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[2410.22009] On uniqueness in structured model learning
Summary
This paper explores the uniqueness in structured model learning for systems of partial differential equations (PDEs), proposing a framework that integrates existing models with learned components from data.
Why It Matters
Understanding uniqueness in structured model learning is crucial for improving the accuracy of physical models derived from data. This research offers insights into how neural networks can be effectively used to approximate unknown model components, which is significant for fields reliant on PDEs, such as physics and engineering.
Key Takeaways
- The paper presents a framework for structured model learning that enhances existing physical models with data-driven components.
- It establishes results on uniqueness and convergence for a wide class of PDEs and neural network approximations.
- The uniqueness result indicates that full, noiseless measurements can lead to a unique identification of unknown model components.
- Convergence results show that learned components from noisy data can approximate regularization-minimizing solutions effectively.
- This research provides a novel approach to model learning, diverging from traditional methods where uniqueness is typically expected.
Mathematics > Optimization and Control arXiv:2410.22009 (math) [Submitted on 29 Oct 2024 (v1), last revised 16 Feb 2026 (this version, v4)] Title:On uniqueness in structured model learning Authors:Martin Holler, Erion Morina View a PDF of the paper titled On uniqueness in structured model learning, by Martin Holler and Erion Morina View PDF HTML (experimental) 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 ...
