machinery may transfer beyond SSMs; the paper does not explore that transfer.
  • Because the symmetry group is supplied in advance, a natural extension would be to score candidate groups by the parameter reduction and by the equivariance residual of the fitted model, which would turn symmetry discovery into a model-selection problem.
  • The theory assumes exact equivariance, so a practical testable extension is to inject a controlled symmetry-breaking perturbation into the data and measure how quickly prediction accuracy degrades; this would quantify how much of the benefit is genuine robustness.
  • The strong performance on the chaotic benchmark, where the strict assumptions around a fixed point are violated, hints that the equivariance constraint acts as a useful regulariser beyond the rigorous domain; comparing low-data equivariant and unconstrained fits could isolate that effect.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper introduces equivariant spectral submanifold (eSSM) reduction, an extension of SSM-based model reduction that imposes exact linear symmetry constraints on the manifold parametrisation and reduced dynamics. The theoretical sections claim that SSMs of equivariant systems are equivariant submanifolds, that the graph parametrisation and reduced vector field inherit the restricted group action, and that the extended normal form is equivariant. The algorithm freezes a reduced representation R0 from the truncated SVD of an orbit-augmented snapshot matrix, then constrains all fitted objects to commute with R0. Numerical experiments on an oscillator chain, the shallow-water equations on a sphere, and a CTF4Science Kuramoto–Sivashinsky forecasting task report reduced parameter counts, faster fits, and competitive accuracy.

Significance. If the central claim is established, the paper offers a principled way to incorporate symmetries into SSM-based model reduction, with exact parameter-count reductions and a concrete algorithm. The theoretical core is largely self-contained and the parameter-count reductions are exact consequences of the group action rather than fitted quantities. The public availability of the code and the inclusion of a standardized benchmark are strengths. However, the unqualified equivariance theorem is false for arbitrary spectral subspaces, the data-driven step that fixes the reduced representation is not protected against wrong similarity classes, and the reported robustness experiment is in-sample. The significance is therefore conditional on correcting these issues.

major comments (3)
  1. [Theorem 2.7] Theorem 2.7(i) is false as stated. For example, let A = λI_2 with λ > 0, let G be generated by the 90-degree rotation, and let E = span{(1,0)}. E is a modal eigenspace and hence a spectral subspace under Definition 2.1, and S commutes with A, but S E ≠ E. The proof of the theorem only shows that each full eigenspace of A is invariant under a commuting S, not that an arbitrary one-dimensional modal eigenspace inside a degenerate eigenspace is invariant. The statement, the abstract's claim that SSMs are 'naturally equivariant submanifolds', and the uniqueness argument for W(E) all need to be restricted to spectral subspaces that are direct sums of complete eigenspaces (or isotypic components). This is not pedantic: the global non-resonance condition explicitly allows repeated eigenvalues, so the counterexample is inside the paper's assumptions.
  2. [Section 4.2, Eq. (22), Lemma 4.5, Eq. (24)] The data-driven bridge freezes R0 from a truncated SVD of the orbit-augmented snapshot matrix. Lemma 4.5 only guarantees G-invariance of the dominant subspace under a strict singular-value gap; it does not guarantee that this subspace is the spectral subspace E of A from the theory, nor that R0 is the correct representation of G on E. If σ_d is close to σ_{d+1}, or if the data are not exactly equivariant under the supplied G (noise, transients, or symmetry breaking), R0 can belong to the wrong similarity class. By Proposition 4.4(ii) and the hard constraint (24), no subsequent continuous optimisation can leave that class, so the advertised parameter reductions and the 'exactly equivariant by construction' claim inherit the faulty R0. The paper provides no diagnostic or safeguard, such as checking the singular-value gap, testing equivariance of residuals, or validating R0 on held-out data.
  3. [Section 5.2, Eq. (22)] The 'Rotated' test metric is not an out-of-sample test. The training procedure in Eq. (22) augments the single training trajectory with all four rotations under G = C4, so the 90-degree-rotated trajectory used for evaluation is exactly one block of the orbit-augmented training matrix. The eSSM model is therefore trained on the rotated test trajectory, while the non-equivariant SSMLearn baseline is not. The reported improvement on the 'Rotated' metric is an in-sample consistency check of the imposed symmetry, not evidence of robustness or generalization. A fair evaluation would use a rotation not contained in the supplied group (for example, 45 degrees) or an initial condition outside the training orbit, or would explicitly exclude the rotated trajectory from the augmented training set.
minor comments (6)
  1. [Section 3.1] In the paragraph introducing the extended normal form, 'the change of variables f and n' should read 'the change of variables t and n'.
  2. [Section 5.1] The figures and Table 1 do not clearly indicate which SSMLearn variant is being compared (the Matlab implementation, SSMLearnPy, or the Python implementation with trivial group). Since runtime comparisons are implementation-dependent, this should be stated explicitly in each figure and table caption.
  3. [Section 5.3] The abstract and conclusions present the CTF4Science benchmark as a success without noting the authors' own caveat in Section 5.3 that the chaotic Kuramoto-Sivashinsky data lie outside the theory's assumption of a hyperbolic fixed point. This limitation should be stated in the abstract or at the first mention of the benchmark.
  4. [Algorithm 1, line 14] The resonant index set uses a strict inequality |Im(m·λ−λ_j)| < δ in Algorithm 1, while Eq. (9) uses ≤ δ. Please make the definition consistent.
  5. [Section 4.2 and Prop. 3.5] The symbol \tilde S is used both for the whitened group action in Step (i.1) and for the diagonalised action W^{-1}(S|_E)W in Proposition 3.5. Different symbols would avoid confusion.
  6. [Section 5] The empirical comparisons in Figures 4, 5, 8, and 9 report wall-clock times and NMTE values from single runs without error bars or repeated trials. At least a small number of repeated fits with different initialisations would make the speedup and accuracy claims more robust.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the equivariance theorems are proved in-text from stated assumptions, the algorithm enforces equivariance as a constraint rather than fitting it, and the benchmarks are external.

full rationale

The central derivation is self-contained. Theorem 2.7 proves G-invariance of the spectral subspace E and of the SSM W(E) from the equivariance identity (2), the resulting commutation SA=AS, flow equivariance, and the uniqueness of the smoothest SSM quoted from the external works Haller-Ponsioen and Haller-Kaszas-Liu-Axas ([27,28]). The present author is not an author of those references, so the load-bearing existence and uniqueness input is not a self-citation. Propositions 3.4 and 3.5 derive equivariance of the chart, reduced vector field, and extended normal form directly from invariance of E and E^c and from the termwise uniqueness of the homological construction (8)-(9); no target conclusion is assumed. Theorem 3.8 and Corollary 3.11 characterize equivariant Taylor coefficients by group averaging and character counts, which are independent mathematical facts. The parameter reductions in Tables 1 and 2 are exact counts of admissible equivariant coefficients, not fitted parameters renamed as predictions. In the eSSM algorithm, equivariance is imposed as a hard constraint: Step (i.1) fixes the reduced representation R0 from the orbit-augmented SVD (eq. 22, Lemma 4.5), and Step (i.2) restricts all fitted coefficients to the nullspace of M_k (eq. 24), so the exact equivariance of the output is a construction, not an empirical discovery. The reported improvement on rotated trajectories follows by construction from the imposed equivariance and is presented as a design benefit rather than as independent validation of the theory. The dependence of R0 on the spectral gap sigma_d > sigma_{d+1} and on exact symmetry of the data is a genuine modeling-assumption and robustness caveat, but it is not a circular derivation: the theory states what holds for genuinely equivariant systems, while the algorithm inherits the quality of its initialization. Self-citations ([47,55,57] and [8,20,22,42]) concern benchmark frameworks and unrelated symmetric-integrator works, and they are not load-bearing for the equivariant-SSM claims. No equation in the paper reduces to its own input, and no fitted quantity is renamed as a prediction.

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

The central theory relies on previously established existence and uniqueness of SSMs (Haller et al. 2023) and on standard compact-group representation theory. The algorithm adds no physical entities; its free parameters are the resonance tolerance, the residual-decay time constant, and truncation/embedding orders, chosen by hand or by tuning.

free parameters (3)
  • resonance tolerance delta = 1e-4 (Example 1); delta*dt for discrete case
    Threshold in (9) and (37) classifying near-resonant monomials; chosen by hand, changes the reduced model and the equivariant support bases.
  • residual decay time constant tau = 500 steps (CTF benchmark)
    Exponential decay rate for the off-manifold residual correction r0 exp(-i/tau); tuned via Ray Tune in Example 3 and strongly influences the short-time score E1.
  • model-order hyperparameters (d, M, M_ROD, nx, q, q_lag, stride) = d=2, M=3/4, M_ROD=2/3, nx=64, q=2, q_lag=2, stride=2
    Truncation and embedding parameters selected per experiment (by hand for Examples 1-2, by Ray Tune for Example 3); they control the number of fitted coefficients and the reported speedups.
assumptions (4)
  • domain assumption Existence, uniqueness, and C-infinity smoothness of the SSM W(E) tangent to a spectral subspace E of a semi-simple linearization under the global non-resonance condition (Theorem 2.3, from Haller et al. 2023).
    Used as the starting point in Theorem 2.7 and Prop. 3.4; the paper cites rather than proves this result.
  • domain assumption G is a compact linear group fixing the origin, acting on R^n (Definition 2.5); the algorithm further restricts to finite groups (Remark 4.1).
    Required for the equivariant inner product (Definition 3.2), the orbit-augmented SVD, and the convergence of averaging operators; excludes nonlinear or non-compact symmetries.
  • standard math Takens-type delay embedding reconstructs the SSM for generic observables with mp >= 2 dim W(E) + 1 (Section 4.1, eq. (16)).
    Justifies the delay-embedding step used in the CTF benchmark; cited from Takens and Sauer et al.
  • ad hoc to paper The CTF4Science chaotic KS data is assumed to be predictable by SSM-based reduction despite lacking a hyperbolic fixed point; the authors explicitly flag this as outside the theory's scope.
    Section 5.3 states the setting lies outside the theory; the benchmark result therefore relies on an unproven extension rather than on the paper's theorems.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Physics-informed reduced-order modelling with equivariant spectral submanifolds." pith.science (2026). https://pith.science/paper/3IKSU2OX

@misc{pith2026260804239,
  author       = {Pith},
  title        = {Pith review of: Physics-informed reduced-order modelling with equivariant spectral submanifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3IKSU2OX}},
  note         = {Machine review of arXiv:2608.04239}
}
read the original abstract

Spectral submanifold (SSM) reduction has emerged as a mathematically principled route to reliable nonlinear reduced-order models, capturing dynamics beyond the reach of linear techniques such as Dynamic Mode Decomposition (DMD). The computation of SSMs, however, remains computationally expensive, particularly for high-dimensional systems. In this work, we introduce equivariant spectral submanifold (eSSM) reduction, a novel extension of the SSM framework that explicitly incorporates symmetries of the full-order model into the reduction process. We establish the mathematical foundations of this approach by showing that SSMs are naturally equivariant submanifolds and that the associated charts and reduced dynamics inherit the appropriate induced group actions. Building on this framework, we develop a novel equivariant SSM reduction algorithm that exploits these symmetries to achieve substantially faster computations while also improving model robustness. We demonstrate the advantages of this approach on several benchmark problems including a test from the Common Task Framework for Science.

Figures

Figures reproduced from arXiv: 2608.04239 by the authors.

Figure 1
Figure 1. Schematic of the chain of oscillators, with nonlinear force at left end. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Projection onto E along Ec . However, generically, the expansion of the vector field r contains coefficients that are messy and partly redundant artefacts of the particular choice of representation of the system (1). Thus [32, 15] advocate for the representation of the reduced dynamics in an extended Poincar´e normal form style (cf. [46, 1, 24]) to 5 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. (a) Leading singular values σi of the orbit-augmented training data. (b) Fitted spectral subspace (purple plane) and SSM (or￾ange surface), together with a training trajectory (blue) [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Prediction benchmark versus the manifold order [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]
Figure 5
Figure 5. Figure 5: Performance of the methods as a function of the ambient dimension [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]
Figure 6
Figure 6. Figure 6: Sensor locations in our SWE experiment at four longitudes and nine latitudinal rings. [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]
Figure 7
Figure 7. Figure 7: Visualisation of the h-perturbation in (31). The trajectory is integrated over t ∈ [0, 400] and sampled at ∆t = 0.25; the first 600 steps are discarded as the off-manifold transient towards the slow SSM. Our observations live in R 72 (two scalar fields) and G ∼= C4. Co…
Figure 8
Figure 8. Figure 8: Performance of the methods as a function of the reduced dimension [PITH_FULL_IMAGE:figures/full_fig_p022_8.png]
Figure 9
Figure 9. Figure 9: Prediction benchmark versus the manifold order [PITH_FULL_IMAGE:figures/full_fig_p022_9.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

59 extracted references · 40 canonical work pages

  1. [34]

    Foundations of Computational Mathematics , pages=

    Symmetric resonance based integrators and forest formulae , author=. Foundations of Computational Mathematics , pages=. 2026 , publisher=

    work page 2026

  2. [1]

    Data-driven modeling and prediction of non-linearizable dynamics via spectral submanifolds , volume =

    Cenedese, Mattia and Ax. Data-driven modeling and prediction of non-linearizable dynamics via spectral submanifolds , volume =. Nature Communications , number =. doi:10.1038/s41467-022-28518-y , id =

    doi:10.1038/s41467-022-28518-y

  3. [2]

    1992 , issn =

    A standard test set for numerical approximations to the shallow water equations in spherical geometry , journal =. 1992 , issn =. doi:https://doi.org/10.1016/S0021-9991(05)80016-6 , author =

    doi:10.1016/s0021-9991(05)80016-6 1992

  4. [3]

    Higher Order Dynamic Mode Decomposition , journal =

    Le Clainche, Soledad and Vega, Jos\'. Higher Order Dynamic Mode Decomposition , journal =. 2017 , doi =

    work page 2017

  5. [4]

    arXiv:1807.05118 , year =

    Tune: A Research Platform for Distributed Model Selection and Training , author =. arXiv:1807.05118 , year =. 1807.05118 , archivePrefix=

    arXiv

  6. [5]

    Physical Review Research , keywords =

    Dedalus: A flexible framework for numerical simulations with spectral methods. Physical Review Research , keywords =. doi:10.1103/PhysRevResearch.2.023068 , primaryClass =

    doi:10.1103/physrevresearch.2.023068

  7. [6]

    Tellus A: Dynamic Meteorology and Oceanography , volume=

    An initial-value problem for testing numerical models of the global shallow-water equations , author=. Tellus A: Dynamic Meteorology and Oceanography , volume=. 2004 , publisher=

    2004

  8. [7]

    Henri Poincar. Les M

    work page

Show all 59 references
  1. [8]

    V. I. Arnold , title =. 1983 , series =

    1983

  2. [9]

    1983 , series =

    John Guckenheimer and Philip Holmes , title =. 1983 , series =

    1983

  3. [10]

    2011 , publisher=

    Halko, Nathan and Martinsson, Per-Gunnar and Tropp, Joel A , journal=. 2011 , publisher=

    2011

  4. [11]

    Faou, Erwan and Maierhofer, Georg and Schratz, Katharina , journal =

  5. [12]

    Mathematics of Computation , year =

    Maierhofer, Georg and Schratz, Katharina , title =. Mathematics of Computation , year =

  6. [13]

    Advances in Neural Information Processing Systems , volume=

    Common task framework for a critical evaluation of scientific machine learning algorithms , author=. Advances in Neural Information Processing Systems , volume=

  7. [14]

    and Budi

    Brunton, Steven L. and Budi. SIAM Review , volume =. 2022 , doi =

    2022

  8. [15]

    An Introductory Guide to Koopman Learning

    Colbrook, Matthew and Drma c , Zlatko and Horning, Andrew. An Introductory Guide to Koopman Learning. Operator Theory. 2016. doi:10.1007/978-3-0348-0692-3_126-1

    2016 doi

  9. [16]

    2025 , issn =

    Group-convolutional extended dynamic mode decomposition , journal =. 2025 , issn =. doi:https://doi.org/10.1016/j.physd.2025.134725 , author =

    2025

  10. [17]

    and D’Souza, Raissa M

    Salova, Anastasiya and Emenheiser, Jeffrey and Rupe, Adam and Crutchfield, James P. and D’Souza, Raissa M. , title =. Chaos: An Interdisciplinary Journal of Nonlinear Science , volume =. 2019 , month =

    2019

  11. [18]

    Nathan and Brunton, Bingni W

    Fasel, Urban and Kutz, J. Nathan and Brunton, Bingni W. and Brunton, Steven L. , title =. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences , volume =. 2022 , doi =

    2022

  12. [19]

    Physical Review Letters , author =

    Model-. Physical Review Letters , author =. 2018 , pages =. doi:10.1103/PhysRevLett.120.024102 , abstract =

    2018 doi

  13. [20]

    Journal of Computer and System Sciences , author =

    On the computational power of circuits of spiking neurons , volume =. Journal of Computer and System Sciences , author =. doi:10.1016/j.jcss.2004.04.001 , abstract =

    2004 doi

  14. [21]

    Jaeger, Herbet , title =

  15. [22]

    Neural computation , volume=

    Long short-term memory , author=. Neural computation , volume=. 1997 , publisher=

    1997

  16. [23]

    and Costa, M

    Coelho, C. and Costa, M. Fernanda P. and Ferr. Applied Mathematics and Computation , volume=. 2024 , publisher=

    2024

  17. [24]

    Variable projection methods for an optimized dynamic mode decomposition , Volume =

    Askham, Travis and Kutz, J Nathan , Journal =. Variable projection methods for an optimized dynamic mode decomposition , Volume =

  18. [25]

    de Silva and J

    Shaowu Pan and Eurika Kaiser and Brian M. de Silva and J. Nathan Kutz and Steven L. Brunton , title =. 2024 , publisher =. doi:10.21105/joss.05881 , url =

    2024 doi

  19. [26]

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

    Lu, Lu and Jin, Pengzhan and Pang, Guofei and Zhang, Zhongqiang and Karniadakis, George Em , journal =. Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators , volume =

  20. [27]

    Hou and Max Tegmark , booktitle=

    Ziming Liu and Yixuan Wang and Sachin Vaidya and Fabian Ruehle and James Halverson and Marin Soljacic and Thomas Y. Hou and Max Tegmark , booktitle=. 2025 , url=

    2025

  21. [28]

    The Eleventh International Conference on Learning Representations , year=

    Effectively Modeling Time Series with Simple Discrete State Spaces , author=. The Eleventh International Conference on Learning Representations , year=

  22. [29]

    The Ninth International Conference on Learning Representations , year=

    Fourier Neural Operator for Parametric Partial Differential Equations , author=. The Ninth International Conference on Learning Representations , year=

  23. [30]

    Chen, Ricky T. Q. and Rubanova, Yulia and Bettencourt, Jesse and Duvenaud, David , booktitle=. 2018 , url=

    2018

  24. [31]

    The Fourteenth International Conference on Learning Representations , year=

    The Seismic Wavefield Common Task Framework , author=. The Fourteenth International Conference on Learning Representations , year=

  25. [32]

    Nathan , journal=

    Riva, Stefano and Introini, Carolina and Cammi, Antonio and Price, Dean and Yermakov, Alexey and Zhao, Yue and Wyder, Philippe M and Goldfeder, Judah and Williams, Jan and Rude, Amy Sara and Tomasetto, Matteo and Germany, Joe and Bakarji, Joseph and Maierhofer, Georg and Cranm...

  26. [33]

    Explicit symmetric low-regularity integrators for the nonlinear Schr

    Feng, Yue and Maierhofer, Georg and Wang, Chushan , journal=. Explicit symmetric low-regularity integrators for the nonlinear Schr. 2025 , publisher=

    2025

  27. [35]

    and Lubich, C

    Hairer, E. and Lubich, C. and Wanner, G. , year=

  28. [36]

    and Stern, Ari , date =

    McLachlan, Robert I. and Stern, Ari , date =. Functional Equivariance and Conservation Laws in Numerical Integration , volume =. Foundations of Computational Mathematics , number =. doi:10.1007/s10208-022-09590-8 , id =

    doi

  29. [37]

    Fast data-driven model reduction for nonlinear dynamical systems , volume =

    Ax. Fast data-driven model reduction for nonlinear dynamical systems , volume =. Nonlinear Dynamics , number =. doi:10.1007/s11071-022-08014-0 , id =

    doi

  30. [38]

    The parameterization method for invariant manifolds I: Manifolds associated to non-resonant subspaces

    Xavier Cabr\'e and Ernest Fontich and Rafael de la Llave. The parameterization method for invariant manifolds I: Manifolds associated to non-resonant subspaces. Indiana Univ. Math. J

  31. [39]

    The parameterization method for invariant manifolds II: regularity with respect to parameters

    Xavier Cabr\'e and Ernest Fontich and Rafael de la Llave. The parameterization method for invariant manifolds II: regularity with respect to parameters. Indiana Univ. Math. J

  32. [40]

    2005 , issn =

    The parameterization method for invariant manifolds III: overview and applications , journal =. 2005 , issn =. doi:10.1016/j.jde.2004.12.003 , author =

    2005 doi

  33. [41]

    Nonlinear normal modes and spectral submanifolds: existence, uniqueness and use in model reduction , volume =

    Haller, George and Ponsioen, Sten , date =. Nonlinear normal modes and spectral submanifolds: existence, uniqueness and use in model reduction , volume =. Nonlinear Dynamics , number =. doi:10.1007/s11071-016-2974-z , id =

    doi

  34. [42]

    Shobhit Jain and Thomas Thurnher and Mingwu Li and George Haller , title =

  35. [43]

    1977 , edition =

    Serre, Jean-Pierre , title =. 1977 , edition =. doi:10.1007/978-1-4684-9458-7 , isbn =

    1977 doi

  36. [44]

    Nonlinear dynamics , volume=

    How to compute invariant manifolds and their reduced dynamics in high-dimensional finite element models , author=. Nonlinear dynamics , volume=. 2022 , publisher=

    2022

  37. [45]

    Chaos: An Interdisciplinary Journal of Nonlinear Science , volume =

    Haller, George and Kaszás, Bálint and Liu, Aihui and Axås, Joar , title =. Chaos: An Interdisciplinary Journal of Nonlinear Science , volume =. 2023 , month =

    2023

  38. [46]

    Science advances , volume=

    Data-driven discovery of partial differential equations , author=. Science advances , volume=. 2017 , publisher=

    2017

  39. [47]

    Schmid, Peter J. , year=. Dynamic mode decomposition of numerical and experimental data , volume=. doi:10.1017/S0022112010001217 , journal=

    doi

  40. [48]

    Tu and Clarence W

    Jonathan H. Tu and Clarence W. Rowley and Dirk M. Luchtenburg and Steven L. Brunton and J. N. Kutz , title =. Journal of Computational Dynamics , doi =. 2014 , pages =

    2014

  41. [49]

    Dynamical Systems and Turbulence, Warwick 1980 , editor =

    Floris Takens , title =. Dynamical Systems and Turbulence, Warwick 1980 , editor =. 1981 , publisher =

    1980

  42. [50]

    and Casdagli, Martin , date =

    Sauer, Tim and Yorke, James A. and Casdagli, Martin , date =. Embedology , volume =. Journal of Statistical Physics , number =. doi:10.1007/BF01053745 , id =

    doi

  43. [51]

    Deyle and George Sugihara , title =

    Ethan R. Deyle and George Sugihara , title =. PLoS ONE , volume =. 2011 , doi =

    2011

  44. [52]

    Capturing the edge of chaos as a spectral submanifold in pipe flows , volume=

    Kaszás, Bálint and Haller, George , year=. Capturing the edge of chaos as a spectral submanifold in pipe flows , volume=. doi:10.1017/jfm.2023.956 , journal=

    2023 doi

  45. [53]

    Nathan and Brunton, Steven L

    Kutz, J. Nathan and Brunton, Steven L. and Brunton, Bingni W. and Proctor, Joshua L. , title =. 2016 , doi =

    2016

  46. [54]

    Brunton and Joshua L

    Steven L. Brunton and Joshua L. Proctor and J. Nathan Kutz , title =. Proceedings of the National Academy of Sciences , volume =. 2016 , doi =

    2016

  47. [55]

    Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences , volume=

    Physics-informed dynamic mode decomposition , author=. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences , volume=. 2023 , publisher=

    2023

  48. [56]

    2017 , doi =

    Model Reduction and Approximation , publisher =. 2017 , doi =

    2017

  49. [57]

    Chaos: An Interdisciplinary Journal of Nonlinear Science , volume =

    Bettini, Leonardo and Kasz\'as, B\'alint and Zybach, Bernhard and Dual, J\"urg and Haller, George , title =. Chaos: An Interdisciplinary Journal of Nonlinear Science , volume =. 2025 , month =

    2025

  50. [58]

    Symmetry-Informed Governing Equation Discovery , volume =

    Yang, Jianke and Rao, Wang and Dehmamy, Nima and Walters, Robin and Yu, Rose , booktitle =. Symmetry-Informed Governing Equation Discovery , volume =. doi:10.52202/079017-2085 , editor =

    doi

  51. [59]

    Model reduction for nonlinearizable dynamics via delay-embedded spectral submanifolds , volume =

    Ax. Model reduction for nonlinearizable dynamics via delay-embedded spectral submanifolds , volume =. Nonlinear Dynamics , number =. doi:10.1007/s11071-023-08705-2 , id =

    doi

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.