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Recent results have shown that for two-layer fully connected neural networks,gradient flow converges to a global optimum in the infinite width limit, bymaking a connection between the mean field dynamics and the Wassersteingradient flow. These results were derived for first-order gradient flow, and anatural question is whether second-order dynamics, i.e., dynamics withmomentum, exhibit a similar guarantee. We show that the answer is positive forthe heavy ball method. In this case, the resulting integro-PDE is a nonlinearkinetic Fokker Planck equation, and unlike the first-order case, it has noapparent connection with the Wasserstein gradient flow. Instead, we study thevariations of a Lyapunov functional along the solution trajectories tocharacterize the stationary points and to prove convergence. While our resultsare asymptotic in the mean field limit, numerical simulations indicate thatglobal convergence may already occur for reasonably small networks.

Global Convergence of Second-order Dynamics in Two-layer Neural Networks