Physics-Informed Structure-Preserving Numerical Approximations of Thermodynamically Consistent Models for Non-equilibrium Phenomena

Non-equilibrium phenomena, including those described by non-isothermal and isothermal hydrodynamic models with applications to complex multiphase fluids, are ubiquitous in science. They require well-developed models to describe their dynamics but pose challenges to the algorithms for their approximations. This project aims to establish a computational framework for models of non-equilibrium phenomena that have the property of being thermodynamically consistent. The algorithms to be designed will preserve the desired properties at the discrete levels. Furthermore, these numerical schemes will be utilized to simulate and investigate the dynamics of several classes of non-equilibrium models in an accurate and efficient way. Software will be developed on high-performance computing platforms and made available to the public. Students will be involved and trained through research involvement in the project. Thermodynamically consistent (TC) partial differential equation (PDE) systems, derivable from the GENERIC formalism (General Equation for Non-Equilibrium Reversible-Irreversible Coupling), encompass a large class of models in science and engineering for non-equilibrium phenomena. The project will (1) establish a paradigm for designing structure-preserving, high order, energy stable, and efficient numerical approximations to solve TCPDE systems by exploiting the mathematical structure of the TC models and reformulating them using the GENERIC formalism; (2) design physics-informed deep neural network frameworks to solve TCPDE models while preserving their structures and properties; (3) apply the numerical framework to investigate several classes of TC models; and (4) develop an object-oriented, open-source, and high-performance software package for hybrid GPU-CPU architectures. The outcomes will advance modeling, analysis, and numerical simulations of non-equilibrium thermodynamic and hydrodynamic models, fostering a deeper understanding of non-equilibrium phenomena in a wide range of applications. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.