On uniqueness in structured model learning

TL;DR

This paper proves that in structured model learning for PDEs, the unknown model components can be uniquely identified and accurately approximated using neural networks under certain conditions, even with noisy data.

Contribution

It establishes theoretical guarantees for uniqueness and convergence in neural network-based structured PDE model learning, a novel insight in the field.

Findings

Uniqueness of model components in the noiseless, full-measurement limit.

Convergence of neural network approximations to regularization-minimizing solutions.

Frameworks where uniqueness can be expected with full measurements.

Abstract

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.