论文标题
基于认证的小波的物理信息神经网络，用于解决参数化偏微分方程的解决方案
A certified wavelet-based physics-informed neural network for the solution of parameterized partial differential equations
论文作者
论文摘要
物理知情的神经网络（PINN）经常用于部分微分方程（PDE）的数值近似。本文的目的是构造PINN以及误差的可计算上限，这与降低参数化PDE（PPDE）特别相关。为此，我们建议使用损耗函数的自适应小波膨胀和误差结合的自适应小波膨胀的加权总和。 此处显示了使用标准变分和最佳稳定的超湿配方的椭圆PPDE的方法。数值示例显示基于小波的误差绑定的定量效果非常好。
Physics Informed Neural Networks (PINNs) have frequently been used for the numerical approximation of Partial Differential Equations (PDEs). The goal of this paper is to construct PINNs along with a computable upper bound of the error, which is particularly relevant for model reduction of Parameterized PDEs (PPDEs). To this end, we suggest to use a weighted sum of expansion coefficients of the residual in terms of an adaptive wavelet expansion both for the loss function and an error bound. This approach is shown here for elliptic PPDEs using both the standard variational and an optimally stable ultra-weak formulation. Numerical examples show a very good quantitative effectivity of the wavelet-based error bound.
