Equilibrium Structure of High-Resolution Differential Equations for Min–Max Optimization

High-resolution differential equations (HRDEs) provide refined continuous-time models for first-order methods in saddle-point and games. In this work, we show that the equilibrium structure of HRDEs need not coincide with the solution set of the underlying problem. We introduce a general framework to characterize equilibria of HRDEs and formalize the notion of spurious equilibria: stationary points of the continuous-time dynamics that are not solutions of the original game, as per their first-order characterization. We show that HRDEs associated with gradient descent–ascent are faithful in this sense, while HRDEs for extragradient and similar methods may admit additional equilibria induced by nonlinear algorithmic correction terms. We derive explicit conditions for the existence of such equilibria and analyze their stability. Our results highlight a structural limitation of continuous-time models beyond standard gradient flows and call for care when using HRDEs or related ODEs to reason about algorithmic solutions or behaviors.