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A Convex Route to Thermomechanics: Learning Internal Energy and Dissipation

This paper introduces a physics-based neural network framework that utilizes input convex neural networks to learn constitutive models for fully coupled thermomechanics by representing internal energy and dissipation potential in terms of deformation and entropy, thereby ensuring thermodynamic admissibility and objectivity without requiring entropy data.

Original authors: Hagen Holthusen, Paul Steinmann, Ellen Kuhl

Published 2026-03-31

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Hagen Holthusen, Paul Steinmann, Ellen Kuhl

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to teach a computer to understand how materials behave when they get hot and get stretched at the same time. Think of a rubber band: when you stretch it, it gets warm. When you heat it, it might shrink or stretch differently. This complex dance between heat and motion is called thermomechanics.

For a long time, scientists have tried to write mathematical "recipes" (constitutive models) to predict this behavior. But these recipes are often messy, hard to write, and sometimes break the fundamental laws of physics (like the law that says you can't create energy out of nothing).

This paper introduces a new, smarter way to teach computers these recipes using Physics-Based Neural Networks. Here is how they did it, explained with some everyday analogies.

1. The Old Way vs. The New Way

The Old Way (The Helmholtz Energy):
Imagine trying to bake a cake where the recipe requires you to make the batter thicker when you add sugar, but thinner when you add flour, and you have to do this in a very specific, contradictory way to make the cake rise. In physics, this is like using "Helmholtz energy." It's a powerful tool, but it forces the computer to follow a "mixed" rule: it must be convex (curving up) in some ways and concave (curving down) in others. This is like trying to balance a pencil on its tip while riding a unicycle—it's mathematically tricky and prone to falling over (failing).

The New Way (Internal Energy & Dissipation):
The authors decided to change the recipe. Instead of the tricky "Helmholtz" cake, they used Internal Energy (the total energy stored in the material) and a Dissipation Potential (the energy lost as heat or friction).

The Analogy: Think of a bank account.
- Internal Energy is your savings. It always goes up when you deposit money (add heat or stretch). It's a simple, "convex" curve (like a bowl).
- Dissipation is the transaction fee or the heat generated when you move money around. It's also a simple, "convex" curve.
- Why it's better: By using two simple, "bowl-shaped" curves instead of one complex, twisted shape, the computer never gets confused. It naturally obeys the laws of physics without needing constant supervision.

2. The Secret Ingredient: Entropy vs. Temperature

In physics, there are two ways to describe heat: Temperature (what you feel with a thermometer) and Entropy (a measure of disorder).

The Problem: Experiments give us Temperature. But the math works best with Entropy. Usually, you have to guess the Entropy based on the Temperature, which is like trying to guess the exact number of people in a room just by looking at the thermostat.
The Solution: The authors built a "smart assistant" (a small neural network) that learns to guess the Entropy based on the Temperature and how much the material is stretched.
- The Metaphor: Imagine a translator. The computer speaks "Entropy," but the human world speaks "Temperature." This translator learns the language perfectly so the computer can do its math in its native tongue while still understanding what the human is saying.

3. The Architecture: Building a "Physics-Proof" Robot

The authors didn't just throw a generic AI at the problem. They built the AI's brain (the neural network) with specific "guardrails" baked into its DNA.

Input Convex Neural Networks (ICNNs): They used a special type of AI that cannot produce a non-convex shape. No matter what data you feed it, it will always output a "bowl" shape. This guarantees that the Second Law of Thermodynamics (entropy always increases) is never broken.
Zero-Anchored: They made sure that if you do nothing (no stretch, no heat change), the energy and stress are exactly zero. It's like a car that won't move unless you press the gas.
Symmetry: They taught the AI that a sphere looks the same from every angle. This prevents the AI from learning weird, impossible behaviors where a material acts differently just because you rotated it.

4. The Training: Learning from Synthetic and Real Data

They tested this new framework on four different scenarios:

Pure Heat Flow: Like heat moving through a wall. The AI learned how heat diffuses without any mechanical stretching.
Porcine Tissue (Pork): Real data from pig tissue. The AI learned how meat gets softer or stiffer as it heats up.
Rubber: Real data from rubber filled with carbon. The AI learned the complex way rubber stiffens or softens with temperature.
The "Spring" Test: A fully coupled simulation where they stretched a spring-like object while heating it.
- The Result: The AI successfully predicted how the object would react to forces and heat, even for shapes and situations it had never seen before. It learned the rules of the material, not just memorized the data points.

5. The Catch (Limitations)

The paper admits that while the AI is great, it's only as good as the data it's fed.

The "Invisible" Effect: In the spring test, the AI struggled to predict tiny temperature changes caused only by stretching (deformation-induced heating).
The Analogy: Imagine trying to hear a whisper in a hurricane. The "whisper" is the tiny heat generated by stretching, and the "hurricane" is the massive heat from the environment. The AI got lost in the noise. To fix this, we need to give the AI data where that whisper is louder, or teach it to listen more carefully.

Summary

This paper presents a new, robust way to teach computers the laws of physics. By switching from a complex, contradictory mathematical recipe to a simpler, "bowl-shaped" one (Internal Energy + Dissipation), and by building "guardrails" directly into the AI's architecture, they created a system that:

Cannot break the laws of thermodynamics.
Learns from real-world data (like rubber and tissue).
Generalizes to new situations it hasn't seen before.

It's like teaching a child to ride a bike not by giving them a physics textbook on gyroscopic stability, but by building a bike with training wheels that physically prevent them from falling over, allowing them to learn balance naturally.

1. Problem Statement

The paper addresses the challenge of discovering constitutive models for fully coupled thermomechanics using data-driven methods. While classical continuum thermomechanics relies on the Helmholtz free energy ( $\psi$ ), this approach presents significant difficulties for neural network-based discovery:

Mixed Convexity Constraints: Thermodynamic stability requires the Helmholtz energy to be convex with respect to deformation but concave with respect to temperature. Enforcing these mixed curvature conditions within neural network architectures is non-trivial and often leads to unstable training or non-physical predictions.
Dissipation Modeling: Existing strongly-enforcing frameworks (e.g., Constitutive Artificial Neural Networks) have primarily focused on thermoelasticity. They struggle to generalize to fully coupled processes involving irreversible dissipation (heat conduction, plasticity) in a general setting.
Data Availability: Experimental data typically provides temperature ( $T$ ) and deformation ( $F$ ), but not entropy ( $s$ ). A framework is needed that respects thermodynamic principles while treating temperature as the primary observable.

2. Methodology

The authors propose a physics-based neural network framework that shifts the primary constitutive potentials from Helmholtz energy to Internal Energy ( $e$ ) and a Dissipation Potential ( $\phi$ ).

A. Theoretical Framework

Constitutive Potentials: The material state is defined by the pair $(F, s)$ $(F, s)$ , where $F$ $F$ is the deformation gradient and $s$ $s$ is the referential entropy density.
- Internal Energy: $e = e(F, s)$
- Dissipation Potential: $\phi = \phi(\dot{F}, \dot{s}, g; F, s)$ , where $g$ is the referential thermal gradient.
Thermodynamic Admissibility: The framework is built upon the Clausius-Duhem inequality. By defining stress ( $P$ ) and temperature ( $T$ ) as derivatives of the internal energy ( $P = \partial e / \partial F$ , $T = \partial e / \partial s$ ), the second law of thermodynamics is satisfied by construction if the potentials are convex.
Temperature as Observable: Although the model is formulated in terms of entropy, temperature is treated as the independent observable. Entropy is inferred internally via the constitutive relation $T = \partial e / \partial s$ .

B. Neural Network Architecture

The authors utilize Input Convex Neural Networks (ICNNs) to guarantee the convexity of the potentials, which is essential for thermodynamic stability.

Internal Energy ( $e$ ): Represented by a Fully Input Convex Neural Network (FICNN). It is formulated as a function of the principal invariants of the Right Cauchy-Green tensor ( $C$ $C$ ) and entropy ( $s$ $s$ ) to ensure objectivity and material symmetry (isotropy). The architecture includes:
- A joint convex branch for $(F, s)$ .
- Separable convex branches for $F$ and $s$ individually.
- A coercive growth term to prevent volumetric collapse.
- Zero-anchoring: Ensures the energy and stress vanish in the reference state.
Dissipation Potential ( $\phi$ ): Represented by a Partially Input Convex Neural Network (PICNN). It is convex with respect to the thermal gradient $g$ but parameterized by the state variables $(F, s)$ . This ensures non-negative dissipation.
Auxiliary Entropy Network: An auxiliary Multilayer Perceptron (MLP) is used during training to approximate entropy $s$ from inputs $(F, T)$ , accelerating the learning process. During inference, an iterative Newton-Raphson solver enforces the exact state law $T = \partial e / \partial s$ .

C. Training Strategy

Loss Function: The training objective is a physics-based loss derived from the weak form of the balance equations (linear momentum and internal energy).
- Physics Loss: Minimizes the residual of the discretized balance laws (Dirichlet and Neumann boundaries).
- Auxiliary Loss: Enforces consistency between the predicted entropy and the state law.
- Regularization: L1 and L2 penalties are applied to encourage sparsity and smoothness.
Data: The framework is tested on both synthetic data (generated from known Helmholtz-based models) and experimental datasets (porcine tissue and carbon-filled rubber).

3. Key Contributions

Convex Route to Thermomechanics: The paper introduces a novel formulation using Internal Energy and Dissipation Potential, avoiding the difficult mixed convexity-concavity constraints of Helmholtz energy. This allows for the use of standard ICNN architectures to guarantee thermodynamic consistency.
Fully Coupled Framework: Unlike previous works limited to thermoelasticity, this framework handles fully coupled thermomechanical processes, including heat conduction and deformation-induced heating, within a single unified architecture.
Architecture-Embedded Physics: Fundamental principles (objectivity, isotropy, convexity, normalization) are hard-coded into the network structure rather than enforced via soft penalties in the loss function.
Handling Unobserved Variables: The method successfully learns constitutive laws using only observable data ( $F, T$ ) by inferring the unobserved entropy ( $s$ ) through the network's internal structure.

4. Results

The framework was validated across four distinct scenarios:

Synthetic Heat Diffusion: The model successfully learned the transient heat conduction behavior (Fourier's law) and accurately predicted heat flux, demonstrating that the dissipation potential correctly captures thermal transport.
Porcine Tissue (Experimental): The model captured temperature-dependent mechanical softening in soft tissue. It implicitly identified a reference temperature consistent with stress-free states, despite the data lacking explicit thermal expansion information.
Carbon-Filled Rubber (Experimental): The model reproduced the temperature-dependent stiffening/softening behavior. While it struggled to capture a specific "thermoelastic inversion" at very high temperatures (likely due to data limitations regarding reference configurations), it performed comparably to Helmholtz-based approaches in other regimes.
Fully Coupled Structural Problem (Synthetic): The model was trained on a plate with an elliptic hole and tested on a spring-like specimen with unseen boundary conditions.
- Success: It accurately predicted mechanical reaction forces and coupled thermal responses under combined loading.
- Limitation: It struggled to capture deformation-induced self-heating (weak coupling effects) when the temperature changes were extremely small ( $\sim 10^{-2}$ K), as these signals were overshadowed by dominant thermal terms during training.

5. Significance and Conclusion

This work represents a significant step forward in Automated Model Discovery for multiphysics problems.

Theoretical Robustness: By shifting to Internal Energy, the authors provide a mathematically rigorous path to ensure thermodynamic admissibility without complex constraint handling.
Practical Applicability: The framework is capable of learning complex, non-linear, and temperature-dependent behaviors from experimental data without requiring explicit knowledge of the underlying constitutive equations.
Future Directions: The study highlights that while the framework is powerful, the quality of the discovered model is strictly limited by the completeness and scale of the training data. Weak coupling effects (like self-heating) require targeted excitation strategies or specialized training regimes to be identifiable.

In summary, the paper establishes a "convex route" that makes the discovery of thermodynamically consistent, fully coupled constitutive models feasible and robust, bridging the gap between data-driven machine learning and classical continuum thermodynamics.