# A Convex Route to Thermoelasticity: Learning Internal Energy and Dissipation

arXiv: https://arxiv.org/abs/2603.28707 · HTML: https://mlsanity.com/paper/2603.28707/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 81 references that are in the map.
- First posted on arXiv 30 Mar 2026.

## Builds on (most cited first, 22 of 22)

- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Respecting causality is all you need for training physics-informed neural networks](https://mlsanity.com/paper/2203.07404/) (2022)
- [Polyconvex anisotropic hyperelasticity with neural networks](https://mlsanity.com/paper/2106.14623/) (2021)
- [A new family of Constitutive Artificial Neural Networks towards automated model discovery](https://mlsanity.com/paper/2210.02202/) (2022)
- [Neural networks meet hyperelasticity: A guide to enforcing physics](https://mlsanity.com/paper/2302.02403/) (2023)
- [Viscoelasticty with physics-augmented neural networks: Model formulation and training methods without prescribed internal variables](https://mlsanity.com/paper/2401.14270/) (2024)
- [Neural network-based multiscale modeling of finite strain magneto-elasticity with relaxed convexity criteria](https://mlsanity.com/paper/2308.16311/) (2023)
- [Viscoelastic Constitutive Artificial Neural Networks (vCANNs) - a framework for data-driven anisotropic nonlinear finite viscoelasticity](https://mlsanity.com/paper/2303.12164/) (2023)
- [Can KAN CANs? Input-convex Kolmogorov-Arnold Networks (KANs) as hyperelastic constitutive artificial neural networks (CANs)](https://mlsanity.com/paper/2503.05617/) (2025)
- [Finite electro-elasticity with physics-augmented neural networks](https://mlsanity.com/paper/2206.05139/) (2022)
- [Theory and implementation of inelastic Constitutive Artificial Neural Networks](https://mlsanity.com/paper/2311.06380/) (2023)
- [Accounting for plasticity: An extension of inelastic Constitutive Artificial Neural Networks](https://mlsanity.com/paper/2407.19326/) (2024)
- [Automated Model Discovery for Tensional Homeostasis: Constitutive Machine Learning in Growth and Remodeling](https://mlsanity.com/paper/2410.13645/) (2024)
- [A generalized dual potential for inelastic Constitutive Artificial Neural Networks: A JAX implementation at finite strains](https://mlsanity.com/paper/2502.17490/) (2025)
- [Inelastic Constitutive Kolmogorov-Arnold Networks: A generalized framework for automated discovery of interpretable inelastic material models](https://mlsanity.com/paper/2602.17750/) (2026)
- [Reduced and All-at-Once Approaches for Model Calibration and Discovery in Computational Solid Mechanics](https://mlsanity.com/paper/2404.16980/) (2024)
- [A physics-augmented neural network framework for finite strain incompressible viscoelasticity](https://mlsanity.com/paper/2511.02959/) (2025)
- [A Complement to Neural Networks for Anisotropic Inelasticity at Finite Strains](https://mlsanity.com/paper/2510.04187/) (2025)
- [Physics-informed Neural Networks for Heterogeneous Poroelastic Media](https://mlsanity.com/paper/2407.03372/) (2024)
- [A dual-stage constitutive modeling framework based on finite strain data-driven identification and physics-augmented neural networks](https://mlsanity.com/paper/2504.15492/) (2025)
- [A data-driven multiscale scheme for anisotropic finite strain magneto-elasticity](https://mlsanity.com/paper/2510.24197/) (2025)

## Built on by (most cited first, 1 of 1)

- [Automatic-Differentiation-Enabled Finite Element Model Updating: Learning Constitutive Neural Networks from Structural Deformation Data](https://mlsanity.com/paper/2606.05199/) (2026)

## Abstract (arXiv)

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.