We consider the ill-posed inverse problem of identifying a nonlinearity in a time-dependent PDE model. The nonlinearity is approximated by a neural network, and needs to be determined alongside other unknown physical parameters and the unknown state. Hence, it is not possible to construct input-output data pairs to perform a supervised training process. Proposing an all-at-once approach, we bypass the need for training data and recover all the unknowns simultaneously. In the general case, the approximation via a neural network can be realized as a discretization scheme, and the training with noisy data can be viewed as an ill-posed inverse problem. Therefore, we study discretization of regularization in terms of Tikhonov and projected Landweber methods for discretization of inverse problems, and prove convergence when the discretization error (network approximation error) and the noise level tend to zero.

翻译：我们认为,在基于时间的PDE模型中,确定非线性是一个错误的反向问题。非线性问题被神经网络所近似,需要与其他未知的物理参数和未知状态一起确定。因此,不可能建立投入-产出数据对对进行监管的培训过程。建议采用全方位方法,我们避免了培训数据的需求,同时回收所有未知数据。在一般情况下,通过神经网络的近似可作为一种离散计划实现,用吵闹数据进行的培训可被视为一个错误的反向问题。 因此,我们研究以Tikhonov和预测的Landweber方法实现非重叠,以分散处理反向问题,并在离散错误(网络近似误差)和噪音水平趋向为零时证明趋同。