Variational Onsager Neural Networks (VONNs): A thermodynamics-based variational learning strategy for non-equilibrium PDEs

TL;DR

This paper introduces VONNs, a neural network approach grounded in thermodynamics, to learn non-equilibrium PDEs by capturing free energy and dissipation potentials from data, ensuring thermodynamic consistency.

Contribution

It presents a novel variational learning framework based on Onsager's principle, applicable to diverse physical processes and capable of learning PDEs through their variational potentials.

Findings

Successfully applied to phase transformation in proteins

Effective in reduced order modeling of viscoelastic solids

Able to learn diffusion models with non-unique potentials

Abstract

We propose a thermodynamics-based learning strategy for non-equilibrium evolution equations based on Onsager's variational principle, which allows to write such PDEs in terms of two potentials: the free energy and the dissipation potential. Specifically, these two potentials are learned from spatio-temporal measurements of macroscopic observables via proposed neural network architectures that strongly enforce the satisfaction of the second law of thermodynamics. The method is applied to three distinct physical processes aimed at highlighting the robustness and versatility of the proposed approach. These include (i) the phase transformation of a coiled-coil protein, characterized by a non-convex free-energy density; (ii) the one-dimensional dynamic response of a three-dimensional viscoelastic solid, which leverages the variational formulation as a tool for obtaining reduced order models; and (iii) linear and nonlinear diffusion models, characterized by a lack of uniqueness of the free energy and dissipation potentials. These illustrative examples showcase the possibility of learning partial differential equations through their variational action density (i.e., a function instead), by leveraging the thermodynamic structure intrinsic to mechanical and multiphysics problems.