A Comparative Investigation of Thermodynamic Structure-Informed Neural Networks

Abstract

Physics-informed neural networks (PINNs) offer a unified framework for solving both forward and inverse problems of differential equations, yet their performance and physical consistency strongly depend on how governing laws are incorporated. In this work, we present a systematic comparison of different thermodynamic structure-informed neural networks by incorporating various thermodynamics formulations, including Newtonian, Lagrangian, and Hamiltonian mechanics for conservative systems, as well as the Onsager variational principle and extended irreversible thermodynamics for dissipative systems. Through comprehensive numerical experiments on representative ordinary and partial differential equations, we quantitatively evaluate the impact of these formulations on accuracy, physical consistency, noise robustness, and interpretability. The results show that Newtonian-residual-based PINNs can reconstruct system states but fail to reliably recover key physical and thermodynamic quantities, whereas structure-preserving formulation significantly enhances parameter identification, thermodynamic consistency, and robustness. These findings provide practical guidance for principled design of thermodynamics-consistency model, and lay the groundwork for integrating more general nonequilibrium thermodynamic structures into physics-informed machine learning.

Figures (9)

• Figure 1: The schematic diagram of the Lagrangian-mechanics, Hamiltonian-mechanics, Onsager's Variational Principle and Extended Irreversible Thermodynamics informed neural networks.
• Figure 2: Results for the forward problem of ideal mass spring. The first column presents the predicted and true Hamiltonian, the second column shows the Lagrangian, the third column is the trajectory of the system, and the last column illustrates the phase diagram of $q$ and $p$. Panels (a-c) correspond to the results obtained using the NM-PINNs, LM-PINNs, and HM-PINNs, respectively.
• Figure 3: Results for the inverse problem of ideal mass spring and ideal pendulum. (a) For the ideal mass spring system, the learned distribution of $k/m$ obtained from the three models after ten random initializations under different noise levels. The black dashed line represents the ground truth value of $k/m$. (b) Corresponding results for ideal pendulum systems.
• Figure 4: Results for the forward problem of ideal pendulum. The first column presents the predicted and true Hamiltonian, the second column shows the Lagrangian, the third column is the trajectory of the system, and the last column illustrates the phase diagram of $q$ and $p$. Panels (a-c) correspond to the results obtained using the NM-PINNs, LM-PINNs, and HM-PINNs, respectively.
• Figure 5: Results for the forward problem of double pendulum. The first column presents the predicted and true Hamiltonian, the second column shows the Lagrangian, and the third column is the trajectory of the system. Panels (a-c) correspond to the results obtained using the NM-PINNs, LM-PINNs, and HM-PINNs, respectively.