Pith. sign in

REVIEW 3 major objections 5 minor 41 references

Discovering Symmetries of ODEs by Symbolic Regression

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read Search-based symbolic regression finds ODE symmetries that computer algebra systems miss.

desk verdict A solid, honestly-scoped heuristic for finding Lie point symmetries by symbolic regression; the method demonstrably works on ten nonlinear ODEs, but the 'existing CASs cannot find' claim is one package wider than the evidence. read the letter →

arxiv 2506.19550 v1 pith:CQEEGONN submitted 2025-06-24 cs.LG

Paul Kahlmeyer , Niklas Merk , Joachim Giesen This is my paper · ORCID
classification cs.LG MSC 34C14
keywords LiepointsymmetriesordinarydifferentialequationssymbolicregressionHamiltoniansymmetryorderreductionnonlinearODEsdeterminingexpressionDAG
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

This paper tries to establish that search-based symbolic regression can replace the PDE-solving step in Lie symmetry analysis of ODEs. The authors build a loss function from the continuous symmetry condition, evaluate it on a few numerically simulated solution curves, and search over symbolic expression graphs for the components of a symmetry generator. They report that this recovers simple symmetry generators for ten nonlinear systems for which an established computer algebra symmetry finder does not succeed, with every recovered generator verified both numerically and symbolically. If this holds, symmetry-based order reduction becomes available for a wider class of nonlinear ODE systems, because the bottleneck moves from solving determining partial differential equations to searching expression space.

What carries the argument

The load-bearing object is the simplified symmetry condition: after removing the Hamiltonian part, a generator $\eta^\star$ is valid exactly when $$0 = \nabla \eta^\star(t,y)\cdot \binom{1}{f(t,y)} - \nabla f(t,y)\cdot \binom{0}{\eta^\star(t,y)},$$ with gradients taken over $t$ and $y$. The paper's loss $L_{D,f}(\eta^\star)$ is the average squared residual of this identity over sample points $D$ from Runge-Kutta trajectories; it is zero when the sampled points satisfy the condition. The search is carried out by a multi-output symbolic regressor that enumerates directed acyclic graph (DAG) skeletons up to a size budget, labels operator nodes with function symbols, computes gradients by automatic differentiation, and discards near-zero generators to avoid the trivial solution $\eta^\star=0$.

What would settle it

Run the pipeline on a nonlinear ODE whose known symmetry generator is too complex for the search budget: if the sampled loss reaches machine precision on the training curves but the best expression fails the exact symmetry condition at densely sampled off-trajectory points, then a small sampled loss is not sufficient evidence of a true global symmetry. A complementary check is to feed the found generator back into an exact symbolic verifier on a held-out set of initial conditions and see whether the reduced ODE system still solves the original problem.

Watch

Extended reading notes

Core claim

The central claim is that the difficult part of symmetry finding, solving the overdetermined PDE system for the generator components, can be bypassed by searching over candidate generator functions with a loss that directly encodes the Lie condition. For a first-order system $y'=f(t,y)$, the paper removes the always-present Hamiltonian symmetry $X_H=\partial_t+\langle f,\nabla_y\rangle$, so only the informative part $\eta^\star$ remains; the corresponding reduced symmetry condition is a pointwise algebraic equation involving $\eta^\star$, its gradients, and $f$. A candidate generator is scored by summing squared residuals of this condition over points from Runge-Kutta trajectories, and a multi-output DAG-based symbolic regressor searches for minimizers. The paper reports that this pipeline finds simple generators for all ten tested systems, and that each found generator satisfies the condition both numerically and symbolically.

Load-bearing premise

The whole approach rests on the assumption that fitting the symmetry condition on a few numerically simulated solution curves is enough to make the search settle on a generator that satisfies the exact symmetry condition everywhere, not just on those curves.

Editorial extensions

If this is right

  • Symmetry finding no longer depends on a computer algebra system solving the determining PDEs; any system whose symmetries have compact symbolic generators becomes reachable by search.
  • Once a generator is found, it provides a coordinate transformation that reduces the ODE dimension by one, as demonstrated on a nonlinear example where two independent generators lead to solvable reduced systems.
  • Explicitly removing the Hamiltonian symmetry makes meaningful independent generators discoverable, so several symmetries can be collected and used separately to simplify a system.
  • The approach extends to higher-order ODEs through the standard first-order reformulation, widening the class of equations whose symmetries can be found automatically.

Reading between the lines

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

  • The paper leaves implicit that the loss could be sampled on random points in state space instead of trajectory points; if that works, the method would depend only on the vector field $f$ and not on choosing informative initial conditions.
  • Because the search space of expression DAGs grows exponentially with generator complexity, the practical advantage over algebraic methods may be limited to symmetries with short formulas; one could test this by adding a complexity penalty to the loss.
  • The same loss is continuous in $\eta^\star$, so the pipeline could in principle detect approximate or partial symmetries on systems that have no exact Lie point symmetry, giving a quantitative measure of symmetry breaking.
  • The Hamiltonian-subtraction trick may transfer to symmetry finding for partial differential equations, where the analogous time-evolution symmetry is equally uninformative.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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. Kahlmeyer et al. propose a search-based symbolic regression method for discovering Lie point symmetry generators of first-order ODE systems. Given a symbolic ODE y' = f(t,y), they simulate RK45 trajectories, then minimize a loss L_{D,f} that enforces the reduced symmetry condition (Eq. 1) at the sampled points, using the DAG-based multi-output symbolic regressor from Kahlmeyer et al. (2024). The Hamiltonian/time-evolution component of the generator is removed so that the search returns only informative, non-redundant symmetries. The paper demonstrates the method on ten nonlinear 2D ODE systems, reporting for each a closed-form generator with near-zero numerical loss and successful symbolic verification (Table 4). It compares with Mathematica's SYM package in automatic and two interactive modes (Table 1), on which SYM mostly fails, and it illustrates the use of two independent generators on one example system.

Significance. Within a properly scoped claim, this is a solid contribution. The algebraic derivation of the reduced symmetry condition (Eq. 1) is correct, and the removal of the trivial Hamiltonian part is algebraically justified. The strongest evidence is Table 4: every reported generator is validated symbolically, not merely by finite-sample loss, so the ten reported findings are not false positives. The circularity concern raised in the stress test does not land, because the known generators enter only through verification and not through the optimized loss. The idea of using a modern multi-output symbolic regressor directly on the Lie condition is a natural but useful adaptation, and the code is released. If the external comparison to computer algebra systems is qualified to the specific package and modes tested, the paper is a meaningful step toward automatic symmetry discovery for nonlinear ODEs.

major comments (3)
  1. [Abstract; Discussion, 'Hard Problems for Mathematica and SYM'] The headline claim that the method finds symmetries 'that existing CASs cannot find' is broader than the evidence presented. Only one package (Mathematica's SYM) in three fixed modes is tested, and the paper gives no details about the SYM version, solving time, or the precise ansatz used in the interactive modes. Table 1 shows that SYM-S, which supplies the known generator as a starting ansatz, already succeeds on seven of the ten problems, so the failures are configuration-dependent rather than an intrinsic property of computer algebra symmetry finding. I recommend either (a) testing at least one further independent symmetry-finding implementation, e.g., Maple's PDEtools-based facilities, or (b) rewriting the title, abstract, and conclusion to say 'symmetries that are not found by Mathematica's SYM package in the configurations described here'.
  2. [Section 'Hamiltonian Symmetry'] The statement that the Hamiltonian symmetry 'cannot be used for simplifying the ODE system' is not correct as stated. For the introductory system f = (-y2, y1), canonical coordinates for X_H = ∂t - y2∂y1 + y1∂y2 give r = sqrt(y1^2 + y2^2), v = t, and s = atan(y1/y2), and the reduced system is r' = 0, s' = 1, which is a genuine order reduction. The exclusion of the Hamiltonian component is better motivated by the search objective of avoiding rediscovery of f, not by an inability to simplify. Please correct or clarify this claim, because it is used to justify restricting the search to generators of the form <η*, ∇y>.
  3. [Section 'Symbolic Regression for Symmetry Finding'; Discussion and Conclusion] The method's general reliability is not certified by the finite-sample loss. L_{D,f} is zero if Eq. (1) holds on finitely many points of a few RK trajectories, and a zero there does not imply the continuous PDE condition. In the reported experiments the subsequent symbolic verification in Table 4 is what rules out false positives, and this should be stated explicitly as part of the method pipeline. The conclusion's sentence that the approach 'enlarges the space of ODEs' whose analytical solution can be approached extrapolates beyond the evidence; a limitations paragraph is needed to describe the sample dependence and the heuristic nature of the search.
minor comments (5)
  1. [Supplementary Material, 'Identifiability'] The text 'see subsection' is a dangling reference; supply the subsection number or title.
  2. [Section 'Symbolic Regression for Symmetry Finding'] The main text describes the data as a single sample D = {t_i, y(t_i)}, while the experiments use three time series (Table 3). State explicitly that D is the union of points over the multiple trajectories.
  3. [Table 4] The column 'Symbolic Loss' with entries 'True/True' should explain the verification procedure, e.g., simplification of the expression to zero by SymPy, and the threshold or simplification method used.
  4. [Figure 5] Figure 5 is hard to read because running times and expression sizes are combined in one panel with two different scales; separate panels or explicit dual-axis labeling would improve clarity.
  5. [References] The reference to 'Richardson and Dale. 2014' is incomplete; include the full author list and title of the book chapter.

Circularity Check

0 steps flagged · score 1.0 of 10

No meaningful circularity: the loss directly encodes the Lie symmetry condition and the found generators are verified symbolically; the self-citation to Kahlmeyer et al. (2024) supplies only the generic search engine.

full rationale

The paper's derivation chain is self-contained. The loss L_D,f is constructed from the known ODE f and the classical symmetry condition (Eq. 1); it contains no fitted parameters and no term that embeds the target generators listed in Table 2. The search minimizes this loss over DAG expressions, and the reported generators are checked symbolically ('We additionally checked, whether the generator condition was fullfilled numerically and symbolically', Table 4), so the results are not equivalent to the loss by construction in the sense of a fitted parameter being renamed as a prediction. The only self-citation, Kahlmeyer et al. (2024), is used to justify that the DAG-based symbolic regressor can handle multi-output problems; that prior work is a generic SR engine, not a result that presupposes the symmetry findings, so the citation is not load-bearing. The finite-sample loss on three Runge-Kutta trajectories cannot by itself certify the continuous symmetry condition, but the paper's separate symbolic verification closes that gap for the reported outputs. The claim that existing CASs cannot find the symmetries is an empirical comparison against one configuration of SYM; that is a correctness/strength-of-evidence concern, not a circularity. No step in the derivation reduces by definition to its own inputs.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central demonstration relies on standard Lie theory, an assumption about the numerical samples, an assumption about the search algorithm's practical effectiveness, and an assumption about the representativeness of the ten test systems. There are no invented entities. The only hand-chosen scalar is the epsilon threshold; other choices are data generation and search hyperparameters.

free parameters (3)
  • epsilon threshold (epsilon = 0.01) = 0.01
    Hand-chosen threshold to discard near-zero generators; not fitted to data but set ad hoc.
  • time series count and intervals = 3 trajectories; intervals in Table 3
    Data generation choices made per problem to ensure identifiability; not fitted, but manually selected.
  • SR search hyperparameters = not reported
    DAG size limit, operator vocabulary, and time budget come from Kahlmeyer et al. 2024 and are not specified here; they control the search space and capacity.
assumptions (5)
  • standard math The Lie symmetry condition in Eq. (1) is necessary and sufficient for eta* grad_y to be the non-trivial part of a point symmetry generator.
    Background from Lie theory (Bluman and Kumei 1989; Stephani 1993), used throughout the derivation.
  • standard math Generators form a vector space, so subtracting xi X_H from X leaves a generator eta* grad_y.
    Invoked in the 'Hamiltonian Symmetry' section to justify the reduced search space.
  • domain assumption The numerically sampled trajectories D are sufficiently accurate and representative for the loss to guide the search toward an exact symmetry.
    The method relies on RK45 samples; the paper does not analyze how sampling density or solver error affect the loss.
  • domain assumption The DAG-based symbolic regressor (Kahlmeyer et al. 2024) can find a near-global minimizer of L_D,f in its expression space.
    The reported successes depend on this empirical property of the search; no guarantees are given.
  • domain assumption The ten test ODE systems and the SYM configuration represent the general landscape of CAS symmetry finding.
    Used to support the broad claim about 'existing CASs'; only one package is tested.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Discovering Symmetries of ODEs by Symbolic Regression." pith.science (2026). https://pith.science/paper/CQEEGONN

@misc{pith2026250619550,
  author       = {Pith},
  title        = {Pith review of: Discovering Symmetries of ODEs by Symbolic Regression},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CQEEGONN}},
  note         = {Machine review of arXiv:2506.19550}
}
read the original abstract

Solving systems of ordinary differential equations (ODEs) is essential when it comes to understanding the behavior of dynamical systems. Yet, automated solving remains challenging, in particular for nonlinear systems. Computer algebra systems (CASs) provide support for solving ODEs by first simplifying them, in particular through the use of Lie point symmetries. Finding these symmetries is, however, itself a difficult problem for CASs. Recent works in symbolic regression have shown promising results for recovering symbolic expressions from data. Here, we adapt search-based symbolic regression to the task of finding generators of Lie point symmetries. With this approach, we can find symmetries of ODEs that existing CASs cannot find.

Figures

Figures reproduced from arXiv: 2506.19550 by the authors.

Figure 1
Figure 1. Three numerical solutions for the example ODEs [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Generator of a Lie symmetry as a tangent vector [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. The symbolic regression framework (Kahlmeyer [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (1 more)
Figure 5
Figure 5. Figure 5: Top: Average running times and 95% confidence [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

41 extracted references · 25 canonical work pages

  1. [1]

    , " * write output.state after.block = add.period write newline

    ENTRY address archivePrefix author booktitle chapter edition editor eid eprint howpublished institution isbn journal key month note number organization pages publisher school series title type volume year label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.a...

  2. [2]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

  3. [3]

    Baumann, G. 2002. Symmetry Analysis of Differential Equations Using MATHLIE. Journal of Mathematical Sciences, 108(6): 1052--1069

    work page 2002

  4. [4]

    W.; and Kumei, S

    Bluman, G. W.; and Kumei, S. 1989. Symmetries and Differential Equations, volume 81 of. Applied Mathematical Sciences Series

    work page 1989

  5. [5]

    Bondesan, R.; and Lamacraft, A. 2019. Learning Symmetries of Classical Integrable Systems

    work page 2019

  6. [6]

    Cheb-Terrab, E.; and von Bülow , K. 1995. A computational approach for the analytical solving of partial differential equations. Computer Physics Communications, 90(1): 102--116

    work page 1995

  7. [7]

    S.; Duarte, L

    Cheb-Terrab , E. S.; Duarte, L. G. S.; and da Mota , L. A. C. P. 1997. Computer Algebra Solving of First Order ODEs Using Symmetry Methods . Computer Physics Communications, 101(3): 254--268

    work page 1997

  8. [8]

    S.; Duarte, L

    Cheb-Terrab , E. S.; Duarte, L. G. S.; and da Mota , L. A. C. P. 1998. Computer Algebra Solving of Second Order ODEs Using Symmetry Methods . Computer Physics Communications, 108(1): 90--114

    work page 1998

Show all 41 references
  1. [9]

    A.; Levinson, N.; and Teichmann, T

    Coddington, E. A.; Levinson, N.; and Teichmann, T. 1956. Theory of ordinary differential equations

    1956

  2. [10]

    Dimas, S.; and Tsoubelis, D. 2005. SYM: A new symmetry-finding package for Mathematica. Group Analysis of Differential Equations, 64--70

    2005

  3. [11]

    Dormand, J.; and Prince, P. 1980. A family of embedded Runge-Kutta formulae. Journal of Computational and Applied Mathematics, 6(1): 19--26

    1980

  4. [12]

    A.; Miranda, M.; Pallarès, J.; and Sales-Pardo, M

    Guimerà, R.; Reichardt, I.; Aguilar-Mogas, A.; Massucci, F. A.; Miranda, M.; Pallarès, J.; and Sales-Pardo, M. 2020. A Bayesian machine scientist to aid in the solution of challenging scientific problems. Science Advances, 6(5): eaav6971

    2020

  5. [13]

    Hirsch, M.; and Smale, S. 1974. Differential equations, dynamical systems, and linear algebra. Number 60 in Pure and applied mathematics. San Diego [u.a.]: Acad. Press

    1974

  6. [14]

    Holland, J. H. 1975. Adaptation in Natural and Artificial Systems. University of Michigan Press. Second edition, 1992

    1975

  7. [15]

    Jin, Y.; Fu, W.; Kang, J.; Guo, J.; and Guo, J. 2020. Bayesian Symbolic Regression. arXiv:1910.08892

    2020 arXiv

  8. [16]

    Kahlmeyer, P.; Giesen, J.; Habeck, M.; and Voigt, H. 2024. Scaling Up Unbiased Search-based Symbolic Regression. In Larson, K., ed., Proceedings of the Thirty-Third International Joint Conference on Artificial Intelligence, IJCAI-24 , 4264--4272. International Joint Conference...

    2024

  9. [17]

    Kamienny, P.-A.; d'Ascoli, S.; Lample, G.; and Charton, F. 2022. End-to-end Symbolic Regression with Transformers. In Oh, A. H.; Agarwal, A.; Belgrave, D.; and Cho, K., eds., Advances in Neural Information Processing Systems

    2022

  10. [18]

    Kommenda, M.; Burlacu, B.; Kronberger, G.; and Affenzeller, M. 2020. Parameter identification for symbolic regression using nonlinear least squares. Genetic Programming and Evolvable Machines, 21(3): 471--501

    2020

  11. [19]

    Koza, J. R. 1994. Genetic programming as a means for programming computers by natural selection. Statistics and computing, 4: 87--112

    1994

  12. [20]

    La Cava, W.; Spector, L.; and Danai, K. 2016. Epsilon-Lexicase Selection for Regression. In Proceedings of the Genetic and Evolutionary Computation Conference 2016, GECCO '16, 741–748. New York, NY, USA: Association for Computing Machinery. ISBN 9781450342063

    2016

  13. [21]

    S.; Yang, J.; Glatt, R.; Santiago, C

    Landajuela, M.; Lee, C. S.; Yang, J.; Glatt, R.; Santiago, C. P.; Aravena, I.; Mundhenk, T.; Mulcahy, G.; and Petersen, B. K. 2022. A Unified Framework for Deep Symbolic Regression. In Koyejo, S.; Mohamed, S.; Agarwal, A.; Belgrave, D.; Cho, K.; and Oh, A., eds., Advances in N...

    2022

  14. [22]

    Lie, S. 1888. Theorie der Transformationsgruppen Abschn. 1. Teubner

  15. [23]

    Liu, Z.; Madhavan, V.; and Tegmark, M. 2022. AI Poincar ' \ e \ 2.0: Machine Learning Conservation Laws from Differential Equations. arXiv preprint arXiv:2203.12610

    2022 arXiv

  16. [24]

    Liu, Z.; and Tegmark, M. 2021. Machine Learning Conservation Laws from Trajectories . Physical Review Letters, 126(18): 180604

    2021

  17. [25]

    Liu, Z.; and Tegmark, M. 2022. Machine-Learning Hidden Symmetries. Physical Review Letters, 128(18): 180201

    2022

  18. [26]

    Lorenz, H. 1993. Nonlinear Dynamical Economics and Chaotic Motion. Lecture notes in economics and mathematical systems. Springer-Verlag. ISBN 9783540568810

    1993

  19. [27]

    Maplesoft, a division of Waterloo Maple Inc. 2019. Maple

    2019

  20. [28]

    P.; Paprocki, M.; Cert \' k, O.; Kirpichev, S

    Meurer, A.; Smith, C. P.; Paprocki, M.; Cert \' k, O.; Kirpichev, S. B.; Rocklin, M.; Kumar, A.; Ivanov, S.; Moore, J. K.; Singh, S.; Rathnayake, T.; Vig, S.; Granger, B. E.; Muller, R. P.; Bonazzi, F.; Gupta, H.; Vats, S.; Johansson, F.; Pedregosa, F.; Curry, M. J.; Terrel, A...

    2017

  21. [29]

    N.; Landajuela, M.; Glatt, R.; Santiago, C

    Mundhenk, T. N.; Landajuela, M.; Glatt, R.; Santiago, C. P.; faissol, D.; and Petersen, B. K. 2021. Symbolic Regression via Deep Reinforcement Learning Enhanced Genetic Programming Seeding. In Beygelzimer, A.; Dauphin, Y.; Liang, P.; and Vaughan, J. W., eds., Advances in Neura...

    2021

  22. [30]

    Noether, E. 1918. Invariante Variationsprobleme. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 1918: 235--257

    1918

  23. [31]

    Paszke, A.; Gross, S.; Chintala, S.; Chanan, G.; Yang, E.; DeVito, Z.; Lin, Z.; Desmaison, A.; Antiga, L.; and Lerer, A. 2017. Automatic differentiation in PyTorch. In NIPS-W

    2017

  24. [32]

    K.; Larma, M

    Petersen, B. K.; Larma, M. L.; Mundhenk, T. N.; Santiago, C. P.; Kim, S. K.; and Kim, J. T. 2021. Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients. In International Conference on Learning Representations

    2021

  25. [33]

    J.; and Dale

    Richardson, M. J.; and Dale. 2014. Complex Dynamical Systems in Social and Personality Psychology: Theory, Modeling, and Analysis. Cambridge University Press

    2014

  26. [34]

    Schmidt, M.; and Lipson, H. 2009. Distilling Free-Form Natural Laws from Experimental Data. Science, 324(5923): 81--85

    2009

  27. [35]

    Stephani, H. 1993. Differentialgleichungen. Symmetrien und L \"o sungsmethoden. Heidelberg: Spektrum , 48

    1993

  28. [36]

    Stephens, T. 2016. Genetic Programming in Python, with a scikit-learn inspired API: gplearn

    2016

  29. [37]

    Strogatz, S. H. 2000. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering. Westview Press

    2000

  30. [38]

    Tenenbaum, M.; and Pollard, H. 1985. Ordinary Differential Equations. Dover Books on Mathematics. Dover Publications. ISBN 9780486649405

    1985

  31. [39]

    Udrescu, S.-M.; and Tegmark, M. 2020. AI Feynman: A physics-inspired method for symbolic regression. Science Advances, 6(16): eaay2631

    2020

  32. [40]

    Virgolin, M.; Alderliesten, T.; Witteveen, C.; and Bosman, P. A. 2021. Improving model-based genetic programming for symbolic regression of small expressions. Evolutionary computation, 29(2): 211--237

    2021

  33. [41]

    Wolfram Research Inc. 2024. Mathematica, V ersion 14.0

    2024

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.