This paper addresses the problem of uniqueness in learning physical laws for systems of partial differential equations (PDEs). Contrary to most existing approaches, it considers a framework of structured model learning, where existing, approximately correct physical models are augmented with components that are learned from data. The main results of the paper are a uniqueness and a convergence result that cover a large class of PDEs and a suitable class of neural networks used for approximating the unknown model components. The uniqueness result shows that, in the limit of full, noiseless measurements, a unique identification of the unknown model components as functions is possible as classical regularization-minimizing solutions of the PDE system. This result is complemented by a convergence result showing that model components learned as parameterized neural networks from incomplete, noisy measurements approximate the regularization-minimizing solutions of the PDE system in the limit. These results are possible under specific properties of the approximating neural networks and due to a dedicated choice of regularization. With this, a practical contribution of this analytic paper is to provide a class of model learning frameworks different to standard settings where uniqueness can be expected in the limit of full measurements.

翻译：本文探讨了偏微分方程（PDE）系统物理规律学习中的唯一性问题。与现有大多数方法不同，本研究采用结构化模型学习框架，即在已有近似正确物理模型的基础上，通过数据学习补充模型组件。论文的主要成果包括唯一性定理与收敛性定理，这些定理适用于一大类偏微分方程及用于逼近未知模型组件的特定神经网络类别。唯一性定理表明：在完整无噪声测量的极限条件下，未知模型组件可作为偏微分方程系统的经典正则化极小化解被唯一地识别为函数。该定理辅以收敛性定理证明：通过含噪声的不完整测量数据学习得到的参数化神经网络模型组件，在极限条件下可逼近偏微分方程系统的正则化极小化解。这些结论的实现依赖于逼近神经网络的特定性质以及专门设计的正则化方法。由此，本理论分析论文的实际贡献在于提出了一类不同于标准设置的模型学习框架，在该框架下，完整测量极限条件下的唯一性是可预期的。