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arxiv: 2606.24106 · v1 · pith:AMEB4R3Nnew · submitted 2026-06-23 · 🧮 math.DS · cs.NA· math.NA

Flexible and Stable Dynamics Discovery with Onsager's Variational Principle

Anthony Gruber , Ritoban Roy-Chowdhury , Irina Tezaur , Nathan M. Urban This is my paper

Pith reviewed 2026-06-25 22:28 UTC · model grok-4.3

classification 🧮 math.DS cs.NAmath.NA
keywords Onsager variational principleenergy stabilitydynamics discoveryvariational discretizationfree energy learningdissipation potentialAllen-Cahn equationCahn-Hilliard equation
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The pith

A variational discretization of Onsager's principle learns uncertain terms in free energy and dissipation from data while guaranteeing unconditional energy stability.

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

The paper establishes a method to discover dynamical models via Onsager's variational principle when parts of the free energy or dissipation potential are uncertain or empirical. It introduces a time discretization based on minimizing a Rayleighan functional and formulates a learning problem that identifies the missing terms directly from observable data. If correct, the resulting models remain provably energy stable for arbitrarily long rollouts and recover standard proximal and gradient flow methods as special cases. A sympathetic reader would care because this supplies a way to build flexible, physics-consistent simulators for dissipative systems without risking instability during extended predictions.

Core claim

Onsager's variational principle characterizes dissipation-dominated phenomena such as phase separation via extremization of an associated functional. When one or more parts of this functional are empirically approximated or uncertain, a novel variational discretization is introduced that recovers previous work as a special case. A learning problem is then formulated to identify the uncertain terms in the free energy and dissipation potential from observable data. The resulting OVP-based models connect directly to prior work in proximal methods, Sobolev and Wasserstein gradient flows, while remaining provably energy-stable under arbitrarily long rollouts. The approach is illustrated on Allen-

What carries the argument

Minimization of a Rayleighan functional inside a variational discretization of Onsager's variational principle, extended to incorporate data-learned free energy and dissipation terms.

If this is right

  • The learned models recover proximal methods and Sobolev and Wasserstein gradient flows as special cases.
  • Provable energy stability holds for arbitrarily long rollouts regardless of the learned terms.
  • Uncertain elements including bulk free-energy densities, nonlocal potentials, and boundary conditions can be identified using polynomials, shallow neural networks, or spectral kernels.
  • The method applies to Allen-Cahn, Fokker-Planck, and Cahn-Hilliard systems while maintaining the variational structure.

Where Pith is reading between the lines

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

  • The same embedding of learning inside the variational step could be tried on time-series data from laboratory experiments rather than synthetic observations.
  • Because stability is unconditional, the models might be directly usable inside optimization loops for long-horizon control of dissipative processes.
  • Similar variational discretizations might be constructed for other dissipation principles to handle uncertain terms outside the Onsager setting.

Load-bearing premise

Uncertain terms in the free energy and dissipation potential can be identified from observable data in a manner that preserves the unconditional energy stability of the proposed variational discretization.

What would settle it

A long-rollout simulation of a model learned on one of the example systems, such as Cahn-Hilliard, in which the discrete energy increases over time steps would show that the stability guarantee fails to hold after learning.

Figures

Figures reproduced from arXiv: 2606.24106 by Anthony Gruber, Irina Tezaur, Nathan M. Urban, Ritoban Roy-Chowdhury.

Figure 1
Figure 1. Figure 1: An illustration depicting Onsager’s Variational Principle [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: OVP-based diffusion with (top) and without (bottom) the essential condition [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Snapshots of the Cahn-Hilliard (CH), Allen-Cahn (AC), and Fokker-Planck (FP) equations, com [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: One cell of the MAC grid used for spatial discretiza￾tion. The PDEs implemented for illustration follow the Wasserstein and Sobolev gradient flow examples in Corollary 2.2 and Theorem 2.4. Two discrete process mappings will be considered: the identity mapping PI (C)W = W in the case of Allen-Cahn (where W ∈ R nx×ny is cell￾centered), and the discrete divergence [P(C)W]i,j = − [PITH_FULL_IMAGE:figures/full… view at source ↗
Figure 5
Figure 5. Figure 5: Three representative Cahn-Hilliard evolutions, differing in their bulk free energy density [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Training and validation loss (mean squared error) curves illustrating the training process in the [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Evolution of the learned free energies Fθ over the duration of training, for the polynomial (red) and neural network (blue) parameterizations. The passage of training epochs is illustrated with a gradient, with later epochs corresponding to darker curves. The ground truth is overlaid with a dashed line in each case [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Evolution of the learned Cahn-Hilliard system in the asymmetric case, starting from an initial [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Boundary potential recovery results. Left: training and validation loss curves as a function of [PITH_FULL_IMAGE:figures/full_fig_p026_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Evolution of the Allen-Cahn system with boundary potential starting from an initial condition in [PITH_FULL_IMAGE:figures/full_fig_p027_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Eigenvalues of the Ohta-Kawasaki potential compared to the learned surrogate in Experiment 1. [PITH_FULL_IMAGE:figures/full_fig_p029_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Evolution of the learned Ohta-Kawasaki system in Experiment 1, starting from an initial condition [PITH_FULL_IMAGE:figures/full_fig_p029_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Spectrum of the Ohta-Kawasaki potential compared to that of the learned surrogate in Experiment [PITH_FULL_IMAGE:figures/full_fig_p030_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Evolution of the learned Ohta-Kawasaki system in Experiment 2, starting from an initial condition [PITH_FULL_IMAGE:figures/full_fig_p031_14.png] view at source ↗
read the original abstract

Variational principles specify the dynamics of a physical system via the extremization of associated functional data. Onsager's variational principle (OVP), which characterizes dissipation-dominated phenomena such as phase separation, admits an unconditionally energy-stable time discretization through the minimization of a Rayleighan functional combining free energy and dissipative effects. The present work considers the case where one or more parts of this functional are empirically approximated or otherwise uncertain. To address this, a novel variational discretization of OVP is introduced which recovers previous work as a special case, and a learning problem is formulated which identifies uncertain terms in the free energy and dissipation potential from observable data. It is shown that the resulting OVP-based models connect directly to previous work in terms of proximal methods, Sobolev and Wasserstein gradient flows, while remaining provably energy-stable under arbitrarily long rollouts. The approach is illustrated on examples including Allen-Cahn, Fokker-Planck, and Cahn-Hilliard system models, where the effects of bulk free-energy densities, nonlocal potentials, and nonstandard boundary conditions are effectively learned with model classes consisting of polynomials, shallow neural networks, and spectral convolution kernels.

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 paper introduces a variational time discretization of Onsager's variational principle (OVP) for dissipative systems that recovers prior proximal and gradient-flow schemes as special cases. When free-energy and dissipation terms are uncertain, a learning problem is posed to identify them from observable data; the central claim is that the resulting models remain unconditionally energy-stable for arbitrarily long rollouts by construction. Connections to Sobolev/Wasserstein gradient flows are drawn, and the method is illustrated on Allen-Cahn, Fokker-Planck, and Cahn-Hilliard systems with polynomial, neural-network, and spectral-kernel parametrizations of the potentials.

Significance. If the stability preservation under data-driven identification holds rigorously, the work would provide a principled route to stable learned models for phase-separation and transport phenomena, directly linking variational discretizations with modern function approximation. The recovery of existing methods as special cases and the explicit handling of nonlocal potentials and nonstandard boundary conditions are positive features that could facilitate adoption in materials modeling.

major comments (2)
  1. [§3] §3 (variational discretization): the proof that the learned free-energy and dissipation functionals preserve the unconditional energy stability of the discrete scheme must be stated explicitly; it is not clear from the abstract whether the learning objective is constrained to maintain the variational structure or whether stability follows automatically from the discretization alone.
  2. [§4] §4 (learning problem): the precise statement of the data-fitting objective (e.g., whether it is a direct regression on observed trajectories or a variational residual) is needed to confirm that it does not introduce terms that could violate the Rayleighian minimization at each step.
minor comments (2)
  1. [Introduction] The abstract claims recovery of 'previous work as a special case' but does not name the specific discretizations recovered; a brief enumeration in the introduction would improve clarity.
  2. Notation for the Rayleighian functional and the learned potentials should be introduced once and used consistently; several symbols appear to be overloaded between the continuous and discrete settings.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment and recommendation of minor revision. We address each major comment below and will revise the manuscript accordingly to improve clarity.

read point-by-point responses

  1. Referee: [§3] §3 (variational discretization): the proof that the learned free-energy and dissipation functionals preserve the unconditional energy stability of the discrete scheme must be stated explicitly; it is not clear from the abstract whether the learning objective is constrained to maintain the variational structure or whether stability follows automatically from the discretization alone.

    Authors: The unconditional energy stability is a direct consequence of the variational discretization itself: at each time step the scheme minimizes a Rayleighian that is constructed from the (possibly learned) free-energy and dissipation functionals, and the proof that this minimization yields a discrete energy dissipation law holds for any admissible functionals. The learning problem is posed over parametrizations that enter the Rayleighian linearly or through convex combinations, thereby preserving the variational structure by construction. We agree that an explicit statement and short proof sketch should appear in §3 rather than being left implicit; this will be added in the revision. revision: yes

  2. Referee: [§4] §4 (learning problem): the precise statement of the data-fitting objective (e.g., whether it is a direct regression on observed trajectories or a variational residual) is needed to confirm that it does not introduce terms that could violate the Rayleighian minimization at each step.

    Authors: The data-fitting objective is formulated as a variational residual that penalizes the mismatch between the observed increments and the minimizer of the discrete Rayleighian; it is not a direct trajectory regression. Because the learned functionals appear only inside the Rayleighian that is subsequently minimized, the resulting scheme remains a valid variational discretization and cannot introduce extraneous terms that would violate the minimization. We will insert the precise mathematical expression of this residual (including the admissible function classes) at the beginning of §4 to make the structure transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The provided abstract and reader's summary describe a variational discretization of Onsager's principle that recovers prior work as a special case, followed by a learning problem that identifies uncertain free-energy and dissipation terms directly from observable data while preserving unconditional energy stability by construction of the discretization. No equations, self-citations, or claims are quoted that reduce the stability result or learned models to fitted inputs by definition, nor is there evidence of self-definitional loops, fitted parameters renamed as predictions, or load-bearing uniqueness theorems imported from the same authors. The derivation is presented as self-contained via the variational structure and external data, consistent with a normal non-circular outcome.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides insufficient detail to enumerate specific free parameters, axioms, or invented entities; no explicit functional forms or fitting procedures are given beyond high-level model classes.

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