A convergence result of a continuous model of deep learning via Łojasiewicz--Simon inequality

This study focuses on a Wasserstein-type gradient flow, which represents an optimization process of a continuous model of a Deep Neural Network (DNN). First, we establish the existence of a minimizer for an average loss of the model under $L^2$-regularization. Subsequently, we show the existence of a curve of maximal slope of the loss. Our main result is the convergence of flow to a critical point of the loss as time goes to infinity. An essential aspect of proving this result involves the establishment of the Łojasiewicz–Simon gradient inequality for the loss. We derive this inequality by assuming the analyticity of NNs and loss functions. Our proofs offer a new approach for analyzing the asymptotic behavior of Wasserstein-type gradient flows for nonconvex functionals.