On uniqueness in structured model learning

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 result of the paper is a uniqueness result that covers 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 idealized setting of full, noiseless measurements, a unique identification of the unknown model components is possible as regularization-minimizing solution of the PDE system. Furthermore, the paper provides a convergence result showing that model components learned on the basis of incomplete, noisy measurements approximate the regularization-minimizing solution 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.