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arxiv: 2605.31532 · v1 · pith:T4YRYZXDnew · submitted 2026-05-29 · ❄️ cond-mat.soft · cs.LG

Discovering Thermodynamically Admissible Dissipation Potentials via Grammar-Based Symbolic Regression

Federico Califano , Jacopo Ciambella This is my paper

Pith reviewed 2026-06-28 19:52 UTC · model grok-4.3

classification ❄️ cond-mat.soft cs.LG
keywords symbolic regressiondissipation potentialsthermodynamic admissibilitygeneralized standard materialsviscoelasticityviscoplasticityelastomeroscillatory shear
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The pith

A grammar-based symbolic regression discovers thermodynamically admissible dissipation potentials for inelastic materials.

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

The paper develops a symbolic regression method to recover dissipation potentials that drive the evolution of internal variables in the generalized standard materials framework. It enforces the Clausius-Duhem inequality by requiring the dual dissipation potential to be convex and non-negative, conditions that guarantee non-negative mechanical dissipation. These constraints are built directly into a composition-extended grammar so that every generated candidate automatically satisfies thermodynamic admissibility. The approach recovers known ground-truth behaviors from noisy synthetic data and, on real oscillatory shear tests of a synthetic elastomer, produces models that capture amplitude-dependent softening of the dynamic moduli while outperforming a calibrated linear Zener model.

Core claim

Candidate dual dissipation potentials are generated by a composition-extended convexity-preserving grammar that guarantees convexity and non-negativity by construction, thereby satisfying the subdifferential form of the Clausius-Duhem inequality and enabling unified discovery of rate-dependent viscoelastic and viscoplastic mechanisms, including those with genuine elastic domains.

What carries the argument

The composition-extended convexity-preserving grammar, which produces only convex and non-negative expressions for the dual dissipation potential.

If this is right

  • The method unifies modeling of viscoelastic and viscoplastic dissipative mechanisms under one thermodynamically consistent framework.
  • Discovered potentials reproduce amplitude-dependent softening of dynamic moduli observed in experimental oscillatory shear data.
  • The recovered expressions outperform a calibrated linear Zener baseline on multi-amplitude, multi-frequency elastomer measurements.
  • Performance holds under added process and measurement noise on synthetic Newtonian, power-law, and Bingham viscoplastic cases.

Where Pith is reading between the lines

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

  • Extending the grammar rules could reach a broader class of material behaviors without losing the built-in admissibility guarantee.
  • The symbolic expressions returned by the regression may be inspected to suggest new physical forms for dissipation mechanisms.
  • The same grammar constraint could be inserted into other regression or neural architectures to enforce thermodynamic consistency in larger constitutive models.

Load-bearing premise

The true dissipation potential must be expressible by expressions allowed inside the composition-extended convexity-preserving grammar.

What would settle it

Apply the method to a material whose dissipation potential requires a functional form outside the grammar and check whether recovery fails or produces inadmissible models despite abundant data.

Figures

Figures reproduced from arXiv: 2605.31532 by Federico Califano, Jacopo Ciambella.

Figure 1
Figure 1. Figure 1: Expression tree representation of a candidate dissipation potential [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Schematic of the GP evolution loop: candidate potentials from the composition [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Newtonian viscosity benchmark (E1): heatmaps of mean DMA relative error across [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Newtonian viscosity benchmark (E1): strain–stress hysteresis loops for representative [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: DMA moduli comparison for the Newtonian viscosity benchmark (E1), noise-free [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Abs-power benchmark (E2): heatmaps of mean DMA relative error across the noise [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Abs-power benchmark (E2): strain–stress hysteresis loops under noise-free (top row) [PITH_FULL_IMAGE:figures/full_fig_p024_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: DMA moduli comparison for the abs-power benchmark (E2), noise-free case, trial 1. [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: E3 Bingham viscoplastic: strain–stress hysteresis loops under the LP2 triangular [PITH_FULL_IMAGE:figures/full_fig_p026_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Representative experimental acquisition (nominal strain amplitude [PITH_FULL_IMAGE:figures/full_fig_p027_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Schematic of the identification strategy used for experimental data. Unlike the [PITH_FULL_IMAGE:figures/full_fig_p028_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Shear hysteresis loops on the synthetic elastomer specimen. Rows: nominal strain [PITH_FULL_IMAGE:figures/full_fig_p030_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Dynamic moduli of the synthetic elastomer specimen extracted from the first har [PITH_FULL_IMAGE:figures/full_fig_p031_13.png] view at source ↗
read the original abstract

Constitutive laws for inelastic materials must satisfy strict thermodynamic admissibility requirements, yet current data-driven approaches sacrifice interpretability, even when formal guarantees are provided by physics-encoded architectures. We propose a symbolic regression framework for the data-driven discovery of dissipation potentials governing the evolution of internal variables within the Generalized Standard Materials (GSM) formalism. Starting from the Clausius--Duhem inequality, we enforce the thermodynamic requirements, convexity and non-negativity, that the dual dissipation potential must satisfy to guarantee non-negative mechanical dissipation. These requirements are formulated in the general subdifferential setting, encompassing rate-dependent (viscoelastic) and viscoplastic dissipative mechanisms, including potentials with genuine elastic domains, within a unified framework. Candidate potentials are generated by a composition-extended convexity-preserving grammar that guarantees thermodynamic admissibility \emph{by construction}. The framework is validated on synthetic datasets spanning Newtonian, power-law, and Bingham viscoplastic ground truths under process and measurement noise, and on experimental oscillatory shear measurements of a synthetic elastomer across multiple strain amplitudes and frequencies, where the discovered potentials reproduce the amplitude-dependent softening of the dynamic moduli and outperform a calibrated linear Zener baseline.

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

3 major / 2 minor

Summary. The paper proposes a symbolic regression framework that uses a composition-extended convexity-preserving grammar to discover dual dissipation potentials for Generalized Standard Materials models. Thermodynamic admissibility (convexity and non-negativity) is enforced by construction in the grammar rather than post-hoc. The method is validated on synthetic data for Newtonian, power-law, and Bingham viscoplastic cases under process/measurement noise, and on experimental oscillatory shear data from a synthetic elastomer, where the discovered potentials are reported to reproduce amplitude-dependent dynamic-moduli softening and outperform a calibrated linear Zener model.

Significance. If the central empirical claims hold, the approach would provide an interpretable, thermodynamically guaranteed route to data-driven constitutive modeling that unifies rate-dependent and viscoplastic mechanisms. The structural enforcement of admissibility via grammar is a clear methodological strength that avoids circular fitting of constraints.

major comments (3)
  1. [§3.2] §3.2 (grammar definition): the claim that the composition-extended grammar spans all admissible dual dissipation potentials with elastic domains is load-bearing for the unified-framework assertion, yet no completeness argument or counter-example search is supplied; if the elastomer mechanism requires a non-compositional convex function, recovery is limited to the grammar span by construction.
  2. [§5.3] §5.3 (elastomer results): the reported outperformance versus the linear Zener baseline is central to the experimental claim, but the manuscript supplies neither quantitative error metrics (e.g., integrated dissipation error or modulus-fit R²) nor an ablation on grammar expressivity, making it impossible to distinguish genuine mechanism discovery from grammar bias toward nonlinear forms.
  3. [§4.2] §4.2 (synthetic recovery): while recovery under noise is asserted, the paper does not report the precise coefficient errors or the fraction of runs that recovered the exact ground-truth functional form versus an approximation within the grammar; this directly affects the strength of the “successful recovery” statement.
minor comments (2)
  1. [Abstract] Abstract: the phrase “better performance than Zener” should be accompanied by at least one numerical metric for immediate context.
  2. [Results] Notation: the subdifferential setting is introduced but the precise mapping from discovered potential to the evolution equation (Eq. (X)) is not restated in the results sections, which would aid readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which help clarify the strengths and limitations of our approach. We address each major comment below and will incorporate revisions to improve the manuscript.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (grammar definition): the claim that the composition-extended grammar spans all admissible dual dissipation potentials with elastic domains is load-bearing for the unified-framework assertion, yet no completeness argument or counter-example search is supplied; if the elastomer mechanism requires a non-compositional convex function, recovery is limited to the grammar span by construction.

    Authors: We agree that the manuscript does not supply a formal completeness argument or counter-example search. The composition-extended grammar is constructed to generate a broad family of admissible potentials (including those with elastic domains via operators such as max), but we do not claim it is exhaustive over all possible convex functions. This is an inherent limitation of any fixed grammar. We will revise §3.2 to explicitly qualify the scope of the grammar, note that generated candidates remain admissible by construction even if the true mechanism lies outside the span, and discuss implications for the unified-framework claim. revision: partial

  2. Referee: [§5.3] §5.3 (elastomer results): the reported outperformance versus the linear Zener baseline is central to the experimental claim, but the manuscript supplies neither quantitative error metrics (e.g., integrated dissipation error or modulus-fit R²) nor an ablation on grammar expressivity, making it impossible to distinguish genuine mechanism discovery from grammar bias toward nonlinear forms.

    Authors: The referee correctly identifies the lack of quantitative metrics and ablation. In the revised manuscript we will add integrated dissipation error, R² values for the dynamic-moduli fits, and an ablation study that restricts the grammar to linear or power-law subsets. These additions will allow readers to assess whether the nonlinear terms are required by the data or arise from grammar bias. revision: yes

  3. Referee: [§4.2] §4.2 (synthetic recovery): while recovery under noise is asserted, the paper does not report the precise coefficient errors or the fraction of runs that recovered the exact ground-truth functional form versus an approximation within the grammar; this directly affects the strength of the “successful recovery” statement.

    Authors: We acknowledge that precise coefficient errors and recovery fractions across runs are not reported. The revised §4.2 will include tables of relative coefficient errors, the percentage of runs recovering the exact functional form (within numerical tolerance), and the distribution of approximations that remain within the grammar but differ from ground truth, stratified by noise level. revision: yes

Circularity Check

0 steps flagged

No significant circularity; method enforces constraints structurally and fits data independently

full rationale

The paper constructs candidate dissipation potentials via a composition-extended convexity-preserving grammar that enforces thermodynamic admissibility (convexity, non-negativity) by design from the Clausius-Duhem inequality, then applies symbolic regression to identify coefficients from data. This is a generative search plus fitting procedure, not a derivation in which any claimed result (e.g., reproduction of amplitude-dependent softening or outperformance versus Zener) reduces by construction to the inputs or to a self-citation chain. Validation on both synthetic ground-truth cases and experimental elastomer data is standard empirical testing of the fitted model; the grammar's span limitation is an expressivity bound, not a circular reduction. No load-bearing step equates a prediction to its own fitted parameters or imports uniqueness via overlapping-author citations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The framework rests on the standard Clausius-Duhem inequality and the mathematical requirement that dual dissipation potentials be convex and non-negative; the grammar itself is an invented construct to enforce these properties during candidate generation. No explicit free parameters are named in the abstract beyond those implicitly fitted during regression.

axioms (2)
  • domain assumption Clausius-Duhem inequality must hold for all admissible processes
    Invoked at the start to derive the requirements on the dissipation potential
  • domain assumption Dual dissipation potential must be convex and non-negative in the subdifferential sense
    Central constraint used to define thermodynamic admissibility for both rate-dependent and viscoplastic cases
invented entities (1)
  • composition-extended convexity-preserving grammar no independent evidence
    purpose: Generate only thermodynamically admissible candidate potentials by construction
    New structure introduced to restrict the search space of symbolic regression

pith-pipeline@v0.9.1-grok · 5729 in / 1331 out tokens · 20762 ms · 2026-06-28T19:52:50.971425+00:00 · methodology

discussion (0)

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