A mean field control formulation using score-based normalizing flows simulates Fokker-Planck equations deterministically, with a convergence theorem for Ornstein-Uhlenbeck processes and experiments on Langevin and chaotic dynamics.
A deep learning algorithm for computing mean field control problems via forward-backward score dynamics
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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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Simulating Fokker-Planck equations via mean field control of score-based normalizing flows
A mean field control formulation using score-based normalizing flows simulates Fokker-Planck equations deterministically, with a convergence theorem for Ornstein-Uhlenbeck processes and experiments on Langevin and chaotic dynamics.