NF-MKV Net: A Constraint-Preserving Neural Network Approach to Solving Mean-Field Games Equilibrium
Keywords: Mean-Field Games, Normalizing Flow, McKean-Vlasov type Forward-Backward Stochastic Differential Equations, Mathematical Constraints Neural Network
TL;DR: This paper explores the neural network solution of MFGs equilibrium from the perspective of stochastic process, while utilizing NF frameworks to solve the MKV FBSDEs fixed point problems, which is equivalent of the MFGs equilibrium.
Abstract: Neural network-based methods for solving Mean-Field Games (MFGs) equilibrium have gained significant attention due to their effectiveness in high-dimensional settings. However, many algorithms face the problem that the density distribution evolution does not satisfy the mathematical constraints in the solution. This paper explores the neural network solution of MFGs equilibrium from the perspective of stochastic process, while coupling process-regularized Normalizing Flow (NF) frameworks and state-policy-connected time series neural networks to solve the McKean-Vlasov type Forward-Backward Stochastic Differential Equations (MKV FBSDEs) fixed point problems, which is equivalent of the MFGs equilibrium. First, we convert the MFGs equilibrium to MKV FBSDEs which introduce the density distribution into equations coefficients within a probabilistic framework, and construct neural networks to approximate value functions and gradients based on these equations. Second, we employ NF architectures—generative neural network models—and by imposing loss constraints on each density transfer function, the algorithm ensures compliance with volumetric-invariance and time-continuity. Additionally, this paper provides theoretical proofs of the algorithm's validity and demonstrates its applicability across various scenarios, showcasing its effective compared to existing approaches.
Primary Area: applications to physical sciences (physics, chemistry, biology, etc.)
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Submission Number: 10193
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