We study the problem of predicting rare critical transition events for a class of slow–fast nonlinear dynamical systems. The state of the system of interest is described by a slow process, whereas a faster process drives its evolution and induces critical transitions. By taking advantage of recent advances in reservoir computing, we present a data-driven method to predict the future evolution of the state. We show that our method is capable of predicting a critical transition event at least several numerical time steps in advance. We demonstrate the success as well as the limitations of our method using numerical experiments on three examples of systems, ranging from low dimensional to high dimensional. We discuss the mathematical and broader implications of our results.

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In stochastic resonance, a periodically forced Brownian particle in a double-well potential jumps between minima at rare increments, the prediction of which pose a major theoretical challenge. Here, we use a path-integral method to predict these transitions by determining the most probable (or "optimal") space-time path of a particle. We characterize the optimal path using a direct comparison principle between the Langevin and Hamiltonian dynamical descriptions, allowing us to express the jump condition in terms of the accumulation of noise around the stable periodic path. In consequence, as a system approaches a rare event these fluctuations approach one of the deterministic minimizers, thereby providing a precursor for predicting the stochastic transition. We demonstrate the method numerically, which allows us to determine whether a state is following a stable periodic path or will experience an incipient jump. The vast range of systems that exhibit stochastic resonance behavior insures broad relevance of our framework, which allows one to extract precursor fluctuations from data.

We use asymptotic methods from the theory of differential equations to obtain an analytical expression for the survival probability of an Ornstein–Uhlenbeck process with a potential defined over a broad domain. We form a uniformly continuous analytical solution covering the entire domain by asymptotically matching approximate solutions in an interior region, centered around the origin, to those in boundary layers, near the lateral boundaries of the domain. The analytic solution agrees extremely well with the numerical solution and takes into account the non-negligible leakage of probability that occurs at short times when the stochastic process begins close to one of the boundaries. Given the range of applications of Ornstein–Uhlenbeck processes, the analytic solution is of broad relevance across many fields of natural and engineering science.