Mean-field neural networks: Learning mappings on Wasserstein space

Area of Science:

• Computational Mathematics
• Machine Learning
• Control Theory

Background:

• Mean-field games and control problems involve operators mapping between probability measures and functions.
• Learning these mean-field functions is computationally challenging.
• Existing methods may lack efficiency or theoretical guarantees.

Purpose of the Study:

• To introduce novel neural network architectures for learning mean-field functions.
• To provide theoretical guarantees for these proposed models.
• To develop and evaluate algorithms for solving time-dependent mean-field problems using these networks.

Main Methods:

• Proposed two classes of neural networks: bin density and cylindrical approximation.
• Established theoretical support using universal approximation theorems.
• Conducted numerical experiments to assess accuracy, efficiency, and generalization error.
• Developed algorithms for solving time-dependent mean-field problems.

Main Results:

• Demonstrated the accuracy and efficiency of the proposed mean-field neural networks.
• Showcased effective generalization across various test distributions.
• Successfully applied the networks to solve a semi-linear partial differential equation in Wasserstein space.

Conclusions:

• The developed mean-field neural networks offer a powerful tool for machine learning in probability measure spaces.
• These networks provide a theoretically sound and practically efficient approach to mean-field problems.
• The study paves the way for advanced applications in game theory, control, and partial differential equations.