论文标题
非线性时间依赖性PDE中的学习信息参数识别
Learning-informed parameter identification in nonlinear time-dependent PDEs
论文作者
论文摘要
我们在全面框架中介绍并分析了一种学习信息的参数识别方法（PDE）。基础PDE模型是在具有三个未知数的相当通用环境中制定的：物理参数，状态和非线性。受到机器学习进展的启发，我们通过神经网络近似非线性，该神经网络的参数是从测量数据中学到的。假定后来作为对未知状态的嘈杂观察，并且状态和物理参数均与神经网络的参数同时识别。此外，与经典方法不同，提出的全面设置避免了通过将状态显式处理作为附加变量来构建参数到状态的映射。使用两种不同的算法设置确认了所提出方法的实际可行性：一种基于分析伴随的函数空间算法以及使用标准机器学习算法的纯粹离散设置。
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 later 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.
