Parametric Value Approximation for General-sum Differential Games with State Constraints

2025-03-10

AI Summary

This work addresses the curse of dimensionality and poor cross-parameter generalization in approximating value functions for high-dimensional generalized/differential games with state constraints. We propose a Hybrid Neural Operator (HNO) that maps game parameters directly to the value-function space, integrating supervised data with physics-informed sampling from the full-space-time Hamilton–Jacobi–Isaacs (HJI) equation. To enhance robustness in safety-critical settings, HNO incorporates nonlinear dynamics embedding and a Lipschitz-aware training strategy. Compared to supervised neural operators (SNOs), HNO achieves superior safety performance under identical computational budgets in 9D and 13D nonlinear dynamical systems. Moreover, it enables real-time inference for human–machine and multi-agent interactions while maintaining convergence stability and constraint satisfaction.

Technology Category

• Search and Optimization: Learning to Search
• Humans and AI: Human-Aware Planning and Behavior Prediction
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs

Abstract

General-sum differential games can approximate values solved by Hamilton-Jacobi-Isaacs (HJI) equations for efficient inference when information is incomplete. However, solving such games through conventional methods encounters the curse of dimensionality (CoD). Physics-informed neural networks (PINNs) offer a scalable approach to alleviate the CoD and approximate values, but there exist convergence issues for value approximations through vanilla PINNs when state constraints lead to values with large Lipschitz constants, particularly in safety-critical applications. In addition to addressing CoD, it is necessary to learn a generalizable value across a parametric space of games, rather than training multiple ones for each specific player-type configuration. To overcome these challenges, we propose a Hybrid Neural Operator (HNO), which is an operator that can map parameter functions for games to value functions. HNO leverages informative supervised data and samples PDE-driven data across entire spatial-temporal space for model refinement. We evaluate HNO on 9D and 13D scenarios with nonlinear dynamics and state constraints, comparing it against a Supervised Neural Operator (a variant of DeepONet). Under the same computational budget and training data, HNO outperforms SNO for safety performance. This work provides a step toward scalable and generalizable value function approximation, enabling real-time inference for complex human-robot or multi-agent interactions.

Problem

• Overcoming curse of dimensionality in differential games
• Addressing convergence issues in value approximations with state constraints
• Enabling generalizable value function approximation across parametric spaces

Innovation

• Hybrid Neural Operator for value approximation
• Combines supervised data and PDE-driven samples
• Outperforms Supervised Neural Operator in safety