基于深度学习和物理信息神经网络的微分方程解

Differential Equation Solution Based on Deep Learning and Physical Information Neural Network

深度神经网络越来越多地被用于解决经典的应用数学问题。由于神经网络的万能函数逼近特性和强大的表现力,深度学习最近成为了科学计算的一种新范式。基于深度学习和物理信息神经网络,以伯格斯方程作为物理模型,通过物理信息神经网络损失的软约束来捕获模型方程的时间演化。结合实际标签数据、边界条件和初值方面的约束和模型方程的残差约束,以检索偏微分方程的实际解,重建了伯格斯方程的反问题。

Deep neural networks are increasingly used to solve classical applied mathematical problems.Due to the universal function approximation and powerful expressiveness of neural network,deep learning has recently become a new paradigm of scientific computing.On the basis of deep learning and physical information neural network,and by taking Burgers equation as the physical model,the time evolution of the model equation is captured by the soft constraint of physical information neural network loss.Combined with the constraints of actual label data,boundary conditions and initial value as well as the residual constraints of the model equation,the actual solution of the partial differential equation is retrieved,and the inverse problem of the Burgers Equation is reconstructed.