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动力学模型所描述的非平衡现象，在科学中是普遍存在的。它们需要完善的模型来描述它们的动态，但对它们的近似算法构成了挑战。该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。要设计的算法将在离散级别上保留所需的属性。此外，这些数值格式将被用来准确和有效地模拟和研究几类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提供。学生将通过参与该项目的研究来参与和培训。热力学一致偏微分方程组(PDE)系统是由一般形式(非平衡可逆-不可逆耦合的一般方程)派生出来的，它包含了科学和工程中关于非平衡现象的一大类模型。该项目将(1)通过利用TC模型的数学结构并使用通用形式对其进行重新描述，建立一个用于设计保结构、高阶、能量稳定和高效的数值逼近的范式来求解TCPDE系统；(2)设计物理信息的深度神经网络框架来求解TCPDE模型，同时保持其结构和性质；(3)应用该数值框架来研究几类TC模型；以及(4)开发一个面向对象的、开源的、高性能的混合GPU-CPU体系结构软件包。这些成果将推进非平衡热力学和流体动力学模型的建模、分析和数值模拟，促进对广泛应用中的非平衡现象的更深入理解。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.