Abstract
In this paper, we put forward a neural network framework to solve the nonlinear hyperbolic systems. This framework, named relaxation neural networks(RelaxNN), is a simple and scalable extension of physics-informed neural networks(PINN). It is shown later that a typical PINN framework struggles to handle shock waves that arise in hyperbolic systems' solutions. This ultimately results in the failure of optimization that is based on gradient descent in the training process. Relaxation systems provide a smooth asymptotic to the discontinuity solution, under the expectation that macroscopic problems can be solved from a microscopic perspective. Based on relaxation systems, the RelaxNN framework alleviates the conflict of losses in the training process of the PINN framework. In addition to the remarkable results demonstrated in numerical simulations, most of the acceleration techniques and improvement strategies aimed at the standard PINN framework can also be applied to the RelaxNN framework.
Abstract (translated)
在本文中，我们提出了一个神经网络框架来解决非线性超几何系统。这个框架名为放松神经网络（RelaxNN），是物理正交神经网络（PINN）的简单且可扩展版本。后来发现，典型的PINN框架很难处理超几何系统解中产生的应激波。这最终导致基于梯度的训练过程中优化失败。放松系统提供了一个平滑的渐进到离散解，只要期望从微观角度解决问题，宏观问题就可以解决。基于放松系统，RelaxNN框架减轻了PINN框架在训练过程中损失的冲突。除了在数值仿真中展示的惊人的结果外，大多数加速技术和改进策略都可以应用到RelaxNN框架中。
URL
https://arxiv.org/abs/2404.01163
