标题： 物理信息神经网络的刚性迁移学习 显示英文标题

标题： Stiff Transfer Learning for Physics-Informed Neural Networks

摘要： 刚性微分方程在各种科学领域中普遍存在，由于其组件的不同时间尺度而带来重大挑战。随着计算能力的增长，物理信息神经网络（PINNs）在建模由微分方程描述的物理过程方面取得了显著进展。尽管它们取得了有希望的结果，但原始PINNs在处理刚性系统时存在局限性，称为失败模式。为了解决这个问题，我们提出了一种新方法，即物理信息神经网络的刚性迁移学习（STL-PINNs），以有效解决刚性常微分方程（ODEs）和偏微分方程（PDEs）。我们的方法涉及在一个低刚度区域内训练一个多头-PINN，并通过迁移学习在高刚度区域内获得最终解。这解决了与PINNs中的刚度相关的失败模式，同时通过计算“一次性”解决方案保持计算效率。与基于PINNs的方法相比，所提出的方法在准确性和速度上表现出优越性，并且在求解刚性参数化的线性和多项式非线性ODEs和PDEs时，计算效率与隐式数值方法相当。此外，我们展示了这种方法的可扩展性以及它在涉及初始条件和强迫函数重新参数化的模拟中提供的优越速度。 显示英文摘要

摘要： Stiff differential equations are prevalent in various scientific domains, posing significant challenges due to the disparate time scales of their components. As computational power grows, physics-informed neural networks (PINNs) have led to significant improvements in modeling physical processes described by differential equations. Despite their promising outcomes, vanilla PINNs face limitations when dealing with stiff systems, known as failure modes. In response, we propose a novel approach, stiff transfer learning for physics-informed neural networks (STL-PINNs), to effectively tackle stiff ordinary differential equations (ODEs) and partial differential equations (PDEs). Our methodology involves training a Multi-Head-PINN in a low-stiff regime, and obtaining the final solution in a high stiff regime by transfer learning. This addresses the failure modes related to stiffness in PINNs while maintaining computational efficiency by computing "one-shot" solutions. The proposed approach demonstrates superior accuracy and speed compared to PINNs-based methods, as well as comparable computational efficiency with implicit numerical methods in solving stiff-parameterized linear and polynomial nonlinear ODEs and PDEs under stiff conditions. Furthermore, we demonstrate the scalability of such an approach and the superior speed it offers for simulations involving initial conditions and forcing function reparametrization.