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arxiv: 2603.28707 · v2 · pith:T6FNO74Qnew · submitted 2026-03-30 · 💻 cs.CE · cs.AI

A Convex Route to Thermomechanics: Learning Internal Energy and Dissipation

Hagen Holthusen , Paul Steinmann , Ellen Kuhl This is my paper

Pith reviewed 2026-05-25 06:32 UTC · model grok-4.3

classification 💻 cs.CE cs.AI
keywords constitutive modelingthermomechanicsinput convex neural networksthermodynamic consistencyinternal energydissipation potentialsoft tissuesrubber
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The pith

Input convex neural networks represent internal energy and dissipation potential to learn thermodynamically consistent constitutive models in thermomechanics from temperature and deformation data.

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

The paper develops a neural network method to discover constitutive models for materials that experience both mechanical deformation and temperature changes. It models the internal energy and a dissipation potential directly as functions of deformation and entropy, using temperature as the observed input while inferring entropy internally. Input convex neural networks enforce the necessary convexity and guarantee compliance with the second law by construction. This setup avoids the mixed convexity-concavity conditions required in traditional Helmholtz-energy approaches. The framework is tested on synthetic data and experiments involving soft tissues and filled rubbers, showing accurate capture of the underlying behavior.

Core claim

By expressing the internal energy and dissipation potential in terms of deformation and entropy and representing both with input convex neural networks, the formulation ensures thermodynamic admissibility, objectivity, material symmetry, and normalization by architecture alone for isotropic materials without preferred directions or internal variables; temperature serves as the independent observable while entropy is recovered through the constitutive relation.

What carries the argument

Input convex neural networks (networks whose outputs remain convex functions of chosen inputs) representing the internal energy and dissipation potential, combined with invariant-based inputs for objectivity and zero-anchored outputs for normalization.

If this is right

  • Thermodynamic admissibility holds automatically without post-hoc checks or penalty terms.
  • No separate entropy measurements are required because entropy is recovered from the temperature-dependent constitutive relations.
  • Objectivity and material symmetry for isotropic cases are satisfied through the choice of invariant inputs rather than added constraints.
  • The same architecture applies to purely thermal problems and to fully coupled thermomechanical loading of soft tissues and rubbers.

Where Pith is reading between the lines

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

  • The method could be tested on materials that develop anisotropy or internal variables by augmenting the network inputs with directional or history-dependent features.
  • If the convexity guarantee remains intact, the approach might integrate directly into finite-element solvers to produce on-the-fly constitutive updates during simulation.
  • Extending the zero-anchored normalization to multiple reference states could allow consistent modeling across wide temperature intervals without retraining.

Load-bearing premise

Input convex neural networks are expressive enough to capture the full response of isotropic materials without preferred directions or internal variables, and entropy can be inferred reliably from temperature observations alone across the relevant range.

What would settle it

Training the networks on a dataset and then checking whether the predicted dissipation potential becomes negative or the inferred entropy leads to a violation of the second law on an independent test set that includes direct entropy measurements.

Figures

Figures reproduced from arXiv: 2603.28707 by Ellen Kuhl, Hagen Holthusen, Paul Steinmann.

Figure 1
Figure 1. Figure 1: Overall network architecture consisting of the subnetworks [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Boundary value problem for the transient heat diffusion example. A cubic specimen with edge length [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Temperature distribution at two exemplary time steps along the wall thickness direction for the transient heat diffusion problem. [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Left: Training history of the physics-based neural network for the transient heat diffusion problem. Shown are the total loss as [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of the heat flux component q1 for the transient heat-conduction problem. Left: Temporal evolution of the heat flux at selected inner and outer locations, comparing the reference solution with the predictions of the auxiliary MLP and the Newton solver. Right: Parity plot of predicted versus reference heat flux values. 4.2. Experimental data: Porcine Tissue Next, we investigate the ability of the … view at source ↗
Figure 6
Figure 6. Figure 6: Left: Training history of the physics-based neural network for the temperature dependent porcine tissue dataset [64]. Shown are [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of the discovered model and the reference data for porcine tissue under incompressible uniaxial loading at different [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Left: Training history of the physics-based neural network for the temperature dependent carbon-filled black rubber dataset [65]. [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of the discovered model and the reference data for carbon-filled black rubber under incompressible uniaxial loading at [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Setup of the boundary value problem and training scenarios for the plate with an elliptic hole. Top: Geometry and dimensions of [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Left: Training history of the physics-based neural network for the plate with an elliptic hole problem. Shown are the total loss [PITH_FULL_IMAGE:figures/full_fig_p019_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Boundary value problem and testing scenarios for the spring-like structure. Left: Geometry and boundary conditions, including [PITH_FULL_IMAGE:figures/full_fig_p020_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Results for Testing #1 of the spring-like structure under coupled thermo-mechanical loading. Shown are five representative time [PITH_FULL_IMAGE:figures/full_fig_p021_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Quantitative comparison of the predicted and reference responses for Testing #1 of the spring-like structure under coupled [PITH_FULL_IMAGE:figures/full_fig_p022_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Results for Testing #2 of the spring-like structure under purely mechanical loading with constant temperature boundary conditions. [PITH_FULL_IMAGE:figures/full_fig_p025_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Quantitative comparison of the predicted and reference responses for Testing #2 of the spring-like structure under purely mechanical [PITH_FULL_IMAGE:figures/full_fig_p026_16.png] view at source ↗
read the original abstract

We present a physics-based neural network framework for the discovery of constitutive models in fully coupled thermomechanics. In contrast to classical formulations based on the Helmholtz energy, we adopt the internal energy and a dissipation potential as primary constitutive functions, expressed in terms of deformation and entropy. This choice avoids the need to enforce mixed convexity--concavity conditions and facilitates a consistent incorporation of thermodynamic principles. In this contribution, we focus on materials without preferred directions or internal variables. While the formulation is posed in terms of entropy, the temperature is treated as the independent observable, and the entropy is inferred internally through the constitutive relation, enabling thermodynamically consistent modeling without requiring entropy data. Thermodynamic admissibility of the networks is guaranteed by construction. The internal energy and dissipation potential are represented by input convex neural networks, ensuring convexity and compliance with the second law. Objectivity, material symmetry, and normalization are embedded directly into the architecture through invariant-based representations and zero-anchored formulations. We demonstrate the performance of the proposed framework on synthetic and experimental datasets, including purely thermal problems and fully coupled thermomechanical responses of soft tissues and filled rubbers. The results show that the learned models accurately capture the underlying constitutive behavior. All code, data, and trained models are made publicly available via https://doi.org/10.5281/zenodo.19248596.

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 / 2 minor

Summary. The paper presents a physics-based neural network framework for discovering constitutive models in fully coupled thermomechanics. It represents the internal energy u(F,s) and dissipation potential using input convex neural networks (ICNNs) to enforce convexity and the second law by architectural construction, infers entropy internally from observed temperature via T=∂u/∂s, and embeds objectivity, isotropy, and normalization via invariants and zero-anchored formulations. The approach is demonstrated on synthetic and experimental datasets for purely thermal problems and thermomechanical responses of soft tissues and filled rubbers, with all code, data, and models released publicly.

Significance. If the central claims hold, the work provides a route to thermodynamically admissible constitutive modeling that avoids explicit enforcement of mixed convexity-concavity conditions. The use of ICNNs supplies built-in guarantees, and the public release of code, data, and trained models is a clear strength that supports reproducibility and community validation in data-driven thermomechanics.

major comments (1)
  1. [Results section] Results section (demonstrations on experimental datasets): the claim that the learned models 'accurately capture the underlying constitutive behavior' is not supported by any reported quantitative metrics (e.g., stress or temperature prediction errors, thermodynamic residual norms, or comparisons to baseline models); reliance on qualitative descriptions alone leaves the validation of the framework's performance on real data load-bearing but incomplete.
minor comments (2)
  1. [Abstract] The abstract would be strengthened by a brief mention of the quantitative performance measures obtained on the synthetic and experimental cases.
  2. [§2-3] Notation for the deformation gradient and entropy arguments in the ICNN definitions should be cross-checked for consistency with the invariant-based representations described later.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive feedback. We address the single major comment below and will revise the manuscript to strengthen the quantitative validation of the experimental results.

read point-by-point responses

  1. Referee: [Results section] Results section (demonstrations on experimental datasets): the claim that the learned models 'accurately capture the underlying constitutive behavior' is not supported by any reported quantitative metrics (e.g., stress or temperature prediction errors, thermodynamic residual norms, or comparisons to baseline models); reliance on qualitative descriptions alone leaves the validation of the framework's performance on real data load-bearing but incomplete.

    Authors: We agree that quantitative metrics are necessary to rigorously support the performance claims on experimental datasets. In the revised manuscript we will augment the Results section with tables reporting stress and temperature prediction errors (e.g., relative L2 norms and RMSE), thermodynamic residual norms confirming second-law compliance, and direct comparisons against baseline constitutive models. These additions will replace reliance on qualitative descriptions alone. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central construction directly represents internal energy u(F,s) and dissipation potential via input convex neural networks, enforcing convexity and the dissipation inequality by architecture rather than by fitting then renaming. Entropy is inferred from the standard thermodynamic relation T = ∂u/∂s with temperature as observable; this is a direct application of the definition, not a self-referential prediction. Objectivity and isotropy are incorporated via invariants, which is a standard embedding and does not reduce the result to its inputs. No load-bearing self-citations, uniqueness theorems from prior author work, or ansatzes smuggled via citation are invoked. The framework is self-contained against external thermodynamic principles and ICNN properties.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard thermodynamic laws and the representational power of ICNNs; no new entities are postulated.

free parameters (1)
  • neural network weights and architecture hyperparameters
    Weights are fitted to data to match observed thermomechanical responses; specific architecture choices (layers, activations) are selected during training.
axioms (2)
  • domain assumption First and second laws of thermodynamics must hold for admissible constitutive models
    Invoked to justify convexity of internal energy and non-negative dissipation; enforced by ICNN construction.
  • domain assumption Materials under consideration are isotropic with no preferred directions or internal variables
    Explicitly stated as the scope of the contribution.

pith-pipeline@v0.9.0 · 5775 in / 1389 out tokens · 38066 ms · 2026-05-25T06:32:02.868443+00:00 · methodology

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Lean theorems connected to this paper

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

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    Thermodynamic admissibility of the networks is guaranteed by construction. The internal energy and dissipation potential are represented by input convex neural networks, ensuring convexity and compliance with the second law.

  • IndisputableMonolith/Cost.lean Jcost_convexity echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    We therefore adopt the internal energy e and describe the material state in terms of the pair (F, s) ... strict convexity of the internal energy with respect to the entropy (∂²e/∂s² > 0)

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

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