论文标题
认证的机器学习：严格的后验错误范围为PDE定义的PINNS
Certified machine learning: Rigorous a posteriori error bounds for PDE defined PINNs
论文作者
论文摘要
对于纯粹的数据驱动和物理信息的方法，机器学习中的预测误差量化已被忽略了神经网络的大多数方法论研究。除了对神经网络的近似功能的统计研究和通用结果，我们还对物理信息信息的预测错误提出了严格的上限。可以在不了解真实解决方案的情况下计算该界限，并且只有通过偏微分方程控制的基本动力系统特征的先验可用信息。我们将其应用于限制的四个问题：传输方程，热方程，navier-stokes方程和klein-gordon方程。
Prediction error quantification in machine learning has been left out of most methodological investigations of neural networks, for both purely data-driven and physics-informed approaches. Beyond statistical investigations and generic results on the approximation capabilities of neural networks, we present a rigorous upper bound on the prediction error of physics-informed neural networks. This bound can be calculated without the knowledge of the true solution and only with a priori available information about the characteristics of the underlying dynamical system governed by a partial differential equation. We apply this a posteriori error bound exemplarily to four problems: the transport equation, the heat equation, the Navier-Stokes equation and the Klein-Gordon equation.
