We introduce and analyze a method of learning-informed parameter identification for partial differential equations (PDEs) in an all-at-once framework. The underlying PDE model is formulated in a rather general setting with three unknowns: physical parameter, state and nonlinearity. Inspired by advances in machine learning, we approximate the nonlinearity via a neural network, whose parameters are learned from measurement data. The latter is assumed to be given as noisy observations of the unknown state, and both the state and the physical parameters are identified simultaneously with the parameters of the neural network. Moreover, diverging from the classical approach, the proposed all-at-once setting avoids constructing the parameter-to-state map by explicitly handling the state as additional variable. The practical feasibility of the proposed method is confirmed with experiments using two different algorithmic settings: A function-space algorithm based on analytic adjoints as well as a purely discretized setting using standard machine learning algorithms.

我们介绍并分析了一种在一次性框架中学习偏微分方程（PDE）参数识别的方法。底层偏微分方程模型是在一个相当通用的环境中制定的，具有三个未知数：物理参数、状态和非线性。受机器学习进步的启发，我们通过神经网络近似非线性，其参数是从测量数据中学习的。假设后者是对未知状态的噪声观测，并且状态和物理参数都与神经网络的参数同时识别。此外，与经典方法不同，所提出的一次性设置通过显式地将状态处理为附加变量来避免构造参数到状态映射。