Approximating Discontinuous Nash Equilibrial Values of Two-Player General-Sum Differential Games

Finding Nash equilibrial policies for two-player differential games requires solving Hamilton-Jacobi-Isaacs PDEs. Recent studies achieved success in circumventing the curse of dimensionality in solving such PDEs with underlying applications to human-robot interactions (HRI), by adopting self-supervised (physics-informed) neural networks as universal value approximators. This paper extends from previous SOTA on zero-sum games with continuous values to general-sum games with discontinuous values, where the discontinuity is caused by that of the players' losses. We show that due to its lack of convergence proof and generalization analysis on discontinuous losses, the existing self-supervised learning technique fails to generalize and raises safety concerns in an autonomous driving application. Our solution is to first pre-train the value network on supervised Nash equilibria, and then refine it by minimizing a loss that combines the supervised data with the PDE and boundary conditions. Importantly, the demonstrated advantage of the proposed learning method against purely supervised and self-supervised approaches requires careful choice of the neural activation function: Among $\texttt{relu}$, $\texttt{sin}$, and $\texttt{tanh}$, we show that $\texttt{tanh}$ is the only choice that achieves optimal generalization and safety performance. Our conjecture is that $\texttt{tanh}$ (similar to $\texttt{sin}$) allows continuity of value and its gradient, which is sufficient for the convergence of learning, and at the same time is expressive enough (similar to $\texttt{relu}$) at approximating discontinuous value landscapes. Lastly, we apply our method to approximating control policies for an incomplete-information interaction and demonstrate its contribution to safe interactions.