Invariant Measures for Data-Driven Dynamical System Identification: Analysis and Application

Jonah Botvinick-Greenhouse

REVIEW 3 major objections 5 minor 51 references

Invariant Measures for Data-Driven Dynamical System Identification: Analysis and Application

T0 review · 3 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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

Pith's one-line read Matching the long-run statistics of a dynamical system in time-delay coordinates identifies it up to topological conjugacy, and with extra observables, uniquely on the attractor.

desk verdict Genuinely new identifiability theorems, but the abstract oversells Theorem 5.2 by omitting a matching-orbit condition that the invariant measures alone don't supply. read the letter →

arxiv 2502.05204 v1 pith:AW3WBMVG submitted 2025-01-31 math.DS cs.LGnlin.CDphysics.data-an

Jonah Botvinick-Greenhouse This is my paper · ORCID

classification math.DScs.LGnlin.CDphysics.data-an MSC 37M1037C4065P99

keywords invariantmeasuressystemidentificationFokker-PlanckequationPerron-Frobeniusoperatortime-delayembeddingTakensdata-adaptivemeshPDE-constrainedoptimization

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 argues that dynamical systems can be identified from their long-run statistics—their invariant measures—instead of from pointwise trajectories, and that this switch repairs three chronic failures of trajectory-based identification: sensitivity to noise, chaos, and slow sampling. The forward model is the stationary solution of a Fokker–Planck equation, computed as the dominant eigenvector of a finite-volume or Galerkin Markov matrix, so the fitting problem becomes a PDE-constrained optimization. The paper's central theoretical contribution is that the inverse problem becomes well-posed when the invariant measure is lifted to time-delay coordinates: equality of delay-coordinate invariant measures forces the two systems to be topologically conjugate on the support, and finitely many such measures from distinct observables, together with a single matching orbit, force them to coincide there. A companion computational contribution makes the approach practical in high dimensions by approximating the Perron–Frobenius operator on a data-adaptive unstructured mesh with Monte–Carlo integration. If the claims hold, modelers can recover attractor dynamics from noisy, sparsely observed data without resolving individual trajectories.

What carries the argument

The time-delay map $Ψ^{{(m)}}$_{(y,T)}(x) = (y(x), y(T(x)), …, y($T^{{m−1}}$(x))) and its pushforward measure ̂$μ^{{(m)}}$_{(y,T)} = $Ψ^{{(m)}}$_{(y,T)#} μ. The argument turns on the fact that an (m+1)-dimensional delay vector determines both $Ψ^{{(m)}}$(x) and $Ψ^{{(m)}}$(T(x)), so equality of the (m+1)-dimensional delay-coordinate measures lets one read off the dynamics' graph and build the conjugacy Θ = ($Ψ^{{(m)}}$_{(y,T)})^{-1} ∘ $Ψ^{{(m)}}$_{(y,S)}. On the computational side, the forward model is the unique fixed point of a column-stochastic Markov matrix M_ε = (1−ε)M + εU—the teleportation-regularized upwind finite-volume discretization of the Fokker–Planck equation—and the high-dimensional extension replaces the uniform mesh with a data-adaptive partition and a regularized Galerkin projection of the Perron–Frobenius operator approximated by Monte–Carlo integration.

What would settle it

Compute delay-coordinate invariant measures for two non-conjugate diffeomorphisms that share a state-coordinate invariant measure (for instance, torus rotations with different rotation parameters or a modified cat map composed with a non-conjugate twist), using a generic observable and embedding dimension above twice the box-counting dimension; the theorems imply the measures must differ, so equality would refute the identifiability claim.

Watch

Extended reading notes

Core claim

The discovery is that the non-uniqueness that plagues invariant-measure-based system identification disappears under a Takens-style time-delay coordinate change. There are two theorems. Theorem 5.1: if two diffeomorphisms have the same invariant measure in (m+1)-dimensional delay coordinates, with m exceeding twice the box-counting dimension of the support, then they are topologically conjugate on the support of that measure, for almost every $C^{1}$ observable. Theorem 5.2: using m delay-coordinate invariant measures from m distinct observables, plus an initial condition in the basin of the measure whose first m−1 iterates agree for the two systems, the systems coincide on the whole support. The key observation is that a single point in (m+1)-dimensional delay coordinates encodes both the current delay-coordinate state and its image under one step of the dynamics, so the invariant measure in those coordinates carries dynamical information that the state-coordinate invariant measure discards.

Load-bearing premise

The uniqueness result depends on knowing an initial condition whose future visits agree for the two systems, and that knowledge is not contained in the invariant measures being matched.

Editorial extensions

If this is right

System identification from invariant statistics becomes well-posed in delay coordinates: the recovered dynamics are determined up to topological conjugacy from a single delay-coordinate invariant measure, and uniquely on the support when several observables and a matching orbit are available.
The Eulerian approach inherits robustness to noise, chaos, and slow or irregular sampling, because occupation measures converge to the invariant measure under mild assumptions, whereas trajectory-matching objectives degrade under these conditions.
The data-adaptive mesh and Galerkin approximation of the Perron–Frobenius operator extend invariant-measure matching to high-dimensional systems with low-dimensional attractors, making it feasible to compare full Markov matrices rather than only dominant eigenvectors.
Because delay-coordinate invariant measures can be formed from scalar time series, the identifiability guarantees apply in partial-observation settings where only one or a few functions of the state are measured.

Reading between the lines

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

The orbit-matching condition in Theorem 5.2 is not obtainable from the invariant measures alone; making the uniqueness result fully data-driven would require identifying such an initial condition from the observed time series, for example by locating a distinguished orbit in the delay embedding, which is a testable extension.
The identifiability guarantee holds for almost every observable in the sense of prevalence, so a specific fixed sensor could in principle be exceptional; one could audit a given sensor by numerically checking injectivity of its delay-coordinate map on the attractor.
The natural synthesis of the paper's two halves—applying the Section 4 data-adaptive mesh and Markov-matrix matching to delay-coordinate invariant measures—is left as future work, and testing it on the 30-dimensional Lorenz-96 example would be a direct benchmark of the combined pipeline.

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.

Desk Editor's Note

read the letter

The core of this paper is worth a serious look. The delay-coordinate identifiability results in Section 5 are new as far as I know: Theorem 5.1 (equality of one delay-coordinate invariant measure implies topological conjugacy on the support) and Theorem 5.2 (several such measures plus an initial condition force equality) are not in the cited literature, and the proofs in the appendix are coherent. Theorem 4.1's convergence guarantee for the regularized Galerkin projection of the Perron–Frobenius operator is also a genuine contribution. The Lorenz-63 experiment in Figure 11 gives a nice concrete demonstration that delay-coordinate matching (loss J2) succeeds where state-coordinate matching (J1) fails.

That said, the paper oversells what is proven. The abstract says the approach can "guarantee unique system identifiability from the invariant measure alone," but Theorem 5.2 requires condition 1: there exists x* in the basin and on the support such that T^k(x*) = S^k(x*) for k=1,...,m-1. The proof uses that condition to align the conjugacies from each observable and to put the orbits on a common track. The invariant measures themselves encode no information about such a common finite orbit segment. The author does acknowledge this inside Section 5.2 ("provided that a suitable initial condition also holds"), but the abstract and the Section 5 introduction drop it. That's a real gap between the advertised claim and the theorem as stated. The theorem itself seems mathematically sound; the framing is what needs correcting.

The numerical support is promising but under-reported: no code, no seeds, no error bars, and the Lorenz-96 objective in (27) leaves open how M^(ε)(v_θ) is estimated during optimization—the mesh cells come from k-means on observed data, but the transition matrix for the parameterized flow needs its own sample base. Also, the reported success metrics in several experiments are the same objectives being minimized, so they partially validate in-sample. Those are fixable issues, not fatal flaws.

This paper deserves peer review. A good referee will want the abstract toned down, the matching-orbit condition made prominent, and the numerical protocols tightened. The theoretical results justify the referee time. I'd bring it to a reading group, though I'd pair it with a careful read of the proof of Theorem 5.2.

Referee Report

3 major / 5 minor

Summary. The paper proposes an Eulerian, invariant-measure-based approach to dynamical system identification. The forward model is the stationary solution of a Fokker-Planck equation, approximated either by an upwind finite-volume discretization (Section 3) or by a data-adaptive Galerkin projection of the Perron-Frobenius operator with Monte Carlo integration (Section 4). The paper's main theoretical contribution is in Section 5, where delay-coordinate invariant measures are shown to provide identifiability: Theorem 5.1 proves topological conjugacy on the support from equality of one delay-coordinate invariant measure, and Theorem 5.2 claims uniqueness on the support from finitely many delay-coordinate invariant measures and a suitable initial condition. Numerical experiments cover the Van der Pol oscillator, the Lorenz-63 system, a Hall-effect thruster signal, and a 30-dimensional Lorenz-96 system.

Significance. If the technical gaps identified below are repaired, the paper makes a valuable contribution: the delay-coordinate invariant measure construction is a genuinely new theoretical tool for invariant-measure-based system identification, and the m+1 dimensional delay-coordinate trick in Theorem 5.1 is elegant. The Galerkin formulation with variance-reducing data-adaptive meshes is also practically motivated and potentially scalable. The proofs are largely self-contained and the paper demonstrates the method on a diverse set of problems. However, the advertised claim of unique identifiability 'from the invariant measure alone' is stronger than Theorem 5.2 actually proves, and the claimed operator-norm convergence in Theorem 4.1 is not established by the proof. The numerical experiments are currently single-run and lack reproducibility details.

major comments (3)

[Abstract and Section 5.1, Theorem 5.2] The abstract and the introduction claim that delay-coordinate invariant measures guarantee unique identifiability 'from the invariant measure alone,' but Theorem 5.2's condition 1 requires the existence of x* in B_{μ,T} ∩ supp(μ) such that T^k(x*) = S^k(x*) for 1 ≤ k ≤ m−1. This matching-orbit condition presumes information about how T and S are aligned at a finite orbit segment and is not encoded in any of the invariant measures listed in condition 2. In the proof in Appendix A.6.2, this condition is what forces each conjugacy Θ_{y_i} to fix x*, and through it the m delay-coordinate measure equalities yield equality of the maps; without condition 1, those equalities yield only m distinct topological conjugacies. Section 5.2 accurately states that uniqueness holds 'provided that a suitable initial condition also holds,' but the abstract and the Section 5 opening paragraph do not carry this caveat. Please either soften the advertised claim or show that the orbit-alignment condition can be obtained from data rather than assumed a priori.
[Section 4.1, Theorem 4.1, and Appendix A.3] Theorem 4.1 states operator-norm convergence, ∥P^{(n,ε)} − P∥_{L^1→L^1} → 0, but the proof establishes at most strong convergence. The final displayed chain in the proof of Theorem 4.1 fixes an arbitrary f with ∥f∥=1 and shows ∥P^{(n,ε)}f − Pf∥_{L^1} → 0 for each f; it never takes a supremum over the unit ball. Pointwise strong convergence of uniformly bounded Markov operators does not imply convergence in operator norm, and the projections Q^{(n)} used in the proof typically do not converge to the identity in operator norm. As written, the claimed norm convergence is not proven. Please either supply a proof of the operator-norm statement under additional assumptions on T and μ, or restate Theorem 4.1 as strong convergence and adjust the subsequent claims accordingly.
[Section 4.3.2, Eq. (27)] The Lorenz-96 experiment is the main numerical evidence for the scalability of the unstructured-mesh method, but Eq. (27) is not fully specified. The paper does not state how M^{(ε)}(v_θ) is estimated during optimization: how many Monte-Carlo sample pairs (x, Φ^{Δt}_{v_θ}(x)) are used per objective evaluation, whether the same fixed observed samples are reused, how the flow map is integrated, or how gradients with respect to θ are computed through the soft partition-of-unity and the k-means cells. The construction of M* from the observed trajectory is also unspecified. In addition, Table 1 and Figures 5, 8, and 11 report single runs without seeds, error bars, or a code release, so the 'comprehensive numerical tests' are not reproducible and the claimed robustness to noise and slow sampling is not quantitatively established. Please provide full experimental details and at least repeated-seed statistics for the main comparisons.

minor comments (5)

[Throughout] There are several typos, including 'observerved' in Section 1, 'correspoinding' in Section 3.2, 'identificaiton' in Section 6, and 'Lebegue's' in Appendix A.3. The notation T|^k_{supp(μ)} in Appendix A.6.2 is also awkward and should be rewritten for clarity.
[Section 3.3.2] The Hall-effect thruster experiment reports no quantitative error between the simulated and observed invariant measures, and the post-training rescaling of v_θ and D is not described. Please add the rescaling procedure and a quantitative comparison, even if only the Wasserstein distance used in Table 1.
[Section 5.3, Figure 11] The loss J2 in the Lorenz-63 comparison is the sum of a state-coordinate term and a delay-coordinate term, so the experiment does not isolate the contribution of the delay-coordinate matching to the successful reconstruction. Consider also reporting results with the delay term only, or with a sweep over the relative weight of the two terms.
[Theorem 4.1 and Appendix A.3] The iterated limit 'lim_{n→∞} lim_{ε→0}' in Theorem 4.1 should be distinguished from simultaneous refinement of the discretization and regularization parameters, since the proof only treats the iterated order. Please state explicitly which mode of convergence is intended and used in later sections.
[Section 4.2] The variance-reduction analysis assumes i.i.d. samples from μ, while the numerical experiments use trajectory data. This distinction should be stated explicitly, and the paper should justify transferring the i.i.d.-based optimal partition principle to correlated trajectory samples.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the uniqueness and convergence theorems are proven from stated assumptions and external embedding results, and the numerical demonstrations are in-sample but do not carry the central claim.

full rationale

The paper's theoretical contributions are self-contained in the relevant sense. Theorem 4.1 is a convergence statement for a regularized Galerkin projection of the Perron-Frobenius operator and is proved in Appendix A.3 from the stated partition and partition-of-unity assumptions; no fitted parameter is involved. Theorems 5.1 and 5.2 derive conjugacy and uniqueness from equality of delay-coordinate invariant measures using the external fractal Takens and Whitney embedding theorems (cited as [13]) and standard measure-support arguments; the proofs in Appendix A.6 do not invoke any of the paper's optimized parameters or fitted velocities. There are no load-bearing self-citations: the author cites prior work (e.g., [2, 11]) for standard ingredients such as teleportation regularization and the Fokker-Planck surrogate, but those works are not by the present author and are not used to force the uniqueness conclusion. The only notable gap is a mismatch between the abstract's phrase "from the invariant measure alone" and Theorem 5.2, which additionally requires Condition 1 (a matching finite orbit segment) and m observables; the paper itself notes "provided that a suitable initial condition also holds" in Section 5.2. This is an overstatement of the proven theorem, not a circular reduction, because the theorem's extra hypothesis is an additional assumption rather than a definition or a fitted value. Numerical experiments (Table 1, Figure 11) evaluate divergences closely related to the training objective, so they are partially in-sample as evidence of effectiveness; however, the central identifiability claim does not rest on those experiments, and no equation of the paper reduces a predicted quantity to its own input. Therefore no significant circularity is present.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new physical entities. The delay-coordinate map and the regularized partition of unity are mathematical constructions, not postulated entities. The main free parameters are discretization and regularization hyperparameters chosen by hand; the theoretical results rest on standard embedding theorems and domain assumptions about the test systems.

free parameters (5)

Diffusion coefficient D = 10 (Lorenz-63), 0.001 (Van der Pol), rescaled after training (HET)
Chosen by hand in the forward Fokker-Planck surrogate; it affects the stationary measure and hence the inverted velocity.
Teleportation parameter ε = 5 (Lorenz-96)
Regularization in the Markov matrix M_ε; chosen to stabilize optimization, and results depend on it.
Number of mesh cells n = 200 (Lorenz-96), 400 (cat map)
Discretization resolution; trade-off between computational cost and approximation accuracy.
Mesh spacing Δx = 2 (Lorenz-63)
Finite-volume grid resolution; coarse due to computational limits, which introduces numerical diffusion.
Embedding dimension m and delay τ = m=3, τ=0.23 (HET)
Chosen via embedding heuristics; affects the delay-coordinate invariant measure and the identifiability guarantees.

assumptions (6)

standard math Fractal Takens' Embedding Theorem (Theorem 2.1, from [13]) including Assumption A.1 on periodic points
Used in the proofs of Theorems 5.1 and 5.2 to ensure injectivity of the delay maps on compact attractors.
standard math Prevalence theory [28] and the property that finite intersections of prevalent sets are prevalent
Used to claim the results hold for almost every observable y in C1(U,R) and almost every vector-valued Y in C1(U,Rm).
domain assumption Existence and uniqueness of physical invariant measures for the test systems (Lorenz-63, Lorenz-96, Van der Pol)
The optimization compares simulated and observed occupation measures; for chaotic systems this requires a unique physical measure, e.g., Lorenz-63 [42].
domain assumption The observed samples in Section 4.2 are i.i.d. from the invariant measure μ
The variance-reduction analysis (Proposition 4.1) uses i.i.d. samples, whereas real trajectory data are time-correlated; the paper assumes this for the theory.
ad hoc to paper Existence of an initial condition x* in B_{μ,T} ∩ supp(μ) with T^k(x*) = S^k(x*) for 1 ≤ k ≤ m−1 (Theorem 5.2 condition)
This matching-orbit condition is required for the uniqueness conclusion and is not guaranteed by the invariant measure data alone.
domain assumption The systems T and S are C1 diffeomorphisms on an open set U with compact invariant support
Required by Takens' theorem and the Whitney embedding results used in the proofs.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Invariant Measures for Data-Driven Dynamical System Identification: Analysis and Application." pith.science (2026). https://pith.science/paper/AW3WBMVG

@misc{pith2026250205204,
  author       = {Pith},
  title        = {Pith review of: Invariant Measures for Data-Driven Dynamical System Identification: Analysis and Application},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AW3WBMVG}},
  note         = {Machine review of arXiv:2502.05204}
}

read the original abstract

We propose a novel approach for performing dynamical system identification, based upon the comparison of simulated and observed physical invariant measures. While standard methods adopt a Lagrangian perspective by directly treating time-trajectories as inference data, we take on an Eulerian perspective and instead seek models fitting the observed global time-invariant statistics. With this change in perspective, we gain robustness against pervasive challenges in system identification including noise, chaos, and slow sampling. In the first half of this paper, we pose the system identification task as a partial differential equation (PDE) constrained optimization problem, in which synthetic stationary solutions of the Fokker-Planck equation, obtained as fixed points of a finite-volume discretization, are compared to physical invariant measures extracted from observed trajectory data. In the latter half of the paper, we improve upon this approach in two crucial directions. First, we develop a Galerkin-inspired modification to the finite-volume surrogate model, based on data-adaptive unstructured meshes and Monte-Carlo integration, enabling the approach to efficiently scale to high-dimensional problems. Second, we leverage Takens' seminal time-delay embedding theory to introduce a critical data-dependent coordinate transformation which can guarantee unique system identifiability from the invariant measure alone. This contribution resolves a major challenge of system identification through invariant measures, as systems exhibiting distinct transient behaviors may still share the same time-invariant statistics in their state-coordinates. Throughout, we present comprehensive numerical tests which highlight the effectiveness of our approach on a variety of challenging system identification tasks.

Figures

Figures reproduced from arXiv: 2502.05204 by the authors.

**Figure 2.** Flowchart describing the paper’s main sections and techniques. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Delay-coordinate invariant measures improve identifiability of the torus rotation [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: As the mesh size of the forward model discretization is refined, we visually observe the [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: Reconstructing the ˙x component of the Lorenz-63 system from its observed, noisy occupation measure. We used a mesh-spacing of ∆x = 2 and a diffusion coefficient of D = 10. In [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

**Figure 6.** Figure 6: Reconstructing the velocity from the embedded Cathode-Pearson signal’s invariant mea [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

**Figure 7.** Figure 7: Comparison between the uniform and unstructured mesh approaches for approximating [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]

**Figure 8.** Figure 8: Reconstructing the first component of the Lorenz-96 system’s velocity ˙x [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]

**Figure 9.** Figure 9: Flowchart of our main results. While the invariant measure [PITH_FULL_IMAGE:figures/full_fig_p021_9.png]

**Figure 10.** Figure 10: Visualization of the state-coordinate (top row) and delay-coordinate (bottom row) [PITH_FULL_IMAGE:figures/full_fig_p022_10.png]

**Figure 11.** Figure 11: While the loss J1, based only on the state-coordinate invariant measure, is insufficient for reconstructing the dynamics (bottom left), the loss J2, which enforces equality of the delaycoordinate invariant measures, yields an accurate reconstruction (bottom right). B…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

51 extracted references · 45 canonical work pages

[1]

Parameterized neural ordinary differential equations: Applications to computational physics problems

Kookjin Lee and Eric J Parish. Parameterized neural ordinary differential equations: Applications to computational physics problems. Proceedings of the Royal Society A , 477(2253):20210162, 2021

work page 2021
[2]

A data-driven approach to model calibration for nonlinear dynamical systems

CM Greve, K Hara, RS Martin, DQ Eckhardt, and JW Koo. A data-driven approach to model calibration for nonlinear dynamical systems. Journal of Applied physics , 125(24):244901, 2019

work page 2019
[3]

Parameter estimation with gravitational waves

Nelson Christensen and Renate Meyer. Parameter estimation with gravitational waves. Rev. Mod. Phys., 94:025001, Apr 2022

work page 2022
[4]

Machine learning in weather prediction and climate analyses—applications and perspectives

Bogdan Bochenek and Zbigniew Ustrnul. Machine learning in weather prediction and climate analyses—applications and perspectives. Atmosphere, 13(2):180, 2022

2022
[5]

Data-driven modeling and learning in science and engineering

Francisco J Mont´ ans, Francisco Chinesta, Rafael G´ omez-Bombarelli, and J Nathan Kutz. Data-driven modeling and learning in science and engineering. Comptes Rendus M´ ecanique, 347(11):845–855, 2019

work page 2019
[6]

Fitting ordinary differential equations to chaotic data

Ellen Baake, Michael Baake, HG Bock, and KM Briggs. Fitting ordinary differential equations to chaotic data. Physical Review A , 45(8):5524, 1992

1992
[7]

Incremental single shooting—a robust method for the estimation of parameters in dynamical systems

Claas Michalik, Ralf Hannemann, and Wolfgang Marquardt. Incremental single shooting—a robust method for the estimation of parameters in dynamical systems. Computers & Chemical Engineering, 33(7):1298–1305, 2009

work page 2009
[8]

Discovering governing equations from data by sparse identification of nonlinear dynamical systems

Steven L Brunton, Joshua L Proctor, and J Nathan Kutz. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences, 113(15):3932–3937, 2016

2016

Show all 51 references

[9]

Stabilized neural ordinary differential equations for long-time forecasting of dynamical systems

Alec J Linot, Joshua W Burby, Qi Tang, Prasanna Balaprakash, Michael D Graham, and Romit Maulik. Stabilized neural ordinary differential equations for long-time forecasting of dynamical systems. Journal of Computational Physics , 474:111838, 2023

2023
[10]

Parameter estimation in ordinary differential equations for biochemical processes using the method of multiple shooting

Martin Peifer and Jens Timmer. Parameter estimation in ordinary differential equations for biochemical processes using the method of multiple shooting. IET systems biology, 1(2):78–88, 2007. 26

2007
[11]

Optimal transport for parameter identification of chaotic dynamics via invariant measures

Yunan Yang, Levon Nurbekyan, Elisa Negrini, Robert Martin, and Mirjeta Pasha. Optimal transport for parameter identification of chaotic dynamics via invariant measures. SIAM Journal on Applied Dynamical Systems , 22(1):269–310, 2023

2023
[12]

Training neural operators to preserve invariant measures of chaotic attractors

Ruoxi Jiang, Peter Y Lu, Elena Orlova, and Rebecca Willett. Training neural operators to preserve invariant measures of chaotic attractors. Advances in Neural Information Processing Systems, 36, 2024

2024
[13]

Embedology

Tim Sauer, James A Yorke, and Martin Casdagli. Embedology. Journal of statistical Physics , 65:579–616, 1991

1991
[14]

The dimension of chaotic attractors

J Doyne Farmer, Edward Ott, and James A Yorke. The dimension of chaotic attractors. Physica D: Nonlinear Phenomena , 7(1-3):153–180, 1983

1983
[15]

Numerical approximation of the frobenius–perron operator using the finite volume method

Richard A Norton, Colin Fox, and Malcolm E Morrison. Numerical approximation of the frobenius–perron operator using the finite volume method. SIAM journal on numerical anal- ysis, 56(1):570–589, 2018

2018
[16]

On the numerical approximation of the perron-frobenius and koopman operator, 2016

Stefan Klus, P´ eter Koltai, and Christof Sch¨ utte. On the numerical approximation of the perron-frobenius and koopman operator, 2016

2016
[17]

Finite approximation for the frobenius-perron operator

Tien-Yien Li. Finite approximation for the frobenius-perron operator. a solution to ulam’s conjecture. Journal of Approximation theory , 17(2):177–186, 1976

1976
[18]

Extensive chaos in the lorenz-96 model

Alireza Karimi and Mark R Paul. Extensive chaos in the lorenz-96 model. Chaos: An inter- disciplinary journal of nonlinear science , 20(4), 2010

2010
[19]

Detecting strange attractors in turbulence

Floris Takens. Detecting strange attractors in turbulence. In Dynamical Systems and Turbu- lence, Warwick 1980 , pages 366–381. Springer, 1981

1980
[20]

Chaos, fractals, and noise: stochastic aspects of dynamics, volume 97

Andrzej Lasota and Michael C Mackey. Chaos, fractals, and noise: stochastic aspects of dynamics, volume 97. Springer Science & Business Media, 2013

2013
[21]

On the numerical approximation of the perron-frobenius and koopman operator

Stefan Klus, P´ eter Koltai, and Christof Sch¨ utte. On the numerical approximation of the perron-frobenius and koopman operator. arXiv preprint arXiv:1512.05997 , 2015

2015 arXiv
[22]

What are SRB measures, and which dynamical systems have them? Journal of statistical physics , 108:733–754, 2002

Lai-Sang Young. What are SRB measures, and which dynamical systems have them? Journal of statistical physics , 108:733–754, 2002

2002
[23]

The Lorenz attractor is mixing

Stefano Luzzatto, Ian Melbourne, and Frederic Paccaut. The Lorenz attractor is mixing. Communications in Mathematical Physics , 260(2):393–401, 2005

2005
[24]

Flow prediction using dynamic mode decomposition with time-delay embedding based on local measurement.Physics of Fluids , 33(9), 2021

Yuan Yuan, Kaiwen Zhou, Wenwu Zhou, Xin Wen, and Yingzheng Liu. Flow prediction using dynamic mode decomposition with time-delay embedding based on local measurement.Physics of Fluids , 33(9), 2021

2021
[25]

Untangling brain-wide dynamics in consciousness by cross-embedding

Satohiro Tajima, Toru Yanagawa, Naotaka Fujii, and Taro Toyoizumi. Untangling brain-wide dynamics in consciousness by cross-embedding. PLoS computational biology, 11(11):e1004537, 2015

2015
[26]

Deep learning delay coordinate dynamics for chaotic attractors from partial observable data

Charles D Young and Michael D Graham. Deep learning delay coordinate dynamics for chaotic attractors from partial observable data. Physical Review E , 107(3):034215, 2023. 27

2023
[27]

Practical method for determining the minimum embedding dimension of a scalar time series

Liangyue Cao. Practical method for determining the minimum embedding dimension of a scalar time series. Physica D: Nonlinear Phenomena , 110(1-2):43–50, 1997

1997
[28]

almost ev- ery

Brian R Hunt, Tim Sauer, and James A Yorke. Prevalence: a translation-invariant “almost ev- ery” on infinite-dimensional spaces. Bulletin of the American mathematical society , 27(2):217– 238, 1992

1992
[29]

Detecting stochastic governing laws with observa- tion on stationary distributions

Xiaoli Chen, Hui Wang, and Jinqiao Duan. Detecting stochastic governing laws with observa- tion on stationary distributions. Physica D: Nonlinear Phenomena , 448:133691, 2023

2023
[30]

Comparison of systems with complex behavior

Igor Mezi´ c and Andrzej Banaszuk. Comparison of systems with complex behavior. Physica D: Nonlinear Phenomena , 197(1-2):101–133, 2004

2004
[31]

Parker, Stephan Hoyer, Volodymyr Kuleshov, Fei Sha, and Leonardo Zepeda-N´ u˜ nez

Yair Schiff, Zhong Yi Wan, Jeffrey B. Parker, Stephan Hoyer, Volodymyr Kuleshov, Fei Sha, and Leonardo Zepeda-N´ u˜ nez. DySLIM: Dynamics stable learning by invariant measure for chaotic systems. In Forty-first International Conference on Machine Learning , 2024

2024
[32]

Controlling the statistical properties of expanding maps

Stefano Galatolo and Mark Pollicott. Controlling the statistical properties of expanding maps. Nonlinearity, 30(7):2737, 2017

2017
[33]

A simple framework to justify linear response theory

Martin Hairer and Andrew J Majda. A simple framework to justify linear response theory. Nonlinearity, 23(4):909, 2010

2010
[34]

When are dynamical sys- tems learned from time series data statistically accurate? In The Thirty-eighth Annual Con- ference on Neural Information Processing Systems , 2024

Jeongjin Park, Nicole Tianjiao Yang, and Nisha Chandramoorthy. When are dynamical sys- tems learned from time series data statistically accurate? In The Thirty-eighth Annual Con- ference on Neural Information Processing Systems , 2024

2024
[35]

Steady states of Fokker–Planck equations: I

Wen Huang, Min Ji, Zhenxin Liu, and Yingfei Yi. Steady states of Fokker–Planck equations: I. existence. Journal of Dynamics and Differential Equations , 27(3):721–742, 2015

2015
[36]

Altan Allawala and J. B. Marston. Statistics of the stochastically forced Lorenz attractor by the Fokker-Planck equation and cumulant expansions. Physical Review E , 94(5), nov 2016

2016
[37]

Efficient grid-based bayesian estimation of nonlinear low-dimensional systems with sparse non-gaussian pdfs

Thomas R Bewley and Atul S Sharma. Efficient grid-based bayesian estimation of nonlinear low-dimensional systems with sparse non-gaussian pdfs. Automatica, 48(7):1286–1290, 2012

2012
[38]

Finite volume methods for hyperbolic problems, volume 31

Randall J LeVeque et al. Finite volume methods for hyperbolic problems, volume 31. Cambridge University Press, 2002

2002
[39]

Pagerank beyond the web

David F Gleich. Pagerank beyond the web. SIAM Review, 57(3):321–363, 2015

2015
[40]

Guckenheimer

J. Guckenheimer. Dynamics of the Van der Pol equation. IEEE Transactions on Circuits and Systems, 27(11):983–989, 1980

1980
[41]

Kingma and Jimmy Ba

Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In ICLR (Poster), 2015

2015
[42]

The Lorenz attractor exists

Warwick Tucker. The Lorenz attractor exists. Comptes Rendus de l’Acad´ emie des Sciences - Series I - Mathematics , 328(12):1197–1202, 1999

1999
[43]

Spatiotem- poral data fusion and manifold reconstruction in hall thrusters

Daniel Eckhardt, Justin Koo, Robert Martin, Michael Holmes, and Kentaro Hara. Spatiotem- poral data fusion and manifold reconstruction in hall thrusters. Plasma Sources Science and Technology, 28(4):045005, 2019. 28

2019
[44]

Optimal partition choice for invariant measure approximation for one-dimensional maps

Rua Murray. Optimal partition choice for invariant measure approximation for one-dimensional maps. Nonlinearity, 17(5):1623, 2004

2004
[45]

Constrained k-means clustering

Paul S Bradley, Kristin P Bennett, and Ayhan Demiriz. Constrained k-means clustering. Microsoft Research, Redmond, 20(0):0, 2000

2000
[46]

Caflisch

Russel E. Caflisch. Monte carlo and quasi-monte carlo methods. Acta Numerica, 7:1–49, 1998

1998
[47]

Interpolating between Optimal Transport and MMD using Sinkhorn Diver- gences

Jean Feydy, Thibault S´ ejourn´ e, Fran¸ cois-Xavier Vialard, Shun’ichi Amari, Alain Trouve, and Gabriel Peyr´ e. Interpolating between Optimal Transport and MMD using Sinkhorn Diver- gences. In The 22nd International Conference on Artificial Intelligence and Statistics , page...

2019
[48]

Introduction to metric and topological spaces

Wilson A Sutherland. Introduction to metric and topological spaces . Oxford University Press, 2009

2009
[49]

Lebesgue almost everywhere

Hassler Whitney. Analytic extensions of differentiable functions defined in closed sets. Hassler Whitney Collected Papers , pages 228–254, 1992. A Appendix A.1 Choice of Objective Function Here we summarize possible choices for D, which is used as a metric or divergence on the...

1992
[50]

The set Ap ⊆ A of p-periodic points satisfies boxdim( Ap) < p/2
[51]

The linearization DT p of each periodic orbit has distinct eigenvalues. We remark that when the diffeomorphism T is given by the time- τ flow map of a Lipschitz continuous vector field, one can choose τ sufficiently small such that Assumption A.1 is satisfied; see [13]. The ge...

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [1]

Parameterized neural ordinary differential equations: Applications to computational physics problems

Kookjin Lee and Eric J Parish. Parameterized neural ordinary differential equations: Applications to computational physics problems. Proceedings of the Royal Society A , 477(2253):20210162, 2021

work page 2021

[2] [2]

A data-driven approach to model calibration for nonlinear dynamical systems

CM Greve, K Hara, RS Martin, DQ Eckhardt, and JW Koo. A data-driven approach to model calibration for nonlinear dynamical systems. Journal of Applied physics , 125(24):244901, 2019

work page 2019

[3] [3]

Parameter estimation with gravitational waves

Nelson Christensen and Renate Meyer. Parameter estimation with gravitational waves. Rev. Mod. Phys., 94:025001, Apr 2022

work page 2022

[4] [4]

Machine learning in weather prediction and climate analyses—applications and perspectives

Bogdan Bochenek and Zbigniew Ustrnul. Machine learning in weather prediction and climate analyses—applications and perspectives. Atmosphere, 13(2):180, 2022

2022

[5] [5]

Data-driven modeling and learning in science and engineering

Francisco J Mont´ ans, Francisco Chinesta, Rafael G´ omez-Bombarelli, and J Nathan Kutz. Data-driven modeling and learning in science and engineering. Comptes Rendus M´ ecanique, 347(11):845–855, 2019

work page 2019

[6] [6]

Fitting ordinary differential equations to chaotic data

Ellen Baake, Michael Baake, HG Bock, and KM Briggs. Fitting ordinary differential equations to chaotic data. Physical Review A , 45(8):5524, 1992

1992

[7] [7]

Incremental single shooting—a robust method for the estimation of parameters in dynamical systems

Claas Michalik, Ralf Hannemann, and Wolfgang Marquardt. Incremental single shooting—a robust method for the estimation of parameters in dynamical systems. Computers & Chemical Engineering, 33(7):1298–1305, 2009

work page 2009

[8] [8]

Discovering governing equations from data by sparse identification of nonlinear dynamical systems

Steven L Brunton, Joshua L Proctor, and J Nathan Kutz. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences, 113(15):3932–3937, 2016

2016