We propose a variational learning strategy for the discovery of non-equilibrium equations, through the variational action density from which these equations may be derived. The strategy is based on the so-called Onsager’s variational principle, which may be written as a function of the free energy and dissipation potential, and utilizes neural network architectures that strongly guarantee thermodynamic consistency. The method is applied to three distinct illustrative examples, aimed at showcasing distinct important features of the strategy proposed. These encompass (i) the phase transformation occurring in coiled-coil protein, where the free energy density is non-convex, (ii) the discovery of a reduced order model for the dynamic response of a viscoelastic material, which utilizes the variational structure as a tool for approximation, and (iii) a linear and nonlinear diffusion model, where both evolution equations may be obtained from distinct free energies and dissipation potentials (i.e., the action is not unique).
 S. Huang, Z. He, B. Chem and C. Reina, Variational Onsager Neural Networks (VONNs): A thermodynamics-based variational learning strategy for non-equilibrium PDEs, Journal of the Mechanics and Physics of Solids 163, 104856-104881 (2022).