A high order explicit time finite element method for the acoustic wave equation with discontinuous coefficients
这是一篇使用 VPinn 来求解 Navier-Stokes 方程的论文。
我们先来看看这篇论文的摘要：
摘要
英文原文：
We present TSA-PINN, a novel Physics-Informed Neural Network (PINN) that leverages a Trainable Sinusoidal Activation (TSA) mechanism to approximate solutions to the Navier-Stokes equations. By incorporating neuronwise sinusoidal activation functions with trainable frequencies and a dynamic slope recovery mechanism, TSAPINN achieves superior accuracy and convergence. Its ability to dynamically adjust activation frequencies enables efficient modeling of complex fluid behaviors, reducing training time and computational cost. Our testing goes beyond canonical problems, to study less-explored and more challenging scenarios, which have typically posed difficulties for prior models. Various numerical tests underscore the efficacy of the TSA-PINN model across five different scenarios. These include steady-state two-dimensional flows in a lid-driven cavity at two different Reynolds numbers; a cylinder wake problem characterized by oscillatory fluid behavior; and two time-dependent three-dimensional turbulent flow cases. In the turbulent cases, the focus is on detailed near-wall phenomenaincluding the viscous sub-layer, buffer layer, and log-law region—as well as the complex interactions among eddies of various scales. Both numerical and quantitative analyses demonstrate that TSA-PINN offers substantial improvements over conventional PINN models. This research advances physics-informed machine learning, setting a new benchmark for modeling dynamic systems in scientific computing and engineering.
翻译：
我们介绍 TSA-PINN，一种新型物理知情神经网络（PINN），利用可训练正弦激活（TSA）机制近似纳维-斯托克斯方程的解。通过结合具有可训练频率的神经元正弦激活功能和动态斜坡恢复机制，TSAPINN 实现了卓越的准确性和收敛性。其动态调节激活频率的能力使复杂流体行为的高效建模成为可能，从而缩短训练时间和计算成本。我们的测试超越了典型问题，还研究了较少被探索且更具挑战性的情景，这些通常对以往模型来说是困难。各种数值测试强调了 TSA-PINN 模型在五种不同情景中的有效性。这些包括在两个不同雷诺数下，盖子驱动腔内的稳态二维流动;一个以振荡流体行为为特征的圆柱尾流问题;以及两个时间相关的三维湍流情况。湍流情况下，重点关注详细的近壁现象，包括粘性亚层、缓冲层和对数定律区域——以及不同尺度涡流之间的复杂相互作用。数值和定量分析均表明，TSA-PINN 相较传统 PINN 模型有显著改进。这项研究推动了基于物理的机器学习，为科学计算和工程中动态系统建模树立了新标杆。