Multi-Agent Dynamics Under Randomness: From MFGs to ML and Their Applications

Dear Colleagues,

This Special Issue of Mathematics foregrounds the analytical characteristics of problems based on the evolution of a vast collection of agents evolving under uncertainty. We seek papers building rigorous bridges from the (S)PDEs and Mean Field Theory to the fine-grained behavior of stochastic dynamical systems, as well as from the uncertainty rigorous quantification to the design of efficient numerical schemes whose convergence and stability both have to be analytically proved. Moreover, we aim at publishing contributions that proactively embed such analytical insights within machine learning and AI solutions, as to obtain math-rigorous tools solving relevant applicative problems.

Accordingly, the contributions should

• Develop theoretical foundations for MFGs and other large-population stochastic control models;
• Design robust and efficient numerical algorithms capable of handling high dimensionality, non-convexities, and complex boundary conditions;
• Quantify and reduce model-form and parametric uncertainty, including rare-event and risk-sensitive analyses;
• Exploit recent advances in AI/ML (e.g., physics-informed neural networks, deep reinforcement learning, and surrogate modeling) to accelerate simulation, calibration, and decision-making.

Original research articles, short communications, and comprehensive reviews are welcome.

Topics of interest include, but are not limited to, the following:

1. Analytical and probabilistic aspects of MFGs
• Existence, uniqueness, and regularity of solutions
• Common vs. idiosyncratic noise, long-time behavior, mean-field type control
2. Stochastic dynamics under uncertainty
• Stochastic differential games and backward SPDEs
• Uncertainty quantification and sensitivity analysis
3. Numerical schemes and high-performance computing
• Sparse grids, multilevel and multiscale methods, Monte Carlo and quasi-Monte Carlo techniques
• Convergence and stability of the proposed numerical schemes
4. AI/ML for stochastic and game-theoretic problems
• Deep learning solvers for high-dimensional (e.g., HJB/Fokker-Planck) systems
• Reinforcement learning approximations of Nash equilibria and MFGs' equilibria
• Mathematical foundations of data-driven models with possible applications
5. Applications
• Economics and finance (e.g., systemic risk, portfolio games, financial-based SPDEs, and AI-boosted numerical schemes for finance)
• Energy systems and climate policy (e.g., electric production/consumption for large agent systems), related MFGs approaches, and forecasting (stochastically based and/or under uncertainty) models
• Epidemiology, crowd and traffic flow, and collective behavior in biology
• Robotics and autonomous systems coordination under uncertainty

Dr. Luca Di Persio
Dr. Viktorya Vardanyan
Guest Editors

Manuscript Submission Information

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• MFGs
• stochastic dynamics
• uncertainty
• numerical schemes
• artificial intelligence / machine learning