pith. sign in

arxiv: 2606.13260 · v1 · pith:BNRY2RIJnew · submitted 2026-06-11 · 💻 cs.LG · q-bio.NC

Extracting Governing Equations from Latent Dynamics via Multi-View Contrastive Learning

Paolo Muratore , Mackenzie Weygandt Mathis This is my paper

Pith reviewed 2026-06-27 07:28 UTC · model grok-4.3

classification 💻 cs.LG q-bio.NC
keywords multi-view contrastive learninglatent dynamical systemsgoverning equationssystem identificationidentifiabilitynoisy observationssymbolic regression
0
0 comments X

The pith

DYSCO uses multi-view contrastive learning to recover latent trajectories and governing equations from noisy observations up to affine indeterminacy.

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

DYSCO is a multi-view temporal contrastive learning method that recovers both latent trajectories and the equations governing their dynamics from high-dimensional noisy data. It does so by using several independent noisy observations of the same process to separate the true signal from noise. The dynamics are expressed in a basis that permits recovering the equations symbolically. The work provides theoretical guarantees that the identification is strong, aside from an affine indeterminacy, even when observations are nonlinear and noisy. This enables extracting scientific models from data like neural recordings where noise is common.

Core claim

The central claim is that DYSCO recovers latent trajectories and governing dynamics from noisy high-dimensional measurements by leveraging multiple independent noisy views to disentangle signal from noise, with dynamics parameterized in a structured functional basis to enable symbolic recovery within an affine gauge, supported by theoretical guarantees for strong identification up to affine indeterminacy in noisy nonlinear settings.

What carries the argument

Multi-view temporal contrastive learning that disentangles signal from noise across independent views, combined with structured functional basis parameterization of the dynamics.

If this is right

  • Accurate recovery of latent trajectories and flow fields in chaotic, oscillatory, and metastable dynamical regimes.
  • Successful performance under Gaussian and Poisson observation noise.
  • Symbolic recovery of the governing equations.
  • Strong identification up to an affine indeterminacy.

Where Pith is reading between the lines

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

  • This approach may generalize to other multi-modal data sources beyond the tested regimes.
  • It could inspire similar contrastive methods for identifiability in related fields like causal discovery.
  • Testing on real-world datasets with unknown ground truth would be a natural next step to validate practical utility.

Load-bearing premise

Multiple independent noisy views of the same underlying dynamical process are available and can be leveraged to disentangle signal from noise.

What would settle it

Demonstrating that the method does not achieve the claimed identification on synthetic data generated from known dynamical systems with multiple independent views under the described noise conditions would falsify the result.

Figures

Figures reproduced from arXiv: 2606.13260 by Mackenzie Weygandt Mathis, Paolo Muratore.

Figure 1
Figure 1. Figure 1: Graphical overview. (a) We consider a latent dynamical system (top) that evolves accord￾ing to an unknown dynamics f : X 7→ X . We observe the system via a non-linear mixing-channel g : R d 7→ R D that produces noisy high-dimensional observations y a t . (b) We solve the identifiabil￾ity problem by simultaneously learning an encoder h : R D 7→ R d and a symbolic dynamics ˆf by minimizing the multi-view tem… view at source ↗
Figure 2
Figure 2. Figure 2: Dynamical system recovery via multi-view contrastive learning. Example phase space portraits of different dynamical systems (top-row) with corresponding affine-aligned recovered sys￾tems (bottom row) for either d = 3 (left) or d = 2 (right). Dynamical system trajectories are color-coded based on the external forcing magnitude ∥ ut ∥. All reported systems are observed via a Poisson observation channel [PIT… view at source ↗
Figure 3
Figure 3. Figure 3: Ablation results. (a) Results for impact of available number of views on the final trajec￾tory (left) and flow-field (right) model performances for the Lorenz dynamical system (observed via a Gaussian channel with standard deviation σ = 1) and for the two noise condition annealed (cyan) and quenched (plum). Markers are averages over 3 repetitions and error bars are the standard error of the mean. (b) Impac… view at source ↗
Figure 4
Figure 4. Figure 4: Ablation on time horizon κ. Ablation result for varying time integration horizons κ = 2, 4, 6, 8, 10 on the Lorenz system with Gaussian noise of intensity σ = 0.3. We trained for 200 epochs and evaluated the model on the validation set every 10 epochs (left column). We report the maximum achieved score on the right column for both the R2 metric (top row) and dynR2 metric (bottom row). Each marker is the av… view at source ↗
read the original abstract

Identifying latent dynamical systems from noisy, high-dimensional measurements is a central problem at the intersection of representation learning, system identification, and scientific discovery. We present DYSCO, a multi-view temporal contrastive learning algorithm that jointly recovers latent trajectories and the governing dynamics from such observations, by leveraging multiple independent noisy views of the same underlying process to disentangle signal from noise. By parameterizing the dynamics in a structured functional basis, our framework further enables symbolic recovery of the governing equations within an affine gauge. We offer theoretical guarantees for strong identification up to an affine indeterminacy, extending prior identifiability results to the realistic setting of noisy nonlinear observations. Empirically, we demonstrate accurate recovery of both latent trajectories and flow fields across a diverse set of dynamical regimes (e.g., chaotic, oscillatory, and metastable) under both Gaussian and Poisson observation noise, the latter being particularly relevant for neural recordings.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The manuscript introduces DYSCO, a multi-view temporal contrastive learning algorithm that jointly recovers latent trajectories and the governing dynamics from noisy high-dimensional observations by leveraging multiple independent noisy views of the same underlying process. The dynamics are parameterized in a structured functional basis to enable symbolic recovery of the governing equations within an affine gauge. The central claims are theoretical guarantees for strong identification up to an affine indeterminacy (extending prior identifiability results to noisy nonlinear observations) together with empirical demonstrations of accurate recovery of latent trajectories and flow fields on chaotic, oscillatory, and metastable regimes under both Gaussian and Poisson noise.

Significance. If the stated theoretical guarantees can be rigorously established, the work would represent a meaningful advance at the intersection of representation learning and system identification. The multi-view contrastive strategy for disentangling signal from noise, combined with the ability to recover symbolic equations, could have practical value for scientific discovery tasks such as analysis of neural recordings. The extension of identifiability results to realistic noisy settings is a substantive contribution if the proofs hold without hidden circularity or overly restrictive assumptions.

minor comments (1)
  1. The abstract refers to 'strong identification up to an affine indeterminacy' and 'parameterizing the dynamics in a structured functional basis'; the full manuscript should explicitly state the precise form of the basis and the exact nature of the affine gauge to allow readers to assess the scope of the symbolic recovery claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and for highlighting the potential significance of DYSCO at the intersection of representation learning and system identification. We note that the provided report lists no specific major comments under the MAJOR COMMENTS section, so we have no individual points to address point-by-point. We remain available to supply additional details on the theoretical guarantees or any other aspect of the work.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The abstract presents DYSCO as a multi-view temporal contrastive learning method that recovers latent trajectories and governing dynamics from noisy observations, with theoretical guarantees for strong identification up to affine indeterminacy by leveraging multiple independent views and a structured functional basis. No derivation steps, equations, or citations are shown that reduce any claimed prediction or result to fitted inputs by construction, self-definitional loops, or load-bearing self-citations. The central claims rest on the multi-view contrastive setup and identifiability extension, which are described as independent of the target outputs. This matches the default case of a self-contained paper with no exhibited circular reductions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on abstract; full paper may introduce additional parameters or assumptions not visible here.

axioms (1)
  • domain assumption Multiple independent noisy views of the same underlying process exist and can be used to disentangle signal from noise
    Central to the contrastive learning step described in the abstract

pith-pipeline@v0.9.1-grok · 5684 in / 1085 out tokens · 18273 ms · 2026-06-27T07:28:42.042019+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

45 extracted references · 15 canonical work pages · 1 internal anchor

  1. [1]

    Self-Supervised Contrastive Learning Performs Non-Linear System Identification , booktitle =

    Laiz, Rodrigo Gonzalez and Schmidt, Tobias and Schneider, Steffen , year = 2025, pages =. Self-Supervised Contrastive Learning Performs Non-Linear System Identification , booktitle =

    2025

  2. [2]

    Mathis, M. W. and Mathis, A. , title =. Nature Reviews Neuroscience , volume =. 2026 , doi =

    2026

  3. [3]

    Proceedings of the National Academy of Sciences , volume =

    Discovering Governing Equations from Data by Sparse Identification of Nonlinear Dynamical Systems , author =. Proceedings of the National Academy of Sciences , volume =. doi:10.1073/pnas.1517384113 , urldate =

    work page doi:10.1073/pnas.1517384113

  4. [4]

    Disentangling

    H. Disentangling. doi:10.48550/arXiv.2106.09620 , urldate =. arXiv , keywords =:2106.09620 , primaryclass =

    work page doi:10.48550/arxiv.2106.09620

  5. [5]

    and Sharma, Yash and Schneider, Steffen and Bethge, Matthias and Brendel, Wieland , year = 2022, month = apr, number =

    Zimmermann, Roland S. and Sharma, Yash and Schneider, Steffen and Bethge, Matthias and Brendel, Wieland , year = 2022, month = apr, number =. Contrastive. doi:10.48550/arXiv.2102.08850 , urldate =. arXiv , keywords =:2102.08850 , primaryclass =

    work page doi:10.48550/arxiv.2102.08850 2022

  6. [6]

    Unsupervised

    Hyvarinen, Aapo and Morioka, Hiroshi , year = 2016, volume =. Unsupervised. Advances in

    2016

  7. [7]

    Nature Methods , volume =

    Inferring Single-Trial Neural Population Dynamics Using Sequential Auto-Encoders , author =. Nature Methods , volume =. doi:10.1038/s41592-018-0109-9 , urldate =

    work page doi:10.1038/s41592-018-0109-9

  8. [8]

    The annals of mathematical statistics , pages=

    Transformations related to the angular and the square root , author=. The annals of mathematical statistics , pages=. 1950 , publisher=

    1950

  9. [9]

    2024 , url =

    Keller Jordan and Yuchen Jin and Vlado Boza and You Jiacheng and Franz Cesista and Laker Newhouse and Jeremy Bernstein , title =. 2024 , url =

    2024

  10. [10]

    arXiv preprint arXiv:1412.6980 , year=

    Adam: A method for stochastic optimization , author=. arXiv preprint arXiv:1412.6980 , year=

    Pith/arXiv arXiv

  11. [11]

    arXiv preprint arXiv:1711.05101 , year=

    Decoupled weight decay regularization , author=. arXiv preprint arXiv:1711.05101 , year=

    Pith/arXiv arXiv

  12. [12]

    Chen, Ricky T. Q. , title=. 2018 , url=

    2018

  13. [13]

    arXiv preprint arXiv:1807.03748 , year=

    Representation learning with contrastive predictive coding , author=. arXiv preprint arXiv:1807.03748 , year=

    Pith/arXiv arXiv

  14. [14]

    Nature , volume =

    Learnable Latent Embeddings for Joint Behavioural and Neural Analysis , author =. Nature , volume =. doi:10.1038/s41586-023-06031-6 , urldate =

    work page doi:10.1038/s41586-023-06031-6

  15. [15]

    arXiv , keywords =:2310.05573 , publisher =

    arXiv

  16. [16]

    Proceedings of the National Academy of Sciences , volume =

    Data-Driven Discovery of Coordinates and Governing Equations , author =. Proceedings of the National Academy of Sciences , volume =. doi:10.1073/pnas.1906995116 , urldate =

    work page doi:10.1073/pnas.1906995116

  17. [17]

    Nonlinear

    Hyvarinen, Aapo and Sasaki, Hiroaki and Turner, Richard , year = 2019, month = apr, pages =. Nonlinear. Proceedings of the

    2019

  18. [18]

    Patterns , volume =

    Nonlinear Independent Component Analysis for Principled Disentanglement in Unsupervised Deep Learning , author =. Patterns , volume =. doi:10.1016/j.patter.2023.100844 , urldate =

    work page doi:10.1016/j.patter.2023.100844 2023

  19. [19]

    and Sharma, Yash and Schneider, Steffen and Bethge, Matthias and Brendel, Wieland , year = 2021, month = jul, pages =

    Zimmermann, Roland S. and Sharma, Yash and Schneider, Steffen and Bethge, Matthias and Brendel, Wieland , year = 2021, month = jul, pages =. Contrastive. Proceedings of the 38th

    2021

  20. [20]

    Indeterminacy in

    Xi, Quanhan and. Indeterminacy in. Proceedings of

  21. [21]

    and Krasa, Benyamin Abramovich and Kunz, Erin and Hahn, Nick and Kamdar, Foram and Avansino, Donald and Hochberg, Leigh R

    Shah, Nishal P. and Krasa, Benyamin Abramovich and Kunz, Erin and Hahn, Nick and Kamdar, Foram and Avansino, Donald and Hochberg, Leigh R. and Henderson, Jaimie M. and Sussillo, David , year = 2025, month = oct, primaryclass =. Improved Interpretability in. doi:10.1101/2025.10.03.680303 , urldate =

    work page doi:10.1101/2025.10.03.680303 2025

  22. [22]

    Physical Review Research , volume =

    Statistical Mechanics of Dynamical System Identification , author =. Physical Review Research , volume =. doi:10.1103/4d98-tdlp , urldate =

    work page doi:10.1103/4d98-tdlp

  23. [23]

    Independent

    Morioka, Hiroshi and H. Independent. Proceedings of

  24. [24]

    and Luo, Thomas Z

    Kim, Timothy D. and Luo, Thomas Z. and Pillow, Jonathan W. and Brody, Carlos D. , year = 2021, month = jul, pages =. Inferring. Proceedings of the 38th

    2021

  25. [25]

    arXiv preprint arXiv:2110.05428 , year=

    Learning temporally causal latent processes from general temporal data , author=. arXiv preprint arXiv:2110.05428 , year=

    arXiv

  26. [26]

    arXiv preprint arXiv:2505.24776 , year=

    Diffusion-Based Symbolic Regression , author=. arXiv preprint arXiv:2505.24776 , year=

    arXiv

  27. [27]

    Artificial intelligence and statistics , pages=

    Bayesian learning and inference in recurrent switching linear dynamical systems , author=. Artificial intelligence and statistics , pages=. 2017 , organization=

    2017

  28. [28]

    Neural computation , volume=

    Variational learning for switching state-space models , author=. Neural computation , volume=. 2000 , publisher=

    2000

  29. [29]

    International conference on machine learning , pages=

    Learning interpretable continuous-time models of latent stochastic dynamical systems , author=. International conference on machine learning , pages=. 2019 , organization=

    2019

  30. [30]

    Discovery of the Kalman filter as a practical tool for aerospace and industry , author=

  31. [31]

    Advances in neural information processing systems , volume=

    Neural ordinary differential equations , author=. Advances in neural information processing systems , volume=

  32. [32]

    International conference on machine learning , pages=

    Data-efficient image recognition with contrastive predictive coding , author=. International conference on machine learning , pages=. 2020 , organization=

    2020

  33. [33]

    arXiv preprint arXiv:1904.05862 , year=

    wav2vec: Unsupervised pre-training for speech recognition , author=. arXiv preprint arXiv:1904.05862 , year=

    arXiv 1904

  34. [34]

    Scientific reports , volume=

    Estimating the intrinsic dimension of datasets by a minimal neighborhood information , author=. Scientific reports , volume=. 2017 , publisher=

    2017

  35. [35]

    Advances in neural information processing systems , volume=

    Maximum likelihood estimation of intrinsic dimension , author=. Advances in neural information processing systems , volume=

  36. [36]

    (No Title) , year=

    Vision: A computational investigation into the human representation and processing of visual information , author=. (No Title) , year=

  37. [37]

    and Ostrow, Mitchell and Zoltowski, David M

    Versteeg, Christopher and McCart, Jonathan D. and Ostrow, Mitchell and Zoltowski, David M. and Washington, Clayton B. and Driscoll, Laura and Codol, Olivier and Michaels, Jonathan A. and Linderman, Scott W. and Sussillo, David and Pandarinath, Chethan , year = 2025, month = feb, primaryclass =. Computation-through-. doi:10.1101/2025.02.07.637062 , urldate =

    work page doi:10.1101/2025.02.07.637062 2025

  38. [38]

    and Sussillo, David and Shenoy, Krishna V

    Vyas, Saurabh and Golub, Matthew D. and Sussillo, David and Shenoy, Krishna V. , year = 2020, month = jul, journal =. Computation. doi:10.1146/annurev-neuro-092619-094115 , urldate =

    work page doi:10.1146/annurev-neuro-092619-094115 2020

  39. [39]

    Opening the

    Sussillo, David and Barak, Omri , year = 2013, month = mar, journal =. Opening the. doi:10.1162/NECO_a_00409 , urldate =

    work page doi:10.1162/neco_a_00409 2013

  40. [40]

    Neuron , volume=

    Linking connectivity, dynamics, and computations in low-rank recurrent neural networks , author=. Neuron , volume=. 2018 , publisher=

    2018

  41. [41]

    Advances in Neural Information Processing Systems , volume=

    Prune and distill: similar reformatting of image information along rat visual cortex and deep neural networks , author=. Advances in Neural Information Processing Systems , volume=

  42. [42]

    doi:10.48550/arXiv.1908.09979 , urldate =

    Yang, Huanrui and Wen, Wei and Li, Hai , year = 2020, month = jan, number =. doi:10.48550/arXiv.1908.09979 , urldate =. arXiv , keywords =:1908.09979 , primaryclass =

    work page doi:10.48550/arxiv.1908.09979 2020

  43. [43]

    Learning Sparse Neural Networks through $L_0$ Regularization

    Louizos, Christos and Welling, Max and Kingma, Diederik P. , year = 2018, month = jun, number =. Learning. doi:10.48550/arXiv.1712.01312 , urldate =. arXiv , keywords =:1712.01312 , primaryclass =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1712.01312 2018

  44. [44]

    Differentiable

    Kolb, Chris and Frost, Laetitia and Bischl, Bernd and R. Differentiable. doi:10.48550/arXiv.2509.23898 , urldate =. arXiv , keywords =:2509.23898 , primaryclass =

    work page doi:10.48550/arxiv.2509.23898

  45. [45]

    International conference on machine learning , pages=

    Understanding contrastive representation learning through alignment and uniformity on the hypersphere , author=. International conference on machine learning , pages=. 2020 , organization=

    2020