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

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.

Variational methods and deep Ritz method for active elastic solids

“…The deep Ritz method mentioned above for the statics of active matter is developed 45 by combining Ritz's variational method of approximation 42 with deep learning methods that are based on deep neural networks and stochastic gradient descent algorithms. Similar deep learning methods can also be developed [48][49][50][74][75][76] for the dynamics of soft matter and active matter by combining the variational method of approximation based on Onsager's variational principle (OVP) 43 with deep learning methods. 77 Furthermore, the input of the neural networks should generally include both spatial and temporal coordinates; the output can be not only displacement fields, but also other slow variables such as polarization, concentration, etc.…”

Empowering engineering with data, machine learning and artificial intelligence: a short introductive review

“…In some cases the learning problem is formulated from the differential form of the GENERIC model [43]. However, variational formulations are also available, as the one of Herglotz (contact geometry) [125] or the one making use of the Onsager variational formulation that involves the so-called Rayleighian [54].…”

Variational methods and deep Ritz method for active elastic solids

“…The deep Ritz method mentioned above for the statics of active matter is developed [44] by combining Ritz's variational method of approximation [42] with deep learning methods that are based on deep neural networks and stochastic gradient descent algorithms. Similar deep learning methods can also be developed [45][46][47][70][71][72] for the dynamics of soft matter and active matter by combining the variational method of approximation based on Onsager's variational principle (OVP) [43] with deep learning methods. Furthermore, the input of the neural networks should generally include both spatial and temporal coordinates; the output can be not only displacement fields, but also other slow variables such as polarization, concentration, etc.…”