Mean-field Approaches, optimal TRansport and artificial Intelligence: a mathematical framework for compleX systems

Descripción del proyecto

Nuevos métodos para modelizar redes caóticas

Muchos sistemas esenciales para la vida cotidiana, como las redes eléctricas, bancarias y logísticas, están profundamente interconectados. Esta interdependencia dificulta su modelización cuando se producen fenómenos de gran magnitud, como una caída repentina de los mercados o un apagón, que pueden alterar el conjunto del sistema. En el proyecto MATRIX, que cuenta con el respaldo de las acciones Marie Skłodowska-Curie, se desarrollará un marco matemático para hacer frente a las limitaciones de los modelos actuales, que aún no logran representar adecuadamente la incertidumbre propia de estos sistemas. Los investigadores diseñarán algoritmos de aprendizaje por refuerzo para calcular estrategias óptimas de control cuando una perturbación común y simultánea afecte a todos los agentes. A continuación, se establecerán fundamentos matemáticos rigurosos mediante distancias de Wasserstein adaptadas, con el fin de garantizar la estabilidad del sistema preservando las relaciones temporales de causalidad. Por último, se emplearán redes neuronales fundamentadas en la física para aproximar soluciones ante perturbaciones graves y discontinuas.

Objetivo

Modern societies rely on interconnected systems in which a large number of complex agents interact amid uncertainty and systemic shocks. In recent years, the mathematical analysis of such stochastic systems has advanced in three closely related areas: mean-field games (MFGs), McKean-Vlasov systems and stochastic partial differential equations (SPDEs). However, multiple roadblocks still prevent the rigorous treatment of intricate real-life problems, such as households in a smart grid, banks in a network, or autonomous vehicles in a city.

The MATRIX project (Mean-field Approaches, optimal TRansport and artificial Intelligence: a mathematical framework for compleX systems) aims to address the limitations of the current models by accounting for shared randomness (events affecting all agents simultaneously, such as policy changes or extreme weather), causal consistency (to prevent information leakage), and non-Gaussian dynamics (such as outages or crashes). A special emphasis will be placed on systems incorporating neural-network-driven agents and on their interaction at the mean-field limit.

The proposed research is organised into three mutually reinforcing work packages (WPs). WP1 involves constructing reinforcement-learning algorithms for MFGs driven by common noise; the aim is to produce scalable representations of optimal controls under shared shocks. WP2 focuses on establishing rigorous foundations for adapted Wasserstein distances in path-dependent McKean-Vlasov systems, providing quantitative stability and propagation-of-chaos results that respect causality. WP3 tackles the well-posedness of McKean-Vlasov SPDEs with Lévy noise and develops physics-informed neural networks that can approximate the underlying solutions despite discontinuities.

The MATRIX project will strengthen the capability to model, predict and control modern complex systems, supporting strategic objectives related to green-energy transition, financial stability and autonomous systems.