Value Approximation for Two-Player General-Sum Differential Games with State Constraints (2311.16520v3)

Abstract: Solving Hamilton-Jacobi-Isaacs (HJI) PDEs numerically enables equilibrial feedback control in two-player differential games, yet faces the curse of dimensionality (CoD). While physics-informed neural networks (PINNs) have shown promise in alleviating CoD in solving PDEs, vanilla PINNs fall short in learning discontinuous solutions due to their sampling nature, leading to poor safety performance of the resulting policies when values are discontinuous due to state or temporal logic constraints. In this study, we explore three potential solutions to this challenge: (1) a hybrid learning method that is guided by both supervisory equilibria and the HJI PDE, (2) a value-hardening method where a sequence of HJIs are solved with increasing Lipschitz constant on the constraint violation penalty, and (3) the epigraphical technique that lifts the value to a higher dimensional state space where it becomes continuous. Evaluations through 5D and 9D vehicle and 13D drone simulations reveal that the hybrid method outperforms others in terms of generalization and safety performance by taking advantage of both the supervisory equilibrium values and costates, and the low cost of PINN loss gradients.

• The paper introduces three innovative methods—hybrid learning, value hardening, and epigraphical learning—to approximate discontinuous game values under state constraints.
• The paper demonstrates the effectiveness of these approaches through 5D, 9D, and 13D simulations, with hybrid learning achieving superior safety and scalability in complex human-robot interactions.
• The paper outlines future research directions, including integrating adaptive activation functions and reinforcement learning with physics-informed machine learning to enhance control in safety-critical robotics.

Understanding the Limits of Current Approaches in Robotics

In the field of robotics, especially where safety is emphasized, controlling the interaction between human and robotic players is essential. A standard approach involves using model predictive control (MPC), where safety is typically ensured by integrating state constraints derived from a zero-sum game formulation. However, limitations of zero-sum games include unnecessary conservatism and slow decision-making due to the need for real-time MPC atop value approximation.

The Challenge of Calculating Game Value

In multiplayer differential games that are not purely adversarial (general-sum), players may possess incomplete information about one another and must constantly reassess their strategies based on observations. Ideally, one would compute the game's value facilitating optimal control while adhering to constraints. Yet, general-sum differential games with restrictions do not have well-characterized solutions, subjecting calculations to the curse of dimensionality (CoD). Although physics-informed machine learning (PIML) approaches have been attempted to avoid CoD, they struggle with learning discontinuous solutions brought about by state constraints.

New Approaches for Value Approximation

Researchers explored three innovative solutions to tackle the problem of discontinuous value approximation:

1. Hybrid Learning: This technique combines equilibrium data and Hamilton-Jacobi-Isaac (HJI) PDE constraints. It utilizes human insights to generate equilibrium demonstrations for discontinuous regions of value.
2. Value Hardening: Influenced by curriculum learning, this method gradually increases the Lipschitz constant on state violation penalties to handle discontinuities.
3. Epigraphical Learning: By lifting the game value to an augmented space through epigraphical technique, a continuous approximation becomes feasible, which PIML can leverage.

Performance Comparisons

Evaluative simulations were conducted using 5D and 9D vehicle and 13D drone interactions. The results revealed that hybrid learning outperformed other methods in terms of safety performance and generalization. It also scaled more effectively to higher-dimensional states within the same computational limits. This marks a substantial advancement in informed decision-making for safer human-robot interactions.

Significance and Future Directions

The hybrid learning approach could revolutionize the speed and safety of decisions in robotics applications involving humans. This integration of supervised learning with PIML provides an effective solution while addressing the computational challenges associated with higher-dimensional problems.

The research invites further investigation into utilizing adaptive activation functions for neural networks and exploring the interplay between PIML and reinforcement learning to fine-tune value-based strategies in safety-critical scenarios. As the field advances, the findings offer a promising direction for developing robust control mechanisms in general-sum differential games.