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

TL;DR

This work tackles real-time, parametric value-function approximation for general-sum differential games with state constraints, addressing the curse of dimensionality and convergence issues of vanilla PINNs. It introduces Hybrid Neural Operator (HNO), an operator-based model that combines informative supervised data with PDE-driven residuals to learn parametric value functions across varying player types, preserving safety in discontinuous regions. Through experiments on 9D and 13D nonlinear dynamics (narrow road, double-lane change, and two-drone scenarios), HNO consistently outperforms a DeepONet-based SNO baseline under the same computational budget, including unseen parameter settings. The results imply that HNO enables scalable, real-time inference for complex human-robot and multi-agent interactions, with insights into activation choices and NTK conditioning that influence safety and generalization.

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.

Figures (6)

• Figure 1: Illustration of Hybrid Neural Operator. The baseline SNO is similar to HNO, but without red dashed box for input data points and loss function terms.
• Figure 2: Narrow road collision avoidance scenario. Simulation shows the ground truth trajectory for player-type configuration $(\theta_1,\theta_2)=(1,1)$.
• Figure 3: Double-lane change scenario. Simulation shows the ground truth trajectory for player-type configuration $(\theta_1,\theta_2)=(1,1)$.
• Figure 4: Ground truth trajectory of two-drone collision avoidance for player-type configuration $(\theta_1,\theta_2)=(1,1)$.
• Figure 5: Safety performance (Col.%) comparison among HNO and SNO for each parameter configuration in $\Theta^2$ across all case studies.