Self-test loss functions for learning weak-form operators and gradient flows

• URL: http://arxiv.org/abs/2412.03506v2
• Date: Fri, 13 Dec 2024 03:46:39 GMT
• Title: Self-test loss functions for learning weak-form operators and gradient flows
• Authors: Yuan Gao, Quanjun Lang, Fei Lu,
• Abstract summary: We introduce self-test loss functions, which employ test functions that depend on the unknown parameters. The proposed self-test loss function conserves energy gradient flows and coincides with the expected log-likelihood ratio for differential equations.
• Score: 5.9739929525316064
• License:
• Abstract: The construction of loss functions presents a major challenge in data-driven modeling involving weak-form operators in PDEs and gradient flows, particularly due to the need to select test functions appropriately. We address this challenge by introducing self-test loss functions, which employ test functions that depend on the unknown parameters, specifically for cases where the operator depends linearly on the unknowns. The proposed self-test loss function conserves energy for gradient flows and coincides with the expected log-likelihood ratio for stochastic differential equations. Importantly, it is quadratic, facilitating theoretical analysis of identifiability and well-posedness of the inverse problem, while also leading to efficient parametric or nonparametric regression algorithms. It is computationally simple, requiring only low-order derivatives or even being entirely derivative-free, and numerical experiments demonstrate its robustness against noisy and discrete data.

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