Pith. sign in

REVIEW 3 major objections 5 minor 18 references

Governing Equation Discovery from Data Based on Differential Invariants

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read Symmetry invariants shrink PDE search space, hitting ~100% discovery

desk verdict A clean symmetry-based trick for shrinking SINDy search spaces, with strong results on four PDEs, but the 'lossless' guarantee only holds on a restricted jet space that excludes mixed derivatives. read the letter →

arxiv 2505.18798 v1 pith:UOY3CKL3 submitted 2025-05-24 cs.LG stat.ML

Lexiang Hu , Yikang Li , Zhouchen Lin This is my paper
classification cs.LGstat.ML MSC 35A30
keywords differentialinvariantsgoverningequationdiscoverySINDyLiepointsymmetryPDEsparseregressionpriorKdV
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

The paper claims that the search space of candidate terms in PDE discovery can be losslessly compressed by choosing terms from the differential invariants of the PDE's Lie point symmetry group. Because any equation admitting that symmetry is equivalent to an equation built only from those invariants, no correct equation is lost, and the discovered skeleton obeys the symmetry by construction. The authors instantiate this idea in DI-SINDy, which feeds invariants into SINDy's sparse regression, and report success rates near 100 percent on KdV, Kuramoto–Sivashinsky, Burgers, and nKdV equations, compared with 0–82 percent for baselines. A reader should care because it turns a soft prior (symmetry as regularization) into a hard structural constraint that also shrinks the search space.

What carries the argument

The central object is the complete set of functionally independent differential invariants of the prolonged symmetry group. A differential invariant is a smooth function of the independent variables, the dependent variable, and derivatives up to order n that is unchanged by the prolonged group action; the paper computes them by prolonging infinitesimal generators, forming characteristic equations, and taking integration constants. Proposition 4.1 is the load-bearing identity: it states that a PDE admits the symmetry group if and only if it can be rewritten as a function of these invariants alone. The invariants act as the candidate-term library for sparse regression, and the proposition guarantees that no invariant-expressible correct equation is excluded.

What would settle it

Find a first-order-in-time PDE with no mixed derivatives, built from $\{t,x,u,u_t,u_x,u_{xx},u_{xxx},u_{xxxx}\}$, that admits the KdV symmetry generators but cannot be written as a function of $\{u_t+uu_x, u_x, u_{xx}, u_{xxx}, u_{xxxx}\}$; such an equation would violate the claimed completeness. Concretely, compute the full set of functionally independent invariants on that restricted space and check whether a fifth independent invariant beyond the listed five exists.

Watch

Extended reading notes

Core claim

The central claim is that differential invariants of the symmetry group are the right candidate terms: given infinitesimal generators, the paper prolongs them, solves characteristic equations to obtain a complete set of functionally independent invariants, and uses them as the equation skeleton for SINDy. Proposition 4.1, restated from the symmetry textbook, supplies the necessity and sufficiency: an n-th order PDE admits the symmetry group if and only if it can be written using only invariants of that group. Therefore the search space is compressed without loss of expressive power, and every discovered equation strictly respects the symmetry. On KdV, Kuramoto–Sivashinsky, Burgers, and nKdV, DI-SINDy reaches 98–100 percent success in 50 runs, including the KS equation where finite-difference baselines fail entirely.

Load-bearing premise

The method assumes the hand-computed invariant sets in Table 2 are complete on the restricted candidate space $\{t,x,u,u_t,u_x,u_{xx},u_{xxx},u_{xxxx}\}$, so that every first-order-in-time PDE with those symmetries and no mixed derivatives can be expressed through them.

Editorial extensions

If this is right

  • DI-SINDy recovers the correct skeleton in nearly 100 percent of runs for KdV, KS, and nKdV and 98 percent for Burgers, compared with 0 percent for KS baselines, so symmetry can rescue discovery where finite-difference error is high.
  • Any SINDy-style method can accept the invariant library, so symmetry becomes a plug-in prior rather than a loss term or an architectural constraint.
  • Because EquivSINDy-c restricts the function library to polynomials and constrains the coefficient space, and EquivSINDy-r needs a tuned regularization weight, DI-SINDy avoids both limitations.
  • Long-term predictions from the discovered equations are more accurate, since the correct skeleton and coefficients reduce integration-error accumulation.

Reading between the lines

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

  • Inference: if symmetry discovery is imperfect, DI-SINDy inherits those errors, so a natural test is feeding it partially correct generator sets and measuring how success decays; the paper notes subgroup invariance but does not quantify this.
  • Inference: the same invariant-as-candidate-terms idea should transfer to systems of PDEs and to mixed-derivative terms, provided a complete invariant basis can be computed for the larger jet space.
  • Inference: because the invariant library is smaller, the method could tolerate more aggressive derivative estimation, which suggests testing it on noisy or sparsely sampled data where finite-difference libraries are unreliable.
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 / 5 minor

Summary. The paper proposes DI-SINDy, a symmetry-informed variant of SINDy for discovering PDEs from data. Given the infinitesimal generators of the PDE's Lie point symmetry group, the method computes differential invariants and uses those invariants, rather than raw derivatives and monomials, as the candidate terms in a sparse regression. The authors invoke Olver's theorem (their Proposition 4.1) to argue that this replacement is lossless: any equation admitting the symmetry can be expressed in terms of a complete set of differential invariants. They instantiate the method on KdV, Kuramoto–Sivashinsky, Burgers, and nKdV equations, reporting substantially higher success rates and lower coefficient RMSE than SINDy and EquivSINDy-r, together with better long-term prediction errors.

Significance. If the losslessness claim is correct, the paper offers a principled way to hard-code symmetry priors into equation discovery without manually specifying the equation skeleton, and it does so in a plug-and-play manner with existing sparse-regression methods. The idea is attractive and the experimental gains on the four benchmark PDEs are large. The paper also deserves credit for grounding the construction in a classical theorem (Olver's functional-basis theorem for differential invariants) and for reporting success-rate and RMSE metrics rather than only qualitative fits. However, the central 'lossless' claim is currently stated for a broader problem than the one actually solved: the invariant sets in Table 2 are complete only on a restricted jet subspace, and the data-generation section is missing, which weakens both the theoretical and empirical support.

major comments (3)
  1. [Abstract, §4.2, Appendix C, Table 2] The losslessness claim is not justified as stated. Proposition 4.1 applies to a complete set of functionally independent invariants on the full jet space X×U^(4), but Appendix C restricts the search space to {t,x,u,u_t,u_x,u_xx,u_xxx,u_xxxx} by assuming a first-order-in-time equation with no mixed derivatives. On the full jet space, the KdV symmetry group {∂_x, ∂_t, t∂_x+∂_u} admits additional invariants involving mixed derivatives; for example, η = u_tx + u u_xx satisfies pr^(2)v_3(η) = t∂_xη + ∂_uη − u_x∂_u_tη − u_xx∂_u_txη = 0, and it is also annihilated by ∂_x and ∂_t. Hence η is a valid second-order differential invariant that is not in Table 2. Consequently, the equation (u_t + u u_x) + (u_tx + u u_xx) = 0 admits the same symmetry group but cannot be expressed using the five invariants listed in Table 2. The claimed 'lossless' compression therefore holds only for an unstated class restriction. The abstract and contributions should either state this restriction explicitly or the invariant computation must be extended to include mixed-derivative invariants on the full jet space.
  2. [Appendix D, Appendix E, §5.1] The experimental data is not actually described. Appendix D, titled 'Data generation', contains the same text as Appendix E and only mentions trajectory samples from 4 initial conditions, the L-BFGS optimizer, and the sparsity threshold; it does not specify the numerical scheme, spatial/temporal discretization, domain, noise level, or the form of the initial conditions for KdV, KS, Burgers, or nKdV. Since the paper's empirical claims (near-100% success rates, long-term prediction errors) are central to the contribution, the data-generation protocol must be reported before the experiments can be reproduced or fully assessed.
  3. [Algorithm 1, §2, §5.1] The end-to-end pipeline with data-driven symmetry discovery is not tested. Algorithm 1 includes a branch that calls a symmetry-discovery method when V(g) is empty, and Section 2 states that the approach 'can be combined with these symmetry discovery methods', but all experiments assume the infinitesimal generators are known a priori. Given that the conclusion acknowledges that inaccurate discovered symmetries may affect accuracy, at least one experiment with a learned symmetry set—or a clear statement that this extension is untested—is needed to support the claimed plug-and-play pipeline.
minor comments (5)
  1. [Appendix C.1, Eq. (23)-(24)] The completeness of the five invariants on the restricted subspace is asserted but not demonstrated. A dimension count or a brief argument that the method of characteristics yields a full functional basis on the restricted jet space would make the claim precise.
  2. [Section 5.1, Eq. (9)] In the definition of L_symm, the notation F is overloaded: it denotes both the PDE and the SINDy skeleton with coefficients W. Please clarify how the norm and expectation are evaluated and how pr^(n)v acts on the learned skeleton during optimization.
  3. [Abstract and Section 1] The word 'losslessly' should be qualified to reflect the restricted class of PDEs considered; as written, it promises more than the method delivers (see major comment).
  4. [References and notation] The reference to Proposition 2.56 in [Olver, 1993] should include the exact statement and its hypotheses (e.g., functional independence on an open dense subset), and the notation Θ(x) in Section 2 should be Θ(·) to avoid confusion with the independent variable.
  5. [Figure 3] The caption states that the MSE is averaged over 4 initial conditions and 50 runs; please specify whether the same 50 discovered equations from Table 3 are used for each initial condition and how the standard deviation is computed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the invariant library is computed from externally supplied symmetry generators and classical invariant theory, not from the target equation or from fitted data.

full rationale

The paper's derivation chain is: symmetry generators -> prolongations -> characteristic equations -> differential invariants -> candidate term library -> sparse regression. Proposition 4.1 is an external classical theorem quoted from Olver (1993), not a self-citation, and the invariants in Table 2 are computed from the symmetry generators alone, not from the target PDE or from the fitted coefficients. In the experiments the symmetry generators are given as prior knowledge, and the coefficient matrix W is fitted by SINDy; success is not guaranteed by construction, as Burgers has 98% rather than 100% success with DI-SINDy. No load-bearing self-citation appears: the cited symmetry generators come from Ko et al. (2024), whose authors do not overlap with the present paper. The only substantive caveat is in Appendix C, where the authors explicitly restrict the search space to first-order-in-time PDEs without mixed derivatives and then list invariants complete on that restricted subspace; the paper does not prove this restriction is without loss for all symmetry-admitting equations, so the abstract's broad 'losslessly reduce the search space' claim outruns the proof. That is a completeness/correctness gap, not circularity: the restricted invariant set is still derived from symmetry theory, and the target equations used in the benchmarks happen to lie inside the restricted class rather than being used to construct the invariants.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The ledger lists the assumptions that the paper's pipeline relies on beyond the data itself: two hand-set hyperparameters, the classical invariant-theoretic proposition borrowed from Olver, the first-order-in-time search space restriction, the unproved completeness of the computed invariant sets, and the assumption that the symmetry generators are known exactly. No new physical entities are introduced.

free parameters (2)
  • sparse regression masking threshold = 0.5 for KdV/KS/nKdV; 5e-3 for Burgers
    Set by hand per equation (Appendix D); it controls which terms survive masking and directly affects success rate.
  • prolongation order n = 4
    Fixed to 4 in all experiments; higher or lower order changes the invariant set and search space.
assumptions (4)
  • standard math Proposition 4.1 (Olver Prop 2.56): a PDE admits G as a symmetry group iff it can be written in terms of a complete set of functionally independent differential invariants of pr^(n)G.
    Invoked in Section 4.2 to justify losslessness; taken from Olver 1993 and not re-proved.
  • domain assumption The target PDEs are first-order in time and contain no mixed derivatives, so the search space can be restricted to {t,x,u,ut,ux,uxx,uxxx,uxxxx}.
    Section 4.2 and Appendix C state this restriction for simplifying the search space, but no argument shows it follows from symmetry or is general.
  • ad hoc to paper The invariant sets listed in Table 2 are complete functionally independent differential invariants on the restricted space.
    The paper computes them by hand in Appendix C but does not prove completeness on the restricted subspace; the 'lossless' claim depends on this.
  • domain assumption The symmetry generators are known exactly in advance.
    Section 5.1 assumes symmetries are known; the paper's own conclusion notes errors in automatically discovered symmetries could reduce accuracy.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Governing Equation Discovery from Data Based on Differential Invariants." pith.science (2026). https://pith.science/paper/UOY3CKL3

@misc{pith2026250518798,
  author       = {Pith},
  title        = {Pith review of: Governing Equation Discovery from Data Based on Differential Invariants},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UOY3CKL3}},
  note         = {Machine review of arXiv:2505.18798}
}
read the original abstract

The explicit governing equation is one of the simplest and most intuitive forms for characterizing physical laws. However, directly discovering partial differential equations (PDEs) from data poses significant challenges, primarily in determining relevant terms from a vast search space. Symmetry, as a crucial prior knowledge in scientific fields, has been widely applied in tasks such as designing equivariant networks and guiding neural PDE solvers. In this paper, we propose a pipeline for governing equation discovery based on differential invariants, which can losslessly reduce the search space of existing equation discovery methods while strictly adhering to symmetry. Specifically, we compute the set of differential invariants corresponding to the infinitesimal generators of the symmetry group and select them as the relevant terms for equation discovery. Taking DI-SINDy (SINDy based on Differential Invariants) as an example, we demonstrate that its success rate and accuracy in PDE discovery surpass those of other symmetry-informed governing equation discovery methods across a series of PDEs.

Figures

Figures reproduced from arXiv: 2505.18798 by the authors.

Figure 1
Figure 1. Comparison between the existing equation discovery method and our differential invariant [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Pipeline of our differential invariant-based equation discovery method. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Long-term prediction errors of different equation discovery methods for the KdV, KS, [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

18 extracted references · 6 canonical work pages

  1. [5]

    Towards multi-spatiotemporal-scale generalized pde modeling

    Jayesh K Gupta and Johannes Brandstetter. Towards multi-spatiotemporal-scale generalized pde modeling. arXiv preprint arXiv:2209.15616,

    arXiv

  2. [9]

    Fourier neural operator for parametric partial differential equations

    Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Fourier neural operator for parametric partial differential equations. arXiv preprint arXiv:2010.08895,

    arXiv 2010

  3. [12]

    Lie Algebra Canonicalization: Equivariant Neural Operators under arbitrary Lie Groups

    Zakhar Shumaylov, Peter Zaika, James Rowbottom, Ferdia Sherry, Melanie Weber, and Carola- Bibiane Schönlieb. Lie algebra canonicalization: Equivariant neural operators under arbitrary Lie groups. arXiv preprint arXiv:2410.02698,

    work page Pith review arXiv

  4. [13]

    Symbolic physics learner: Discovering governing equations via Monte Carlo tree search

    Fangzheng Sun, Yang Liu, Jian-Xun Wang, and Hao Sun. Symbolic physics learner: Discovering governing equations via Monte Carlo tree search. arXiv preprint arXiv:2205.13134,

    arXiv

  5. [14]

    Physics-based deep learning

    Nils Thuerey, Philipp Holl, Maximilian Mueller, Patrick Schnell, Felix Trost, and Kiwon Um. Physics-based deep learning. arXiv preprint arXiv:2109.05237,

    arXiv

  6. [15]

    Generalized Lie Symmetries in Physics-Informed Neural Operators

    Amy Xiang Wang, Zakhar Shumaylov, Peter Zaika, Ferdia Sherry, and Carola-Bibiane Schönlieb. Gen- eralized Lie symmetries in physics-informed neural operators. arXiv preprint arXiv:2502.00373,

    work page Pith review arXiv

  7. [17]

    3D steer- able CNNs: Learning rotationally equivariant features in volumetric data

    Maurice Weiler, Mario Geiger, Max Welling, Wouter Boomsma, and Taco S Cohen. 3D steer- able CNNs: Learning rotationally equivariant features in volumetric data. Advances in Neural Information Processing Systems, 31, 2018a. Maurice Weiler, Fred A Hamprecht, and Martin Storath. Learning steerable filters for rotation equivariant CNNs. In Proceedings of the ...

    arXiv

  8. [18]

    14 A Applications of symmetry Symmetry plays an important role in both traditional mathematical physics problems and the field of deep learning. For the mathematical solution of differential equations, symmetry can guide variable substitutions to reduce their order [Olver, 1993, McLachlan, 1995, Ibragimov, 1999, Hydon, 2000, Bluman and Anco, 2008, Bluman,...

    work page 1993

Show all 18 references
  1. [1993]

    Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients

    Brenden K Petersen, Mikel Landajuela, T Nathan Mundhenk, Claudio P Santiago, Soo K Kim, and Joanne T Kim. Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients. arXiv preprint arXiv:1912.04871,

    1912 arXiv

  2. [2010]

    Lie point symmetry data augmentation for neural PDE solvers

    Johannes Brandstetter, Max Welling, and Daniel E Worrall. Lie point symmetry data augmentation for neural PDE solvers. In International Conference on Machine Learning, pages 2241–2256. PMLR, 2022a. Johannes Brandstetter, Daniel Worrall, and Max Welling. Message passing neural ...

    arXiv

  3. [2018]

    Contemporary symbolic regression methods and their relative performance

    William La Cava, Bogdan Burlacu, Marco Virgolin, Michael Kommenda, Patryk Orzechowski, Fabrício Olivetti de França, Ying Jin, and Jason H Moore. Contemporary symbolic regression methods and their relative performance. Advances in Neural Information Processing Systems, 2021 (DB1):1,

    2021

  4. [2019]

    Parameterized physics-informed neural networks for parameterized PDEs

    Woojin Cho, Minju Jo, Haksoo Lim, Kookjin Lee, Dongeun Lee, Sanghyun Hong, and Noseong Park. Parameterized physics-informed neural networks for parameterized PDEs. arXiv preprint arXiv:2408.09446,

    arXiv

  5. [2020]

    Learning symbolic physics with graph networks

    Miles D Cranmer, Rui Xu, Peter Battaglia, and Shirley Ho. Learning symbolic physics with graph networks. arXiv preprint arXiv:1909.05862,

    1909 arXiv

  6. [2021]

    Active learning for neural PDE solvers

    Daniel Musekamp, Marimuthu Kalimuthu, David Holzmüller, Makoto Takamoto, and Mathias Niepert. Active learning for neural PDE solvers. arXiv preprint arXiv:2408.01536,

    arXiv

  7. [2022]

    Physics-guided data augmentation for learning the solution operator of linear differential equations

    Ye Li, Yiwen Pang, and Bin Shan. Physics-guided data augmentation for learning the solution operator of linear differential equations. In 2022 IEEE 8th International Conference on Cloud Computing and Intelligent Systems (CCIS), pages 543–547. IEEE,

    2022

  8. [2023]

    Learning infinitesimal generators of continuous symmetries from data

    Gyeonghoon Ko, Hyunsu Kim, and Juho Lee. Learning infinitesimal generators of continuous symmetries from data. arXiv preprint arXiv:2410.21853,

    arXiv

  9. [2024]

    Interpretable machine learning for science with PySR and SymbolicRegression

    Miles Cranmer. Interpretable machine learning for science with PySR and SymbolicRegression. jl. arXiv preprint arXiv:2305.01582,

    arXiv

  10. [2025]

    Incorporating symmetry into deep dynamics models for improved generalization

    Rui Wang, Robin Walters, and Rose Yu. Incorporating symmetry into deep dynamics models for improved generalization. arXiv preprint arXiv:2002.03061,

    2002 arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.