Mean-field neural networks: learning mappings on Wasserstein space

• URL: http://arxiv.org/abs/2210.15179v3
• Date: Mon, 18 Sep 2023 07:34:11 GMT
• Title: Mean-field neural networks: learning mappings on Wasserstein space
• Authors: Huy\^en Pham and Xavier Warin
• Abstract summary: We study the machine learning task for models with operators mapping between the Wasserstein space of probability measures and a space of functions. Two classes of neural networks are proposed to learn so-called mean-field functions. We present different algorithms relying on mean-field neural networks for solving time-dependent mean-field problems.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the machine learning task for models with operators mapping between the Wasserstein space of probability measures and a space of functions, like e.g. in mean-field games/control problems. Two classes of neural networks, based on bin density and on cylindrical approximation, are proposed to learn these so-called mean-field functions, and are theoretically supported by universal approximation theorems. We perform several numerical experiments for training these two mean-field neural networks, and show their accuracy and efficiency in the generalization error with various test distributions. Finally, we present different algorithms relying on mean-field neural networks for solving time-dependent mean-field problems, and illustrate our results with numerical tests for the example of a semi-linear partial differential equation in the Wasserstein space of probability measures.

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