Numerical Analysis on Neural Network Projected Schemes for Approximating One Dimensional Wasserstein Gradient Flows
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866916139130421248 |
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| author | Zuo, Xinzhe Zhao, Jiaxi Liu, Shu Osher, Stanley Li, Wuchen |
| author_facet | Zuo, Xinzhe Zhao, Jiaxi Liu, Shu Osher, Stanley Li, Wuchen |
| contents | We provide a numerical analysis and computation of neural network projected schemes for approximating one dimensional Wasserstein gradient flows. We approximate the Lagrangian mapping functions of gradient flows by the class of two-layer neural network functions with ReLU (rectified linear unit) activation functions. The numerical scheme is based on a projected gradient method, namely the Wasserstein natural gradient, where the projection is constructed from the $L^2$ mapping spaces onto the neural network parameterized mapping space. We establish theoretical guarantees for the performance of the neural projected dynamics. We derive a closed-form update for the scheme with well-posedness and explicit consistency guarantee for a particular choice of network structure. General truncation error analysis is also established on the basis of the projective nature of the dynamics. Numerical examples, including gradient drift Fokker-Planck equations, porous medium equations, and Keller-Segel models, verify the accuracy and effectiveness of the proposed neural projected algorithm. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_16821 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Numerical Analysis on Neural Network Projected Schemes for Approximating One Dimensional Wasserstein Gradient Flows Zuo, Xinzhe Zhao, Jiaxi Liu, Shu Osher, Stanley Li, Wuchen Numerical Analysis Optimization and Control We provide a numerical analysis and computation of neural network projected schemes for approximating one dimensional Wasserstein gradient flows. We approximate the Lagrangian mapping functions of gradient flows by the class of two-layer neural network functions with ReLU (rectified linear unit) activation functions. The numerical scheme is based on a projected gradient method, namely the Wasserstein natural gradient, where the projection is constructed from the $L^2$ mapping spaces onto the neural network parameterized mapping space. We establish theoretical guarantees for the performance of the neural projected dynamics. We derive a closed-form update for the scheme with well-posedness and explicit consistency guarantee for a particular choice of network structure. General truncation error analysis is also established on the basis of the projective nature of the dynamics. Numerical examples, including gradient drift Fokker-Planck equations, porous medium equations, and Keller-Segel models, verify the accuracy and effectiveness of the proposed neural projected algorithm. |
| title | Numerical Analysis on Neural Network Projected Schemes for Approximating One Dimensional Wasserstein Gradient Flows |
| topic | Numerical Analysis Optimization and Control |
| url | https://arxiv.org/abs/2402.16821 |