A Machine Learning Framework for Solving High-Dimensional Mean Field Game and Mean Field Control Problems (1912.01825v3)

Abstract: Mean field games (MFG) and mean field control (MFC) are critical classes of multi-agent models for efficient analysis of massive populations of interacting agents. Their areas of application span topics in economics, finance, game theory, industrial engineering, crowd motion, and more. In this paper, we provide a flexible machine learning framework for the numerical solution of potential MFG and MFC models. State-of-the-art numerical methods for solving such problems utilize spatial discretization that leads to a curse-of-dimensionality. We approximately solve high-dimensional problems by combining Lagrangian and Eulerian viewpoints and leveraging recent advances from machine learning. More precisely, we work with a Lagrangian formulation of the problem and enforce the underlying Hamilton-Jacobi-BeLLMan (HJB) equation that is derived from the Eulerian formulation. Finally, a tailored neural network parameterization of the MFG/MFC solution helps us avoid any spatial discretization. Our numerical results include the approximate solution of 100-dimensional instances of optimal transport and crowd motion problems on a standard work station and a validation using an Eulerian solver in two dimensions. These results open the door to much-anticipated applications of MFG and MFC models that were beyond reach with existing numerical methods.

• The paper presents a novel machine learning framework that combines Lagrangian and Eulerian approaches to solve high-dimensional mean field game and control problems.
• It leverages neural network-based parameterization to overcome the curse of dimensionality, with successful 100-dimensional simulations demonstrating its effectiveness.
• Open-source Julia implementations and competitive numerical results underline its practical potential for real-world multi-agent systems in economics, finance, and engineering.

A Machine Learning Framework for Solving High-Dimensional Mean Field Game and Mean Field Control Problems

The paper introduces a novel machine learning framework aimed at solving high-dimensional Mean Field Game (MFG) and Mean Field Control (MFC) problems, which have broad applications across economics, finance, and engineering. The focus is on overcoming the computational challenges associated with traditional numerical methods, particularly the curse of dimensionality encountered in high-dimensional problems.

MFGs and MFCs represent a continuum of interacting agents where the optimal strategies are commonly determined via solving coupled Hamilton-Jacobi-BeLLMan (HJB) and continuity equations. However, traditional grid-based methods that discretize these spatial domains face scalability issues in high-dimensional spaces due to their exponential growth in computational complexity.

The authors circumvent these challenges by adopting a machine learning framework that engineers a bespoke combination of Lagrangian and Eulerian perspectives. By utilizing neural networks for solution parameterization, the paper introduces a method that bypasses spatial discretization entirely. This approach facilitates the resolution of up to 100-dimensional instances of optimal transport and crowd motion problems on standard computational platforms.

Notably, the paper details how the Lagrangian viewpoint is leveraged to solve characteristics of the continuity equation without constructing a mesh, which is seamlessly paired with a neural network approximation. The framework incorporates recent advancements in machine learning for numerical computations: the potential functions are expressed as outputs of bespoke crafted neural networks. This approach directly reduces spatial dimensionality issues as grid-based computation becomes unnecessary.

Strong numerical results are presented in the paper, highlighting the efficacy of the proposed method. For instance, they achieve successful implementation in simulating 100-dimensional scenarios using their neural network architectures. The authors validate the solutions through comparison with lower-dimensional Eulerian solvers, observing competitive or superior performance, which underscores the framework’s potential to handle complex, high-dimensional problems effectively.

Contrary to other approaches constrained by explicit or cumbersome formulations, the authors’ method allows solving a diverse set of MFGs where mean field terms are analytically intractable. Their Lagrangian formulation includes potential penalties for HJB violations, advancing the scheme’s adaptability and correctness.

The research contributes significantly to both theoretical and practical fields. It demonstrates the feasibility of employing machine learning to approximate high-dimensional dynamics, which offers intriguing possibilities for future developments in AI. These include improving neural network training methodologies for enhanced stability and convergence, parallelizing computations further for efficiency gains, and addressing even broader classes of differential games.

Moreover, by providing open-source implementation through Julia, the authors facilitate exploration and adaptation of this method across various domains, encouraging advancements in solving real-world MFG and MFC problems.

In conclusion, the paper effectively marries modern machine learning techniques with classical numerical methods to tackle high-dimensional computational challenges in MFGs and MFCs. Its implications suggest a promising pathway toward more scalable and efficient frameworks for complex multi-agent systems, contributing valuable insights into the intersection of AI, optimal control, and differential equations.