论文标题
认证的机器学习：物理知识神经网络的后验误差估计
Certified machine learning: A posteriori error estimation for physics-informed neural networks
论文作者
论文摘要
物理知识的神经网络（PINN）是一种流行的方法，可以将有关物理系统的先验知识纳入学习框架。众所周知，针对较小的训练集，提出更好的概括问题，并且训练更快。在本文中，我们表明，与纯数据驱动的神经网络相比，使用PINN不仅有利于训练性能，而且使我们能够提取有关近似解决方案质量的重要信息。假设PINN训练的基本微分方程是一个普通的微分方程，我们会在PINN预测误差上获得严格的上限。该界限甚至适用于未包含在训练阶段的输入数据，并且没有有关真实解决方案的任何先验知识。因此，我们的后验误差估计是证明PINN的必要步骤。我们将误差估计器示例性应用于两个学术玩具问题，其中一个属于模型预测性控制类别，从而显示了派生结果的实际使用。
Physics-informed neural networks (PINNs) are one popular approach to incorporate a priori knowledge about physical systems into the learning framework. PINNs are known to be robust for smaller training sets, derive better generalization problems, and are faster to train. In this paper, we show that using PINNs in comparison with purely data-driven neural networks is not only favorable for training performance but allows us to extract significant information on the quality of the approximated solution. Assuming that the underlying differential equation for the PINN training is an ordinary differential equation, we derive a rigorous upper limit on the PINN prediction error. This bound is applicable even for input data not included in the training phase and without any prior knowledge about the true solution. Therefore, our a posteriori error estimation is an essential step to certify the PINN. We apply our error estimator exemplarily to two academic toy problems, whereof one falls in the category of model-predictive control and thereby shows the practical use of the derived results.
