Speaker
Description
In this talk, we address the problem of uniqueness in learning physical laws for systems of partial differential equations (PDEs) [1]. Contrary to most existing approaches, we consider a framework of structured model learning, where existing, approximately correct physical models are augmented with components that are learned from data. Our main result 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 the regularization-minimizing solution of the PDE system. Furthermore, we provide a convergence result showing that model components learned on the basis of incomplete, noisy measurements approximate the ground truth model component 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 is to provide a class of model learning frameworks different to standard settings where uniqueness can be expected in the limit of full measurements.
Reference:
[1] Martin Holler and Erion Morina. On uniqueness in structured model learning, 2024. [ArXiv preprint https://arxiv.org/abs/2410.22009 ]
