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We consider a general class of mean field control problems described by stochastic delayed differential equations of McKean-Vlasov type. Two numerical algorithms are provided based on deep learning techniques, one is to directly parameterizing the optimal control using neural networks, the other is based on numerically solving the McKean-Vlasov forward anticipated backward stochastic differential equation (MV-FABSDE) system. In addition, we establish the necessary and sufficient stochastic maximum principle of this class of mean field control problems with delay based on the differential calculus on function of measures, and the exis- tence and uniqueness results are proved for the associated MV-FABSDE system under suitable conditions.
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This content will become publicly available on June 5, 2026
Counting the number of stationary solutions of partial differential equations via infinite dimensional sampling
This paper is concerned with the problem of counting solutions of stationary nonlinear Partial Differential Equations (PDEs) when the PDE is known to admit more than one solution. We suggest tackling the problem via a sampling-based approach. The method allows one to find solutions that are stable, in the sense that they are stable equilibria of the associated time-dependent PDE. We test our proposed methodology on the McKean–Vlasov PDE, more precisely on the problem of determining the number of stationary solutions of the McKean–Vlasov equation. This article is part of the theme issue ‘Partial differential equations in data science’.
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- Award ID(s):
- 2111278
- PAR ID:
- 10643428
- Publisher / Repository:
- Royal Society Publishing
- Date Published:
- Journal Name:
- Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
- Volume:
- 383
- Issue:
- 2298
- ISSN:
- 1364-503X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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