A Convex Route to Thermomechanics: Learning Internal Energy and Dissipation
2026-03-30 • Computational Engineering, Finance, and Science
Computational Engineering, Finance, and ScienceArtificial Intelligence
AI summaryⓘ
The authors developed a neural network approach to model how materials respond when both temperature changes and mechanical forces are involved. Instead of using common energy descriptions, they focus on internal energy and energy lost to heat (dissipation), which helps follow the rules of thermodynamics more easily. Their method uses special neural networks designed to guarantee physically correct behavior and respects material symmetries. They tested their approach with both simulated and real data, showing it can accurately predict material responses without needing direct measurements of entropy. The authors also provided all their data and code for others to use.
thermomechanicsconstitutive modelinternal energydissipation potentialentropyconvex neural networksthermodynamic consistencymaterial symmetryobjectivitysoft tissue mechanics
Authors
Hagen Holthusen, Paul Steinmann, Ellen Kuhl
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.