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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2412.03506 |
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| _version_ | 1866912760480137216 |
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| author | Gao, Yuan Lang, Quanjun Lu, Fei |
| author_facet | Gao, Yuan Lang, Quanjun Lu, Fei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_03506 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Self-test loss functions for learning weak-form operators and gradient flows Gao, Yuan Lang, Quanjun Lu, Fei Machine Learning 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. |
| title | Self-test loss functions for learning weak-form operators and gradient flows |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2412.03506 |