A Convex Route to Thermomechanics: Learning Internal Energy and Dissipation

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.

Figures (16)

• Figure 1: Overall network architecture consisting of the subnetworks $\mathrm{MLP}_{s}$, $\mathrm{FICNN}_{s}$, $\mathrm{FICNN}_{\hbox{\boldmath{\scriptsize{$F$}}}{}}$, $\mathrm{FICNN}_{\hbox{\boldmath{\scriptsize{$F$}}}{},s}$, and $\mathrm{PICNN}_{\hbox{\boldmath{\scriptsize{$g$}}}{}}$, together with their interactions. In a first step, the entropy $s$ is predicted by $\mathrm{MLP}_{s}$. Subsequently, $s$ and the deformation gradient $\hbox{\boldmath $F$}{}$ are provided as inputs to the respective subnetworks, from which the first Piola--Kirchhoff stress $\hbox{\boldmath $P$}{}$ is obtained via differentiation. In parallel, the referential heat gradient $\hbox{\boldmath $g$}{}$ is processed by $\mathrm{PICNN}_{\hbox{\boldmath{\scriptsize{$g$}}}{}}$, which is additionally parameterized by $(\hbox{\boldmath $F$}{},s)$, yielding the referential heat flux $\hbox{\boldmath $q$}{}$ through differentiation.
• Figure 2: Boundary value problem for the transient heat diffusion example. A cubic specimen with edge length $1\,\mathrm{mm}$ is subjected to a time-dependent temperature at the exterior boundary and a constant temperature at the interior boundary. The prescribed loading is given by $T(t)=\left[300-293.15\right]\sin\!\left(\frac{\pi}{4}t\right)+293.15,$ while the initial temperature is $T_{\mathrm{init}}=293.15\,\mathrm{K}$. The domain is discretized using $4\times 4\times 4$ elements with edge length $0.25\,\mathrm{mm}$.
• Figure 3: Temperature distribution at two exemplary time steps along the wall thickness direction for the transient heat diffusion problem. Although the temperatures at the interior and exterior boundaries are identical, a nonlinear temperature profile emerges within the wall as a result of transient effects. The edge length of one finite element is $0.25$ mm.
• Figure 4: Left: Training history of the physics-based neural network for the transient heat diffusion problem. Shown are the total loss as well as its individual contributions, including the physics-based residual losses, the loss of the auxiliary network, and the regularization term, over the course of training iterations. Right: Relative activity according to \ref{['eq:activity']} of the individual subnetworks for the transient heat diffusion problem. The diagram shows the normalized contributions of internal energy, dissipation, and auxiliary MLP to the overall activity. It can be observed that the thermal response is dominated by dissipative effects, while the remaining contributions are of significantly smaller magnitude.
• Figure 5: Comparison of the heat flux component $q_1$ 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.