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.

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用于求解参数化偏微分方程的经认证的基于小波的物理信息神经网络

物理信息神经网络 (PINN) 经常用于偏微分方程 (PDE) 的数值近似。本文的目标是构造 PINN 以及可计算的误差上限，这与参数化偏微分方程 (PPDE) 的模型简化特别相关。为此，我们建议在损失函数和误差界限的自适应小波展开方面使用残差展开系数的加权和。此处展示了使用标准变分和最佳稳定超弱公式的椭圆 PPDE 的方法。数值例子表明基于小波的误差界具有很好的定量有效性。