StringNET: Neural Network Based Variational Method for Transition Pathways

Rare transition events in metastable systems under noisy fluctuations are crucial for many non-equilibrium processes, where the primary contributions to reactive flux are near the transition pathways between two metastable states. Efficient computation of these paths remains a numerical challenge particularly in high dimensions. In this work, we examine the temperature-dependent maximum flux path, the minimum energy path, and the minimum action path at zero temperature and propose the StringNET method for training these paths by building the variational formulations. Unlike traditional nudged elastic band and string method driven by the tangent-projected gradient force, StringNET seeks the intrinsic variational formulation in path space and parametrizes the paths by neural network functions. We show that the loss function for the maximum flux path serves as a softmax approximation to the numerically challenging minimax problem of the minimum energy path, and develop the pre-training strategy that includes the maximum flux path loss in the early training stage, significantly accelerating the computation of minimum energy and action paths. We demonstrate the superior performance of this method through various analytical and chemical examples, as well as the two- and four-dimensional Ginzburg-Landau functional energy.