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arxiv: 2605.16724 · v1 · pith:2UUCGQROnew · submitted 2026-05-16 · 🧬 q-bio.QM

Discovering interpretable low-dimensional dynamics using maximum entropy

Michael C. Chung , Tarran Mohan , Purushottam D. Dixit , Juan Guan This is my paper

Pith reviewed 2026-05-19 20:01 UTC · model grok-4.3

classification 🧬 q-bio.QM
keywords maximum entropydimensionality reductionsymbolic dynamicslatent spacemodel discoveryinterpretable modelsbiological time seriesdata compression
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The pith

Edwin compresses high-dimensional observations into low-dimensional symbolic dynamics using maximum entropy.

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

The paper introduces Edwin, a framework that reduces high-dimensional data to a latent space via the dynamic maximum entropy principle while simultaneously finding sparse symbolic equations that govern the dynamics there and link them to external metadata. It is demonstrated on simulated cases such as diffusion, Ornstein-Uhlenbeck processes, particle self-assembly, neural spiking, and recurrent networks, plus noisy experimental data on RNA-liposome aggregation. The resulting models remain physically interpretable and continue to predict behavior under conditions absent from the training data. This matters because it supplies a single procedure that turns complex measurements into understandable, testable rules without sacrificing either accuracy or clarity.

Core claim

Edwin simultaneously performs dimensionality reduction using the dynamic maximum entropy (DME) principle and discovers sparse symbolic models governing latent dynamics, as well as the coupling between learned features and external metadata. Across all tested systems it recovers low-dimensional symbolic models that are physically interpretable and generalize to unseen conditions.

What carries the argument

Dynamic maximum entropy (DME) compression that produces a latent space in which sparse symbolic dynamics and their coupling to metadata can be discovered.

If this is right

  • Low-dimensional symbolic governing equations can be extracted directly from high-dimensional time series in both simulated and experimental settings.
  • The extracted models remain accurate when external conditions change after training.
  • Coupling between latent coordinates and observable metadata appears in the same symbolic form as the dynamics.
  • The same procedure handles stochastic diffusion, neural population activity, and noisy aggregation experiments.

Where Pith is reading between the lines

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

  • The method may allow researchers to derive mechanistic insight from large-scale recordings without first choosing a specific reduced-order basis by hand.
  • If the latent symbolic equations preserve conservation laws or other invariants, they could support long-term forecasting beyond the observed time window.
  • Extending the framework to spatial fields or networked systems would test whether the same compression principle scales to higher structural complexity.

Load-bearing premise

High-dimensional observations can be compressed by the dynamic maximum entropy principle into a latent space whose dynamics admit sparse symbolic descriptions that incorporate external metadata.

What would settle it

On a held-out external condition the symbolic model recovered by Edwin produces trajectories in the latent space that deviate substantially from the measured evolution of the original high-dimensional observations.

Figures

Figures reproduced from arXiv: 2605.16724 by Juan Guan, Michael C. Chung, Purushottam D. Dixit, Tarran Mohan.

Figure 1
Figure 1. Figure 1: Algorithmic overview of Edwin. Edwin takes as input nonnegative, normalized high-dimensional data, 𝒑𝒑𝒊𝒊𝒊𝒊, and libraries of model terms, 𝜣𝜣𝒁𝒁 and 𝜣𝜣𝒀𝒀. Here 𝐱𝐱 denotes external feature metadata associated with each index 𝒊𝒊 (e.g., spatial coordinates for a spatiotemporal system, or cluster size for a self-assembly system). Using Dynamic Maximum Entropy (DME) for compression and sparse regression for model … view at source ↗
Figure 2
Figure 2. Figure 2: Performance of the SLIC-Sparse Regression algorithm (Algorithm 1 in Appendix 8.1) in correctly recovering the governing equations of several nonlinear dynamical systems from data. The y-axis of each subplot is the accuracy (in the sense of a classification task) of the recovered model; 𝟎𝟎 means that none of the correct terms are identified (no true positives or negatives) and 𝟏𝟏 means the model is perfectl… view at source ↗
Figure 3
Figure 3. Figure 3: The diverse systems studied in this work, which [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Reconstruction accuracy across seven methods at the latent dimension [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Reconstruction loss 𝓛𝓛𝑲𝑲 𝑲𝑲 as a function of latent dimension 𝑲𝑲 for all methods. Factorization-based methods (el￾POD, NMF, Dyn-NMF) require substantially larger 𝑲𝑲 to approach the same reconstruction quality, and ultimately fall below the noise floor at high 𝑲𝑲, indicating overfitting to noise in 𝒑𝒑𝒊𝒊𝒊𝒊 rather than genuine compression. AE-based methods (AE, Koop-AE, SINDy-AE) saturate at the same noise fl… view at source ↗
Figure 6
Figure 6. Figure 6: Latent variables and inferred models for [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Robustness of inferred latent models to random [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Effect of mild over-specification of latent dimensionality (𝑲𝑲 + 𝟏𝟏). Across systems, the additional coordinate primarily introduces redundant or weakly active directions rather than qualitatively new dynamics. For the Brownian and self-assembly systems, the extra latent becomes strongly correlated with existing coordinates. In the neuronal system, the three-dimensional trajectories clearly exhibit attract… view at source ↗
Figure 9
Figure 9. Figure 9: Generalization of discovered dynamics to unseen conditions via limited adaptation. (A) Example predictions for each [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Recovery of intrinsic low-rank dynamics in recurrent networks. (A) Recurrent dynamics 𝒙𝒙̇ 𝒊𝒊 = −𝒙𝒙𝒊𝒊 + ∑𝒋𝒋 𝑾𝑾𝒊𝒊 𝝋𝝋(𝒙𝒙𝒋𝒋) with 𝑾𝑾 = 𝟏𝟏 𝑵𝑵 𝑼𝑼𝑼𝑼⊤ + 𝜺𝜺 √𝑵𝑵 𝑮𝑮, where 𝑼𝑼 ∈ ℝ𝑵𝑵× defines an -dimensional structured subspace and 𝑮𝑮𝒊𝒊 ∼ 𝓝𝓝(𝟎𝟎, 𝟏𝟏) adds full-rank noise. The intrinsic coordinates are 𝜿𝜿( ) = 𝟏𝟏 𝑵𝑵 𝑼𝑼⊤𝝋𝝋(𝒙𝒙( )). (B) log𝟏𝟏 KLD as a function of candidate latent dimension 𝑲𝑲 (x-axis) for ground-truth ran… view at source ↗
Figure 11
Figure 11. Figure 11: Learned dynamics of experimental RNA-nanoparticle system. (A) Sweep of KLDs. (B) Learned models for time- and feature-dependent latents. (C) Data versus reconstruction. Top row is a log-log plot of the data and k-mer index. Bottom row is a heatmap for distribution over time. (D) Inferred power-law model accurately predicts several moments of the distribution, which were not constrained by the DME procedur… view at source ↗
Figure 12
Figure 12. Figure 12: Flow schematic for Algorithm [alg:main]. Inputs: 𝐏𝐏, 𝛩𝛩𝑍𝑍, 𝛩𝛩𝑌𝑌, 𝐾𝐾, 𝜂𝜂, 𝑚𝑚 , 𝑀𝑀 Outputs: 𝐙𝐙, 𝐘𝐘, 𝛯𝛯𝑍𝑍, 𝛯𝛯𝑌𝑌 𝑖𝑖 ← 1 𝐙𝐙 ← 𝐙𝐙 − 𝜂𝜂 𝜕𝜕ℒ 𝜕𝜕𝐙𝐙 𝐘𝐘 ← 𝐘𝐘 − 𝜂𝜂 𝜕𝜕ℒ 𝜕𝜕𝐘𝐘 𝛯𝛯𝑍𝑍 ← 𝑆𝑆 𝑆𝑆 (𝑍𝑍,𝛩𝛩𝑍𝑍) 𝛯𝛯𝑌𝑌 ← 𝑆𝑆 𝑆𝑆 (𝑌𝑌,𝛩𝛩𝑌𝑌) 𝛯𝛯𝑍𝑍 ← 𝛩𝛩𝑍𝑍 † (𝐙𝐙)𝐙𝐙̇ 𝛯𝛯𝑌𝑌 ← 𝛩𝛩𝑌𝑌 † (𝐱𝐱)𝐘𝐘 𝑖𝑖 ← 𝑖𝑖 + 1 In Algorithm 2, 𝜕𝜕ℒ/𝜕𝜕𝐙𝐙 and 𝜕𝜕ℒ/𝜕𝜕𝐘𝐘 denote partial derivatives of the loss function with respect to 𝐙𝐙 and 𝐘𝐘, respectively. We compute them dire… view at source ↗
Figure 13
Figure 13. Figure 13: Runtime scaling of the primary algorithm [PITH_FULL_IMAGE:figures/full_fig_p020_13.png] view at source ↗
read the original abstract

Models (i.e., governing equations) are fundamental to science and engineering. Advances in data acquisition now make it possible to extract interpretable, low dimensional descriptions from high dimensional observations. However, existing approaches sacrifice either interpretability for reconstruction accuracy or infer symbolic dynamics without relating latent coordinates to physically meaningful observables. Here we present Edwin (maximum entropy driven compression with interpretable nonlinear model discovery), a unified framework that simultaneously performs dimensionality reduction using the dynamic maximum entropy (DME) principle and discovers sparse symbolic models governing latent dynamics, as well as the coupling between learned features and external metadata. We validate Edwin on diverse simulated systems, including stochastic diffusion, the Ornstein-Uhlenbeck process, self assembling particles, spiking neural populations, and low rank recurrent neural networks, as well as on a noisy experimental time series of aggregating RNA-liposome complexes. Across all systems, Edwin recovers low dimensional symbolic models that are physically interpretable and generalize to unseen conditions. Together, these results establish Edwin as a powerful framework for inferring interpretable, low dimensional dynamics directly from high dimensional data.

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

2 major / 2 minor

Summary. The manuscript introduces Edwin, a unified framework that performs dimensionality reduction of high-dimensional observations via the dynamic maximum entropy (DME) principle while simultaneously discovering sparse symbolic governing equations for the latent dynamics and their coupling to external metadata. Validation is reported across simulated systems with known ground truth (stochastic diffusion, Ornstein-Uhlenbeck process, self-assembling particles, spiking neural populations, low-rank recurrent neural networks) and one noisy experimental time series of aggregating RNA-liposome complexes, with the central claim that the method recovers physically interpretable low-dimensional models that generalize to unseen conditions.

Significance. If the claims hold after addressing identifiability concerns, the work could offer a useful bridge between maximum-entropy compression and symbolic regression for extracting interpretable dynamics from complex data in biophysics and neuroscience. The multi-system validation including ground-truth cases is a constructive element that provides external grounding beyond pure self-consistency.

major comments (2)
  1. [Abstract and §3] Abstract and §3 (DME compression): the central claim requires that DME produces a latent space whose dynamics admit sparse symbolic descriptions forced by the principle. However, maxent distributions are fixed only once constraint functions and Lagrange multipliers are chosen; if these are selected or optimized jointly with the symbolic regression step, multiple latent representations are possible. The manuscript should provide an identifiability analysis, uniqueness result, or ablation showing that alternative constraint sets yield equivalent symbolic forms. This is load-bearing for the claim that recovered equations reflect intrinsic structure rather than library choice or regularization.
  2. [Results on experimental RNA-liposome series] Results on experimental RNA-liposome series: the generalization claims for the noisy experimental dataset lack reported error bars, data exclusion criteria, or quantitative fitting procedures. Without these, it is difficult to evaluate whether the sparse symbolic models are robust or post-hoc. This directly affects the strength of the 'across all systems' claim.
minor comments (2)
  1. [Figures] Figure captions and legends should explicitly state the number of independent runs, random seeds, and any hyperparameter values used for the symbolic discovery step to improve reproducibility.
  2. [Notation] The notation distinguishing latent coordinates, metadata coupling terms, and Lagrange multipliers could be summarized in a table for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment of our work and for the constructive comments, which have helped us improve the manuscript. We address each major comment point by point below, indicating revisions made where appropriate.

read point-by-point responses

  1. Referee: [Abstract and §3] Abstract and §3 (DME compression): the central claim requires that DME produces a latent space whose dynamics admit sparse symbolic descriptions forced by the principle. However, maxent distributions are fixed only once constraint functions and Lagrange multipliers are chosen; if these are selected or optimized jointly with the symbolic regression step, multiple latent representations are possible. The manuscript should provide an identifiability analysis, uniqueness result, or ablation showing that alternative constraint sets yield equivalent symbolic forms. This is load-bearing for the claim that recovered equations reflect intrinsic structure rather than library choice or regularization.

    Authors: We thank the referee for raising this important identifiability concern, which bears directly on the interpretability of the recovered models. In the Edwin framework, the DME step first determines the latent coordinates by maximizing entropy subject to a fixed set of dynamic and metadata constraints derived from the data; symbolic regression is then applied to these coordinates without further joint optimization of the constraints. While we acknowledge that alternative constraint choices could in principle produce different latent spaces, the combination of the maximum-entropy objective and the sparsity-promoting symbolic regression favors parsimonious, physically consistent equations. To address the referee's request, we have added a new ablation study (Supplementary Note 4 and Figure S5) that systematically varies the constraint set (e.g., number of moments and metadata couplings) and shows that the recovered symbolic forms are equivalent up to reparameterization across these choices. This supports that the equations reflect intrinsic structure. revision: yes

  2. Referee: [Results on experimental RNA-liposome series] Results on experimental RNA-liposome series: the generalization claims for the noisy experimental dataset lack reported error bars, data exclusion criteria, or quantitative fitting procedures. Without these, it is difficult to evaluate whether the sparse symbolic models are robust or post-hoc. This directly affects the strength of the 'across all systems' claim.

    Authors: We agree that additional quantitative details on the experimental validation are needed to strengthen the robustness claims. In the revised manuscript we have added error bars to all generalization metrics (computed across five random train-test splits of the time series), specified the data exclusion criteria (time points with SNR < 5 were removed prior to analysis), and provided a full description of the quantitative fitting procedure, including the exact loss function, regularization schedule, and convergence criteria used for symbolic regression. These additions appear in §5.3 and the associated supplementary text. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via external validation

full rationale

The paper introduces Edwin as a unified framework that applies the dynamic maximum entropy principle for dimensionality reduction while simultaneously discovering sparse symbolic models for latent dynamics and metadata coupling. Validation occurs on multiple simulated systems with known ground-truth dynamics (stochastic diffusion, Ornstein-Uhlenbeck, self-assembling particles, spiking networks, low-rank RNNs) plus one experimental RNA-liposome time series, with explicit claims of generalization to unseen conditions. No equations or steps in the provided description reduce a claimed prediction or uniqueness result to a fitted parameter or self-citation by construction. The central claims rest on empirical recovery and interpretability across independent test systems rather than internal redefinition or load-bearing self-reference.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of the dynamic maximum entropy principle to latent dynamics and the existence of sparse symbolic representations; these are domain assumptions rather than derived results.

free parameters (1)
  • sparsity or regularization parameter in symbolic discovery
    Symbolic model discovery typically requires a tunable parameter to control model complexity and prevent overfitting.
axioms (1)
  • domain assumption High-dimensional observations arise from low-dimensional latent dynamics governed by sparse symbolic equations that couple to external metadata
    This premise enables both the DME compression step and the subsequent symbolic recovery.

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Reference graph

Works this paper leans on

90 extracted references · 90 canonical work pages · 5 internal anchors

  1. [1]

    Single cells make big data: New challenges and opportunities in transcriptomics

    P. Angerer, L. Simon, S. Tritschler, F.A. Wolf, D. Fischer, F.J. Theis, "Single cells make big data: New challenges and opportunities in transcriptomics", Current Opinion in Systems Biology 4, 85 –91, (2017)

    work page 2017

  2. [2]

    Eleven grand challenges in single-cell data science

    D. Lähnemann, J. Köster, E. Szczurek, D.J. McCarthy, S.C. Hicks, M.D. Robinson, et al., "Eleven grand challenges in single-cell data science", Genome Biology 21(1), 31, (2020)

    work page 2020

  3. [3]

    The evolution of Big Data in neuroscience and neurology

    L. Dipietro, P. Gonzalez-Mego, C. Ramos-Estebanez, L.H. Zukowski, R. Mikkilineni, R.J. Rushmore, et al., "The evolution of Big Data in neuroscience and neurology", Journal of Big Data 10(1), 116, (2023)

    work page 2023

  4. [4]

    Neuroscience: Big brain, big data

    E. Landhuis, "Neuroscience: Big brain, big data", Nature 541(7638), 559–561, (2017)

    work page 2017

  5. [5]

    Big data and machine learning for materials science

    J.F. Rodrigues, L. Florea, M.C.F. de Oliveira, D. Diamond, O.N. Oliveira, "Big data and machine learning for materials science", Discover Materials 1(1), 12, (2021)

    work page 2021

  6. [6]

    The low -rank hypothesis of complex systems

    V. Thibeault, A. Allard, P. Desrosiers, "The low -rank hypothesis of complex systems", Nature Physics 20(2), 294–302, (2024)

    work page 2024

  7. [7]

    Distilling Free-Form Natural Laws from Experimental Data

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

    work page 2009

  8. [8]

    Interpretable Machine Learning for Science with PySR and SymbolicRegression.jl

    M. Cranmer, "Interpretable Machine Learning for Science with PySR and SymbolicRegression.jl", arXiv(arXiv:2305.01582), (2023)

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  9. [9]

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

    B.K. Petersen, M. Landajuela, T.N. Mundhenk, C.P. Santiago, S.K. Kim, J.T. Kim, "Deep symbolic regression: Recovering mathematical expressions from data via risk -seeking policy gradients", arXiv(arXiv:1912.04871), (2021)

    work page arXiv 1912

  10. [10]

    End-to-end symbolic regression with transformers

    P.A. Kamienny, S. d’Ascoli, G. Lample, F. Charton, "End-to-end symbolic regression with transformers", arXiv(arXiv:2204.10532), (2022)

    work page arXiv 2022

  11. [11]

    Extrapolation and learning equations

    G. Martius, C.H. Lampert, "Extrapolation and learning equations", arXiv(arXiv:1610.02995), (2016)

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  12. [12]

    AI Feynman: A physics-inspired method for symbolic regression

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

    work page 2020

  13. [13]

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

    S.L. Brunton, J.L. Proctor, J.N. 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)

    work page 2016

  14. [14]

    Ensemble-SINDy: Robust sparse model discovery in the low -data, high -noise limit, with active learning and control

    U. Fasel, J.N. Kutz, B.W. Brunton, S.L. Brunton, "Ensemble-SINDy: Robust sparse model discovery in the low -data, high -noise limit, with active learning and control", Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 478(2260), 20210904, (2022)

    work page 2022

  15. [15]

    Weak SINDy: Galerkin-Based Data -Driven Model Selection

    D.A. Messenger, D.M. Bortz, "Weak SINDy: Galerkin-Based Data -Driven Model Selection", Multiscale Modeling \& Simulation 19(3), 1474 – 1497, (2021)

    work page 2021

  16. [16]

    Robust learning from noisy, incomplete, high-dimensional experimental data via physically constrained symbolic regression

    P.A.K. Reinbold, L.M. Kageorge, M.F. Schatz, R.O. Grigoriev, "Robust learning from noisy, incomplete, high-dimensional experimental data via physically constrained symbolic regression", Nature Communications 12(1), 3219, (2021)

    work page 2021

  17. [17]

    Sparse nonlinear models of chaotic electroconvection

    Y. Guan, S.L. Brunton, I. Novosselov, "Sparse nonlinear models of chaotic electroconvection", Royal Society Open Science 8(8), 202367, (2021)

    work page 2021

  18. [18]

    Constrained sparse Galerkin regression

    J.C. Loiseau, S.L. Brunton, "Constrained sparse Galerkin regression", Journal of Fluid Mechanics 838, 42–67, (2018)

    work page 2018

  19. [19]

    Physics -constrained, low -dimensional models for magnetohydrodynamics: First -principles and data -driven approaches

    A.A. Kaptanoglu, K.D. Morgan, C.J. Hansen, S.L. Brunton, "Physics -constrained, low -dimensional models for magnetohydrodynamics: First -principles and data -driven approaches", Physical Review E 104(1), 015206, (2021)

    work page 2021

  20. [20]

    Discovering governing equations from partial measurements with deep delay autoencoders

    J. Bakarji, K. Champion, J. Nathan Kutz, S.L. Brunton, "Discovering governing equations from partial measurements with deep delay autoencoders", Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 479(2276), 20230422, (2023)

    work page 2023

  21. [21]

    Data -driven discovery of intrinsic dynamics

    D. Floryan, M.D. Graham, "Data -driven discovery of intrinsic dynamics", Nature Machine Intelligence 4(12), 1113–1120, (2022)

    work page 2022

  22. [22]

    Data-driven discovery of coordinates and governing equations

    K. Champion, B. Lusch, J.N. Kutz, S.L. Brunton, "Data-driven discovery of coordinates and governing equations", Proceedings of the National Academy of Sciences 116(45), 22445–22451, (2019)

    work page 2019

  23. [23]

    Minimum Kullback entropy approach to the Fokker -Planck equation

    A. R. Plastino, H.G. Miller, A. Plastino, "Minimum Kullback entropy approach to the Fokker -Planck equation", Physical Review E 56(4), 3927– 3934, (1997)

    work page 1997

  24. [24]

    Dynamic maximum entropy provides accurate approximation of structured population dynamics

    K. Bod’ová, E. Szép, N.H. Barton, "Dynamic maximum entropy provides accurate approximation of structured population dynamics", PLOS Computational Biology 17(12), e1009661, (2021)

    work page 2021

  25. [25]

    Principles of maximum entropy and maximum caliber in statistical physics

    S. Pressé, K. Ghosh, J. Lee, K.A. Dill, "Principles of maximum entropy and maximum caliber in statistical physics", Reviews of Modern Physics 85(3), 1115– 1141, (2013)

    work page 2013

  26. [26]

    The Maximum Caliber Variational Principle for Nonequilibria

    K. Ghosh, P.D. Dixit, L. Agozzino, K.A. Dill, "The Maximum Caliber Variational Principle for Nonequilibria", Annual Review of Physical Chemistry 71, 213–238, (2020)

    work page 2020

  27. [27]

    Perspective: Maximum caliber is a general variational principle for dynamical systems

    P.D. Dixit, J. Wagoner, C. Weistuch, S. Pressé, K. Ghosh, K.A. Dill, "Perspective: Maximum caliber is a general variational principle for dynamical systems", The Journal of Chemical Physics 148(1), 010901, (2018)

    work page 2018

  28. [28]

    Thermodynamic inference of data manifolds

    P.D. Dixit, "Thermodynamic inference of data manifolds", Physical Review Research 2(2), 023201, (2020)

    work page 2020

  29. [29]

    SiGMoiD: A super - statistical generative model for binary data

    X. Zhao, G. Plata, P.D. Dixit, "SiGMoiD: A super - statistical generative model for binary data", PLOS Computational Biology 17(8), e1009275, (2021)

    work page 2021

  30. [30]

    An introduction to the maximum entropy approach and its application to inference problems in biology

    A. De Martino, D. De Martino, "An introduction to the maximum entropy approach and its application to inference problems in biology", Heliyon 4(4), e00596, (2018)

    work page 2018

  31. [31]

    Inferring spatial source of disease outbreaks using maximum entropy

    M. Ansari, D. Soriano -Paños, G. Ghoshal, A.D. White, "Inferring spatial source of disease outbreaks using maximum entropy", Physical Review E 106(1), 014306, (2022)

    work page 2022

  32. [32]

    A maximum entropy framework for nonexponential distributions

    J. Peterson, P.D. Dixit, K.A. Dill, "A maximum entropy framework for nonexponential distributions", Proceedings of the National Academy of Sciences 110(51), 20380–20385, (2013)

    work page 2013

  33. [33]

    A General Approximation for the Dynamics of Quantitative Traits

    K. Bod’ová, G. Tkačik, N.H. Barton, "A General Approximation for the Dynamics of Quantitative Traits", Genetics 202(4), 1523–1548, (2016)

    work page 2016

  34. [34]

    Maximum entropy models for antibody diversity

    T. Mora, A.M. Walczak, W. Bialek, C.G. Callan, "Maximum entropy models for antibody diversity", Proceedings of the National Academy of Sciences 107(12), 5405–5410, (2010)

    work page 2010

  35. [35]

    Statistical mechanics for natural flocks of birds

    W. Bialek, A. Cavagna, I. Giardina, T. Mora, E. Silvestri, M. Viale, et al., "Statistical mechanics for natural flocks of birds", Proceedings of the National Academy of Sciences 109(13), 4786–4791, (2012)

    work page 2012

  36. [36]

    Searching for Collective Behavior in a Large Network of Sensory Neurons

    G. Tkačik, O. Marre, D. Amodei, E. Schneidman, W. Bialek, M.J.B. Ii, "Searching for Collective Behavior in a Large Network of Sensory Neurons", PLOS Computational Biology 10(1), e1003408, (2014)

    work page 2014

  37. [37]

    Latent Dynamical Variables Produce Signatures of Spatiotemporal Criticality in Large Biological Systems

    M.C. Morrell, A.J. Sederberg, I. Nemenman, "Latent Dynamical Variables Produce Signatures of Spatiotemporal Criticality in Large Biological Systems", Physical Review Letters 126(11), 118302, (2021)

    work page 2021

  38. [38]

    EMBED: Essential MicroBiomE Dynamics, a dimensionality reduction approach for longitudinal microbiome studies

    M. Shahin, B. Ji, P.D. Dixit, "EMBED: Essential MicroBiomE Dynamics, a dimensionality reduction approach for longitudinal microbiome studies", npj Systems Biology and Applications 9(1), 26, (2023)

    work page 2023

  39. [39]

    The Maximum Entropy Principle in Cosmic Ray Transport Theory

    P. Hick, G. Stevens, J. van Rooijen, "The Maximum Entropy Principle in Cosmic Ray Transport Theory", The Sun and the Heliosphere in Three Dimensions, 355–358, (1986)

    work page 1986

  40. [40]

    The information bottleneck method

    N. Tishby, F.C. Pereira, W. Bialek, "The information bottleneck method", arXiv(arXiv:physics/0004057), (2000)

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  41. [41]

    A Generalized Information Bottleneck Theory of Deep Learning

    C. Westphal, S. Hailes, M. Musolesi, "A Generalized Information Bottleneck Theory of Deep Learning", arXiv(arXiv:2509.26327), (2026)

    work page arXiv 2026

  42. [42]

    Past -future information bottleneck in dynamical systems

    F. Creutzig, A. Globerson, N. Tishby, "Past -future information bottleneck in dynamical systems", Physical Review E 79(4), 041925, (2009)

    work page 2009

  43. [43]

    Information theory for data-driven model reduction in physics and biology

    M.S. Schmitt, M. Koch- Janusz, M. Fruchart, D.S. Seara, M. Rust, V. Vitelli, "Information theory for data-driven model reduction in physics and biology", bioRxiv, (2024)

    work page 2024

  44. [44]

    The Elements of Statistical Learning

    T. Hastie, R. Tibshirani, J. Friedman, "The Elements of Statistical Learning", Springer, (2009)

    work page 2009

  45. [45]

    Model -order selection: a review of information criterion rules

    P. Stoica, Y. Selen, "Model -order selection: a review of information criterion rules", IEEE Signal Processing Magazine 21(4), 36–47, (2004)

    work page 2004

  46. [46]

    On the Convergence of the SINDy Algorithm

    L. Zhang, H. Schaeffer, "On the Convergence of the SINDy Algorithm", arXiv(arXiv:1805.06445), (2018)

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  47. [47]

    Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations

    F. Locatello, S. Bauer, M. Lucic, G. Raetsch, S. Gelly, B. Schölkopf, et al., "Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations", Proceedings of the 36th International Conference on Machine Learning, 4114–4124, (2019)

    work page 2019

  48. [48]

    A Bayesian machine scientist to aid in the solution of challenging scientific problems

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

    work page 2020

  49. [49]

    Stochastic thermodynamics, fluctuation theorems, and molecular machines

    U. Seifert, "Stochastic thermodynamics, fluctuation theorems, and molecular machines", Reports on Progress in Physics 75(12), 126001, (2012)

    work page 2012

  50. [50]

    A tutorial on kernel density estimation and recent advances

    Y.C. Chen, "A tutorial on kernel density estimation and recent advances", Biostatistics \ & Epidemiology 1(1), 161–187, (2017)

    work page 2017

  51. [51]

    The role of interparticle and external forces in nanoparticle assembly

    Y. Min, M. Akbulut, K. Kristiansen, Y. Golan, J. Israelachvili, "The role of interparticle and external forces in nanoparticle assembly", Nature Materials 7(7), 527–538, (2008)

    work page 2008

  52. [52]

    Engineering precision nanoparticles for drug delivery

    M.J. Mitchell, M.M. Billingsley, R.M. Haley, M.E. Wechsler, N.A. Peppas, R. Langer, "Engineering precision nanoparticles for drug delivery", Nature Reviews Drug Discovery 20(2), 101–124, (2021)

    work page 2021

  53. [53]

    RNA aggregates harness the danger response for potent cancer immunotherapy

    H.R. Mendez- Gomez, A. DeVries, P. Castillo, C.v. Roemeling, S. Qdaisat, B.D. Stover, et al., "RNA aggregates harness the danger response for potent cancer immunotherapy", Cell 187(10), 2521 – 2535.e21, (2024)

    work page 2024

  54. [54]

    Multi - Step Assembly of an RNA -Liposome Nanoparticle Formulation Revealed by Real -Time, Single -Particle Quantitative Imaging

    M.C. Chung, H.R. Mendez -Gomez, D. Soni, R. McGinley, A. Zacharia, J. Ashbrook, et al., "Multi - Step Assembly of an RNA -Liposome Nanoparticle Formulation Revealed by Real -Time, Single -Particle Quantitative Imaging", Advanced Science 12(12), 2414305, (2025)

    work page 2025

  55. [55]

    Chapter 6 - Modelling of aggregation processes

    M. Elimelech, J. Gregory, X. Jia, R.A. Williams, "Chapter 6 - Modelling of aggregation processes", Particle Deposition \& Aggregation, 157–202, (1995)

    work page 1995

  56. [56]

    A unifying perspective on neural manifolds and circuits for cognition

    C. Langdon, M. Genkin, T.A. Engel, "A unifying perspective on neural manifolds and circuits for cognition", Nature Reviews Neuroscience 24(6), 363– 377, (2023)

    work page 2023

  57. [57]

    A neural manifold view of the brain

    M.G. Perich, D. Narain, J.A. Gallego, "A neural manifold view of the brain", Nature Neuroscience 28(8), 1582–1597, (2025)

    work page 2025

  58. [58]

    Attractor and integrator networks in the brain

    M. Khona, I.R. Fiete, "Attractor and integrator networks in the brain", Nature Reviews Neuroscience 23(12), 744–766, (2022)

    work page 2022

  59. [59]

    Motor cortex activity across movement speeds is predicted by network -level strategies for generating muscle activity

    S. Saxena, A.A. Russo, J. Cunningham, M.M. Churchland, "Motor cortex activity across movement speeds is predicted by network -level strategies for generating muscle activity", eLife 11, e67620, (2022)

    work page 2022

  60. [60]

    Motor Cortex Embeds Muscle -like Commands in an Untangled Population Response

    A.A. Russo, S.R. Bittner, S.M. Perkins, J.S. Seely, B.M. London, A.H. Lara, et al., "Motor Cortex Embeds Muscle -like Commands in an Untangled Population Response", Neuron 97(4), 953 –966.e8, (2018)

    work page 2018

  61. [61]

    Neural population dynamics during reaching

    M.M. Churchland, J.P. Cunningham, M.T. Kaufman, J.D. Foster, P. Nuyujukian, S.I. Ryu, et al., "Neural population dynamics during reaching", Nature 487(7405), 51–56, (2012)

    work page 2012

  62. [62]

    Neural dynamics and circuit mechanisms of decision -making

    X.J. Wang, "Neural dynamics and circuit mechanisms of decision -making", Current Opinion in Neurobiology 22(6), 1039–1046, (2012)

    work page 2012

  63. [63]

    A Recurrent Network Mechanism of Time Integration in Perceptual Decisions

    K.F. Wong, X.J. Wang, "A Recurrent Network Mechanism of Time Integration in Perceptual Decisions", Journal of Neuroscience 26(4), 1314 – 1328, (2006)

    work page 2006

  64. [64]

    Probabilistic Decision Making by Slow Reverberation in Cortical Circuits

    X.J. Wang, "Probabilistic Decision Making by Slow Reverberation in Cortical Circuits", Neuron 36(5), 955–968, (2002)

    work page 2002

  65. [65]

    The dynamics and geometry of choice in the premotor cortex

    M. Genkin, K.V. Shenoy, C. Chandrasekaran, T.A. Engel, "The dynamics and geometry of choice in the premotor cortex", Nature, 1–9, (2025)

    work page 2025

  66. [66]

    A State -Space Approach to Dynamic Nonnegative Matrix Factorization

    N. Mohammadiha, P. Smaragdis, G. Panahandeh, S. Doclo, "A State -Space Approach to Dynamic Nonnegative Matrix Factorization", IEEE Transactions on Signal Processing 63(4), 949 –959, (2015)

    work page 2015

  67. [67]

    Data -Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control

    S.L. Brunton, J.N. Kutz, "Data -Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control", Cambridge University Press, (2019)

    work page 2019

  68. [68]

    The Why and How of Nonnegative Matrix Factorization

    N. Gillis, "The Why and How of Nonnegative Matrix Factorization", arXiv(arXiv:1401.5226), (2014)

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  69. [69]

    Deep learning for universal linear embeddings of nonlinear dynamics

    B. Lusch, J.N. Kutz, S.L. Brunton, "Deep learning for universal linear embeddings of nonlinear dynamics", Nature Communications 9(1), 4950, (2018)

    work page 2018

  70. [70]

    Approximation capabilities of multilayer feedforward networks

    K. Hornik, "Approximation capabilities of multilayer feedforward networks", Neural Networks 4(2), 251 – 257, (1991)

    work page 1991

  71. [71]

    Definitions, methods, and applications in interpretable machine learning

    W.J. Murdoch, C. Singh, K. Kumbier, R. Abbasi -Asl, B. Yu, "Definitions, methods, and applications in interpretable machine learning", Proceedings of the National Academy of Sciences 116(44), 22071 – 22080, (2019)

    work page 2019

  72. [72]

    Models with higher effective dimensions tend to produce more uncertain estimates

    A. Puy, P. Beneventano, S.A. Levin, S. Lo Piano, T. Portaluri, A. Saltelli, "Models with higher effective dimensions tend to produce more uncertain estimates", Science Advances 8(42), eabn9450, (2022)

    work page 2022

  73. [73]

    Learning the intrinsic dynamics of spatio-temporal processes through Latent Dynamics Networks

    F. Regazzoni, S. Pagani, M. Salvador, L. Dede’, A. Quarteroni, "Learning the intrinsic dynamics of spatio-temporal processes through Latent Dynamics Networks", Nature Communications 15(1), 1834, (2024)

    work page 2024

  74. [74]

    Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering

    S.H. Strogatz, "Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering", CRC Press, (2018)

    work page 2018

  75. [75]

    Time series forecasting for nonlinear and non -stationary processes: a review and comparative study

    C. Cheng, A. Sa -Ngasoongsong, O. Beyca, T. Le, H. Yang, Z. Kong, et al., "Time series forecasting for nonlinear and non -stationary processes: a review and comparative study", IIE Transactions 47(10), 1053– 1071, (2015)

    work page 2015

  76. [76]

    Linking Connectivity, Dynamics, and Computations in Low-Rank Recurrent Neural Networks

    F. Mastrogiuseppe, S. Ostojic, "Linking Connectivity, Dynamics, and Computations in Low-Rank Recurrent Neural Networks", Neuron 99(3), 609 –623.e29, (2018)

    work page 2018

  77. [77]

    Drei Vortrage uber Diffusion, Brownsche Bewegung und Koagulation von Kolloidteilchen

    M.V. Smoluchowski, "Drei Vortrage uber Diffusion, Brownsche Bewegung und Koagulation von Kolloidteilchen", Zeitschrift fur Physik 17, 557 –585, (1916)

    work page 1916

  78. [78]

    Structure and kinetics of lipid – nucleic acid complexes

    N. Dan, D. Danino, "Structure and kinetics of lipid – nucleic acid complexes", Advances in Colloid and Interface Science 205, 230–239, (2014)

    work page 2014

  79. [79]

    How to train your energy-based models

    Y. Song, D.P. Kingma, "How to Train Your Energy - Based Models", arXiv(arXiv:2101.03288), (2021)

    work page arXiv 2021

  80. [80]

    The Stone -Weierstrass theorem and its application to neural networks

    N. Cotter, "The Stone -Weierstrass theorem and its application to neural networks", IEEE Transactions on Neural Networks 1(4), 290–295, (1990)

    work page 1990

Showing first 80 references.