pith. sign in

arxiv: 2605.16211 · v1 · pith:EM4MOEQ2new · submitted 2026-05-15 · 💻 cs.LG · math.DS

Hypothesis-driven construction of mesoscopic dynamics

Zhuoyuan Li , Aiqing Zhu , Qianxiao Li This is my paper

Pith reviewed 2026-05-20 20:37 UTC · model grok-4.3

classification 💻 cs.LG math.DS
keywords mesoscopic dynamicsOnsager principlehypothesis classspatio-temporal evolutiondynamics learninginterpretable modelingphysics-informed learning
0
0 comments X

The pith

Mesoscopic dynamics are learned by selecting members from a hypothesis class defined by the generalized Onsager principle, which supplies uniform guarantees before any data is used.

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

The authors shift from deriving fixed equations for each problem to working inside a single hypothesis class built from a generalized Onsager principle. This class covers both dissipative and conservative mesoscopic evolution equations and carries uniform proofs of global well-posedness, asymptotic stability, unique factorization identifiability, and discrete energy dissipation that hold for every candidate equation before data arrives. Observed data then identifies which member of the class best matches the target behavior. The framework is checked against known continuum PDEs and against microscopic chain data where no exact meso-scale equation is available in advance. The resulting models remain accurate, robust, and physically interpretable by construction.

Core claim

A hypothesis class constructed via the generalized Onsager principle admits uniform and a priori guarantees of global well-posedness, asymptotic stability, unique factorization identifiability, and discrete energy dissipation for every spatio-temporal evolution equation inside the class, prior to all learning stages. Data from individual problem instances is used only afterward to select the appropriate member of the class, producing accurate and interpretable dynamical models.

What carries the argument

The generalized Onsager principle, which supplies a unified hypothesis class for both dissipative and conservative mesoscopic dynamics and enables the listed uniform guarantees to hold for all members before data fitting begins.

If this is right

  • Every candidate equation inside the class is globally well-posed and asymptotically stable by construction.
  • Unique factorization inside the class guarantees that the recovered factors are identifiable from data.
  • Discrete energy dissipation holds for all models in the class even after data-driven selection.
  • Data is used only to choose among already-guaranteed candidates rather than to invent arbitrary functional forms.

Where Pith is reading between the lines

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

  • The same constrained-class strategy could be tried with other variational principles to generate guaranteed models for different classes of physical systems.
  • Because the guarantees are uniform and data-independent, the approach may reduce the risk of unstable or non-physical extrapolations common in purely data-driven dynamical models.
  • Testing the class on additional microscopic-to-mesoscopic transitions, such as those in biological or social systems, would clarify how far the uniform guarantees extend in practice.

Load-bearing premise

The generalized Onsager principle defines a hypothesis class that is broad enough to represent the desired mesoscopic behavior yet narrow enough for the uniform guarantees to apply to every possible equation inside it without seeing data.

What would settle it

A concrete counterexample would be a microscopic simulation whose long-term statistics cannot be reproduced to high accuracy by any member of the hypothesis class, or a fitted member that fails to preserve discrete energy dissipation or loses asymptotic stability.

Figures

Figures reproduced from arXiv: 2605.16211 by Aiqing Zhu, Qianxiao Li, Zhuoyuan Li.

Figure 1
Figure 1. Figure 1: Comparison of our framework and classical modeling framework. Classical approaches [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: fig. 2. Detailed description about the baselines as well as the evaluation metric can be found in [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of long-time prediction performance [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Identifiability of the PDE dynamics. (a) relationship between the learned potential and the [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Inference step size scaling for the Allen–Cahn PDE models. [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Illustration of microscopic chain models and their coarse-graining. [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: One-step variation of the learned potentials [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Evolution of the learned potentials Vθ for the microscopic chain models. We use colors to distinguish different trajectories. Firstly, we evaluate the learned components V0 and V2 on a randomly sampled initial state u0 scaled by a varying amplitude a in fig. 9. Two critical observations emerge from this scaling. For both cases, V0(au0) carries the dominant amplitude dependence, whereas V2(au0) varies on a … view at source ↗
Figure 9
Figure 9. Figure 9: Scaling property of the components V0 and V2 for the microscopic chain models. For each test u0, we run a log-log linear fit of V0 w.r.t. the scalar a; the resulting scaling exponent α and the the coefficients of determination R2 are appended in the legend. 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00 u 15 10 5 0 5 10 15 energy density (shifted) FPUT pointwise density of V0 F(u) F(0) V0(u) V0(0) region whe… view at source ↗
Figure 10
Figure 10. Figure 10: The pointwise component F(u) and V0(u) of the learned potentials Vθ 20 [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
read the original abstract

Traditional scientific modeling typically begins with fixed, instance-wise effective equations and then carries out equation-specific analysis and computation, a procedure that becomes exceptionally challenging in complex applications such as multiscale systems. We propose an alternative paradigm by learning mesoscopic dynamics within a mathematically constrained hypothesis class. Building upon a generalized Onsager principle, we introduce a unified framework encompassing both dissipative and conservative mesoscopic dynamics. We establish uniform and a priori theoretical guarantees, including global well-posedness, asymptotic stability, unique factorization identifiability, and discrete energy dissipation, applicable to all spatio-temporal evolution equations within this hypothesis class prior to all learning stages. Data from each problem instance is then used to guide the identification of members within our hypothesis class, giving rise to accurate, robust and interpretable dynamical models. We empirically validate this framework on both data from continuum PDE models as a check, and on data arising from microscopic chain models for which exact meso-scale models are unknown. The proposed approach not only acts as an effective dynamics learner, but also offers vital interpretable diagnostics of the underlying physics.

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

1 major / 0 minor

Summary. The paper proposes a hypothesis-driven paradigm for constructing mesoscopic dynamics via a generalized Onsager principle. It introduces a unified hypothesis class that encompasses both dissipative and conservative spatio-temporal evolution equations, claims uniform a priori theoretical guarantees (global well-posedness, asymptotic stability, unique factorization identifiability, and discrete energy dissipation) that hold for every member of the class before any data fitting or learning occurs, and then uses instance-specific data to identify accurate, interpretable models within the class. Empirical validation is performed on continuum PDE data and microscopic chain models.

Significance. If the uniform a priori guarantees can be established rigorously for the full breadth of the claimed hypothesis class, the framework would provide a significant methodological advance for multiscale modeling by supplying physically constrained, theoretically grounded mesoscopic models directly from data while offering built-in diagnostics of underlying physics.

major comments (1)
  1. Abstract and statement of main results: the manuscript asserts that uniform guarantees of asymptotic stability and discrete energy dissipation apply to all members of the hypothesis class prior to learning. The class is explicitly described as encompassing both dissipative and conservative mesoscopic dynamics. Conservative dynamics conserve energy (rather than dissipate it) and exhibit Lyapunov stability with trajectories remaining on level sets, not asymptotic stability. This creates an apparent internal inconsistency in the uniformity claim unless the generalized Onsager principle or the hypothesis class definition implicitly excludes genuine conservative cases or the theorems contain unstated case distinctions. Please cite the precise theorem(s) establishing these properties and clarify the mapping from the Onsager principle to the class that permits uniform application to every valid

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying this important point of clarification regarding the scope of the uniform guarantees. We address the concern directly below.

read point-by-point responses

  1. Referee: Abstract and statement of main results: the manuscript asserts that uniform guarantees of asymptotic stability and discrete energy dissipation apply to all members of the hypothesis class prior to learning. The class is explicitly described as encompassing both dissipative and conservative mesoscopic dynamics. Conservative dynamics conserve energy (rather than dissipate it) and exhibit Lyapunov stability with trajectories remaining on level sets, not asymptotic stability. This creates an apparent internal inconsistency in the uniformity claim unless the generalized Onsager principle or the hypothesis class definition implicitly excludes genuine conservative cases or the theorems contain unstated case distinctions. Please cite the precise theorem(s) establishing these properties and clarify the mapping from the Onsager principle to the class that permits uniform application to every valid

    Authors: We appreciate the referee's observation and agree that the current wording in the abstract and main results section is imprecise. The generalized Onsager principle is constructed with a dissipation operator that is positive semi-definite; when this operator is identically zero the dynamics are conservative. Global well-posedness and discrete energy conservation (i.e., the energy dissipation identity reduces to exact conservation) hold uniformly for the entire class. Asymptotic stability is proved only when the dissipation operator is strictly positive definite; when it vanishes we obtain Lyapunov stability with trajectories confined to level sets of the conserved energy. The theorems therefore already contain the necessary case distinction, but it is not stated explicitly in the high-level claims. We will revise the abstract, the statement of main results, and the relevant theorem statements (specifically Theorems 3.1 and 3.2) to make the distinction clear and to add a short remark mapping the Onsager structure (skew-symmetric part for conservative transport, symmetric positive semi-definite part for dissipation) to the two regimes. The revised text will cite the precise theorems and will no longer assert asymptotic stability for every member of the class. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation; a priori guarantees claimed prior to fitting

full rationale

The paper defines a hypothesis class from a generalized Onsager principle and states that uniform mathematical guarantees (well-posedness, stability, identifiability, energy dissipation) are established for every member of that class before any data or learning occurs. No quoted step reduces a claimed prediction or theorem to a fitted parameter, self-citation chain, or definitional tautology; the guarantees are presented as consequences of the class construction itself. The derivation is therefore self-contained and does not exhibit any of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on the assumption that a generalized Onsager principle can be used to define a hypothesis class with the listed properties; no free parameters or invented entities are named in the abstract.

axioms (1)
  • domain assumption A generalized Onsager principle can be formulated so that every member of the resulting hypothesis class satisfies global well-posedness, asymptotic stability, and discrete energy dissipation before any data fitting occurs.
    Invoked in the abstract as the foundation for the uniform guarantees that hold prior to learning.

pith-pipeline@v0.9.0 · 5712 in / 1286 out tokens · 61876 ms · 2026-05-20T20:37:22.798060+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

56 extracted references · 56 canonical work pages

  1. [1]

    Journal of Computational and Applied Mathematics , volume=

    Learning Hamiltonians of constrained mechanical systems , author=. Journal of Computational and Applied Mathematics , volume=. 2023 , publisher=

    work page 2023

  2. [2]

    arXiv preprint arXiv:2505.20370 , year=

    Learning mechanical systems from real-world data using discrete forced Lagrangian dynamics , author=. arXiv preprint arXiv:2505.20370 , year=

    work page arXiv

  3. [3]

    1955 , institution =

    Studies of the nonlinear problems , author =. 1955 , institution =

    work page 1955

  4. [4]

    Industrial & Engineering Chemistry Fundamentals , volume =

    Kinetic theory and rheology of dilute suspensions of finitely extendible dumbbells , author =. Industrial & Engineering Chemistry Fundamentals , volume =. 1972 , doi =

    work page 1972

  5. [5]

    2010 , doi=

    Verification and Validation in Scientific Computing , author =. 2010 , doi=

    work page 2010

  6. [6]

    2008 , doi=

    Multiscale Methods: Averaging and Homogenization , author =. 2008 , doi=

    work page 2008

  7. [7]

    2011 , publisher =

    Principles of Multiscale Modeling , author =. 2011 , publisher =

    work page 2011

  8. [8]

    Bayesian calibration of computer models , author =. J. R. Stat. Soc. Ser. B. Stat. Methodol. , volume =. 2001 , issn =. doi:10.1111/1467-9868.00294 , publisher =

    work page doi:10.1111/1467-9868.00294 2001

  9. [9]

    Equation-free multiscale computation: Algorithms and applications , author =. Annu. Rev. Phys. Chem. , volume =. 2009 , doi =

    work page 2009

  10. [10]

    2008 , publisher =

    The Mathematical Theory of Finite Element Methods , author =. 2008 , publisher =

    work page 2008

  11. [11]

    2010 , publisher =

    Partial Differential Equations , author =. 2010 , publisher =

    work page 2010

  12. [12]

    Interaction of ``solitons'' in a collisionless plasma and the recurrence of initial states , author =. Phys. Rev. Lett. , volume =. 1965 , doi=

    work page 1965

  13. [13]

    On metastability in

    Bambusi, Dario and Ponno, Antonio , journal =. On metastability in. 2006 , doi =

    work page 2006

  14. [14]

    Gallagher, Isabelle and Saint-Raymond, Laure and Texier, Benjamin , year =. From. doi:10.4171/129 , publisher =

    work page doi:10.4171/129

  15. [15]

    Stochastic

    Bertini, Lorenzo and Giacomin, Giambattista , journal =. Stochastic. 1997 , doi =

    work page 1997

  16. [16]

    Corwin, Ivan , journal =. The. 2012 , doi =

    work page 2012

  17. [17]

    2019 , publisher =

    Data-driven science and engineering: Machine learning, dynamical systems, and control , author =. 2019 , publisher =

    work page 2019

  18. [18]

    The discovery of dynamics via linear multistep methods and deep learning: error estimation , author =. SIAM J. Numer. Anal. , volume =. 2022 , doi =

    work page 2022

  19. [19]

    2025 , doi =

    Chen, Junfeng and Wu, Kailiang and Xiu, Dongbin , journal =. 2025 , doi =

    work page 2025

  20. [20]

    Proceedings of the 33rd International Conference on Neural Information Processing Systems , pages=

    Greydanus, Sam and Dzamba, Misko and Yosinski, Jason , title =. Proceedings of the 33rd International Conference on Neural Information Processing Systems , pages=. 2019 , publisher =

    work page 2019

  21. [21]

    On learning

    Bertalan, Tom and Dietrich, Felix and Mezi. On learning. Chaos , volume =. 2019 , month =

    work page 2019

  22. [22]

    Proceedings of the 35th International Conference on Neural Information Processing Systems , pages =

    Lee, Kookjin and Trask, Nathaniel and Stinis, Panos , title =. Proceedings of the 35th International Conference on Neural Information Processing Systems , pages =. 2021 , publisher =

    work page 2021

  23. [23]

    Variational

    Huang, Shenglin and He, Zequn and Reina, Celia , journal =. Variational. 2022 , publisher =

    work page 2022

  24. [24]

    2022 , issn =

    Zhang, Zhen and Shin, Yeonjong and Em Karniadakis, George , journal =. 2022 , issn =. doi:10.1098/rsta.2021.0207 , publisher =

    work page doi:10.1098/rsta.2021.0207 2022

  25. [25]

    Structure-preserving method for reconstructing unknown Hamiltonian systems from trajectory data , author =. SIAM J. Sci. Comput. , volume =. 2020 , doi =

    work page 2020

  26. [26]

    2025 , issn =

    Huang, Shenglin and He, Zequn and Dirr, Nicolas and Zimmer, Johannes and Reina, Celia , journal =. 2025 , issn =. doi:10.1016/j.jmps.2024.105908 , publisher =

    work page doi:10.1016/j.jmps.2024.105908 2025

  27. [27]

    Learning nonlinear operators via

    Lu, Lu and Jin, Pengzhan and Pang, Guofei and Zhang, Zhongqiang and Karniadakis, George Em , journal =. Learning nonlinear operators via. 2021 , doi =

    work page 2021

  28. [28]

    2021 , month =

    Yu, Haijun and Tian, Xinyuan and E, Weinan and Li, Qianxiao , journal =. 2021 , month =

    work page 2021

  29. [29]

    Constructing custom thermodynamics using deep learning , author =. Nat. Comput. Sci. , volume =. 2024 , doi =

    work page 2024

  30. [30]

    Chen, Ricky T. Q. and Rubanova, Yulia and Bettencourt, Jesse and Duvenaud, David , title =. Proceedings of the 32nd International Conference on Neural Information Processing Systems , pages =. 2018 , publisher =

    work page 2018

  31. [31]

    International Conference on Learning Representations , year=

    Fourier Neural Operator for Parametric Partial Differential Equations , author=. International Conference on Learning Representations , year=

    work page

  32. [32]

    Brunton and Joshua L

    Steven L. Brunton and Joshua L. Proctor and J. Nathan Kutz , title =. Proc. Natl. Acad. Sci. USA , volume =. 2016 , doi =

    work page 2016

  33. [33]

    Nathan Kutz and Steven L

    Kathleen Champion and Bethany Lusch and J. Nathan Kutz and Steven L. Brunton , title =. Proc. Natl. Acad. Sci. USA , volume =. 2019 , doi =

    work page 2019

  34. [34]

    Reciprocal Relations in Irreversible Processes

    Onsager, Lars , journal =. Reciprocal Relations in Irreversible Processes. 1931 , month =

    work page 1931

  35. [35]

    arXiv preprint arXiv:2510.24160 , year =

    Identifiable learning of dissipative dynamics , author =. arXiv preprint arXiv:2510.24160 , year =

    work page arXiv

  36. [36]

    doi:10.1088/0953-8984/23/28/284118 , year =

    Doi Masao , title =. doi:10.1088/0953-8984/23/28/284118 , year =

    work page doi:10.1088/0953-8984/23/28/284118

  37. [37]

    doi:10.1088/1674-1056/24/2/020505 , year =

    Doi Masao , title =. doi:10.1088/1674-1056/24/2/020505 , year =

    work page doi:10.1088/1674-1056/24/2/020505

  38. [38]

    arXiv preprint arXiv:2412.10354 , year =

    Jean Kossaifi and Nikola Kovachki and Zongyi Li and David Pitt and Miguel Liu-Schiaffini and Robert Joseph George and Boris Bonev and Kamyar Azizzadenesheli and Julius Berner and Valentin Duruisseaux and Anima Anandkumar , title =. arXiv preprint arXiv:2412.10354 , year =

    work page arXiv

  39. [39]

    Phase‐Field Methods in Materials Science and Engineering , doi =

    Nikolas Provatas and Ken Elder , publisher =. Phase‐Field Methods in Materials Science and Engineering , doi =

    work page

  40. [40]

    Message Passing Neural

    Johannes Brandstetter and Daniel Worrall and Max Welling , booktitle=. Message Passing Neural

    work page

  41. [41]

    2025 , issn =

    Differentiability in unrolled training of neural physics simulators on transient dynamics , journal =. 2025 , issn =. doi:10.1016/j.cma.2024.117441 , author =

    work page doi:10.1016/j.cma.2024.117441 2025

  42. [42]

    Gaussian error linear units (

    Hendrycks, Dan and Gimpel, Kevin , journal=. Gaussian error linear units (

    work page

  43. [43]

    Dedalus: A flexible framework for numerical simulations with spectral methods , author =. Phys. Rev. Res. , volume =. 2020 , month =

    work page 2020

  44. [44]

    doi:10.1137/140951758 , journal =

    An Adaptive Shifted Power Method for Computing Generalized Tensor Eigenpairs , author =. doi:10.1137/140951758 , journal =

    work page doi:10.1137/140951758

  45. [45]

    Nick Higham , title =

    work page

  46. [46]

    Kolda and Ali Pinar , eprint =

    Chengbin Peng and Tamara G. Kolda and Ali Pinar , eprint =. Accelerating Community Detection by Using

    work page

  47. [47]

    and Zhang, Shanrong and Merritt, Matthew E

    Woessner, Donald E. and Zhang, Shanrong and Merritt, Matthew E. and Sherry, A. Dean , title =. Magnetic Resonance in Medicine , doi =

    work page

  48. [48]

    2003 , eid =

    Properties of Highly Clustered Networks , author =. 2003 , eid =. doi:10.1103/PhysRevE.68.026121 , journal =

    work page doi:10.1103/physreve.68.026121 2003

  49. [49]

    Clawpack Software , author =

    work page

  50. [50]

    : A Document Preparation System

    Leslie Lamport. : A Document Preparation System. 1986

    work page 1986

  51. [51]

    Frank Mittlebach and Michel Goossens , title =

    work page

  52. [52]

    and Van Loan, Charles F

    Golub, Gene H. and Van Loan, Charles F. , title =

    work page

  53. [53]

    Paul Dawkins , title =

    work page

  54. [54]

    User's Guide for the

    work page

  55. [55]

    Michael Downes , title =

    work page

  56. [56]

    Christian Feuers\"anger , title =

    work page