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

非平衡现象，包括应用于复杂多相流体的非等温和等温流体力学模型所描述的非平衡现象，在科学中无处不在。它们需要完善的模型来描述它们的动力学，但对它们的近似算法提出了挑战。本项目旨在为具有热力学一致性的非平衡现象模型建立一个计算框架。所设计的算法将在离散水平上保持所需的特性。此外，这些数值格式将被用于精确和有效地模拟和研究几种非平衡模型的动力学。软件将在高性能计算平台上开发，并向公众开放。学生将参与并通过研究参与项目进行培训。热力学一致（TC）偏微分方程（PDE）系统可以从一般形式（非平衡可逆-不可逆耦合的一般方程）推导出来，包含了科学和工程中非平衡现象的大量模型。该项目将(1)通过利用TC模型的数学结构并使用通用形式主义对其进行重新表述，建立一个设计结构保持、高阶、能量稳定和高效的数值近似来求解TCPDE系统的范例；(2)设计基于物理的深度神经网络框架来求解TCPDE模型，同时保留其结构和特性；(3)应用数值框架研究了几种不同类型的TC模型；(4)开发面向对象、开源、高性能的GPU-CPU混合架构软件包。研究结果将促进非平衡热力学和水动力模型的建模、分析和数值模拟，促进对非平衡现象在广泛应用中的更深入理解。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.