A deep learning algorithm for computing mean field control problems via forward-backward score dynamics
We propose a deep learning approach to compute mean field control problems with individual noises. The problem consists of the Fokker-Planck (FP) equation and the Hamilton-Jacobi-Bellman (HJB) equation. Using the differential of the entropy, namely the score function, we first formulate the determin...
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| Published in | Research in the mathematical sciences Vol. 12; no. 3; p. 42 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Cham
Springer International Publishing
01.09.2025
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 2522-0144 2197-9847 |
| DOI | 10.1007/s40687-025-00531-9 |
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| Abstract | We propose a deep learning approach to compute mean field control problems with individual noises. The problem consists of the Fokker-Planck (FP) equation and the Hamilton-Jacobi-Bellman (HJB) equation. Using the differential of the entropy, namely the score function, we first formulate the deterministic forward-backward characteristics for the mean field control system, which is different from the classical forward-backward stochastic differential equations (FBSDEs). We further apply the neural network approximation to fit the proposed deterministic characteristic lines. Numerical examples, including the control problem with entropy potential energy, the linear quadratic regulator, and the systemic risks, demonstrate the effectiveness of the proposed method. |
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| AbstractList | We propose a deep learning approach to compute mean field control problems with individual noises. The problem consists of the Fokker-Planck (FP) equation and the Hamilton-Jacobi-Bellman (HJB) equation. Using the differential of the entropy, namely the score function, we first formulate the deterministic forward-backward characteristics for the mean field control system, which is different from the classical forward-backward stochastic differential equations (FBSDEs). We further apply the neural network approximation to fit the proposed deterministic characteristic lines. Numerical examples, including the control problem with entropy potential energy, the linear quadratic regulator, and the systemic risks, demonstrate the effectiveness of the proposed method. |
| ArticleNumber | 42 |
| Author | Osher, Stanley Li, Wuchen Zhou, Mo |
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| SubjectTerms | Algorithms Applications of Mathematics Brownian motion Computational Mathematics and Numerical Analysis Deep learning Differential equations Entropy Lagrange multiplier Linear quadratic regulator Machine learning Mathematics Mathematics and Statistics Neural networks Potential energy Shadow prices Velocity Viscosity |
| Title | A deep learning algorithm for computing mean field control problems via forward-backward score dynamics |
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