OAC Core: Geometry-aware and Deep Learning-based Cyberinfrastructure for Scalable Modeling of Solids and Fluids

固体和流体力学中的许多现象都是用复偏微分方程(PDE)来模拟的。由于通过传统的数值方法来求解这些偏微分方程组的费用高得令人望而却步，因此越来越多地使用诸如深度神经网络(DNN)之类的仿真器来逼近偏微分方程组的解。虽然在这方面已经花费了大量的努力，但现有的技术提供了昂贵的解决方案，这些解决方案不能在不同的应用程序之间转移，也不能扩展到复杂的PDE。该项目旨在使用分而治之的方法解决这些限制。在将为该项目开发的框架中，该项目将首先建立一个DNN库，以解决称为基因组的小域上的单物理PDE系统。然后，为了解决大型不可见区域上的多物理偏微分方程组，该项目将开发一种自适应方法，将DNN耦合并组装它们的基因组预测，以便在整个领域满足控制方程。该项目预计，预先训练的DNN和耦合机制将使学者在没有获得科学机器学习所需的硬件或知识的情况下大大受益。该框架的可转移性有可能减少与使用DNN来解决偏微分方程的现有技术相关的高计算成本的碳足迹，为科学研究和整个社会提供巨大的好处。该项目将建立学习基因组算子(LEGO)，使用贝叶斯强化学习(BRL)进行泛化，即：(1)模拟多物理系统，和/或(2)实现时空可转移性和可扩展性。这项工作的贡献有望通过预先训练的DNN实现对固体和流体行为的动态逼近，从而在提高准确性和可扩展性的同时消除长时间的训练时间。乐高框架是假设驱动的，并利用了独特地利用并行和异构机的域分解方法的数学。该框架通过首先将区域分解成称为基因组的小的子域来求解具有任意初始和边界条件的大区域中的偏微分方程组。然后，通过预先训练的乐高积木来近似每个基因组中的解，使得基因组预测的集合近似于大区域中的解。本质上，乐高积木模拟基因组中的不同物理现象(例如，材料变形或流体流动)，而BRL代理耦合乐高积木以模拟多物理现象和/或在时空上扩展乐高积木的预测，同时保持基因组中溶液的一致性。为了实现实时和稳健的性能以及高可转移性和可扩展性，该框架(1)使用混合精度计算和硬件加速器，(2)结合几何感知学习算法，(3)在数学上估计解决方案汇编期间传播的误差。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。

Many phenomena in solid and fluid mechanics are modeled via complex partial differential equations (PDEs). Since solving these PDEs via traditional numerical methods is prohibitively expensive, emulators such as deep neural networks (DNNs) are increasingly employed to approximate PDE solutions. While significant effort has been expended in this direction, existing technologies provide expensive solutions that are not transferable across different applications or scalable to complex PDEs. This project aims to address these limitations using a divide and conquer approach. In the framework that will be developed for the project, the project will first build a library of DNNs that solve single-physics PDE systems over small domains called genomes. Then, to solve multi-physics PDEs over large unseen domains, the project will develop an adaptive method that couples the DNNs and assembles their genome-wise predictions such that the governing equations are satisfied in the entire domain. The project expects that the pre-trained DNNs and coupling mechanism will greatly benefit scholars without access to the hardware or knowledge that are needed for scientific machine learning. The transferability of the framework has the potential to reduce the carbon footprint of the high computing costs that are associated with existing technologies that use DNNs to solve PDEs, providing great benefits to both scientific research and to society as a whole.The project will build LEarned Genomic Operators (LEGOs) that use Bayesian reinforcement learning (BRL) for generalization, i.e., for (1) emulating multi-physics systems, and/or (2) achieving spatiotemporal transferability and scalability. The contributions of this work are expected to enable on-the-fly approximation of the behavior of solids and fluids via pre-trained DNNs, thus eliminating long training times while increasing accuracy and scalability. The LEGO framework is hypothesis-driven and leverages the mathematics of domain decomposition methods that uniquely exploit parallel and heterogeneous machines. The framework solves a PDE system in a large domain with arbitrary initial and boundary conditions by first decomposing the domain into small subdomains called genomes. Then, the solution in each genome is approximated via pre-trained LEGOs such that the assembly of the genome-wise predictions approximates the solution in the large domain. In essence, the LEGOs model different physical phenomena (e.g., material deformation or fluid flow) in genomes while the BRL agent couples the LEGOs to model multi-physics phenomena and/or spatiotemporally extends the predictions of LEGOs while preserving solution consistency across the genomes. To achieve real-time and robust performance with high transferability and scalability, the framework (1) uses mixed-precision computing and hardware accelerators, (2) incorporates geometry-aware learning algorithms, and (3) mathematically estimates the propagated errors during solution assembly.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.