Predicting stochastic dynamical systems remains a challenge: noise can induce qualitatively new phenomena, such as noise-induced transitions between states. Existing approaches either require an exact model (e.g., the Fokker-Planck equation) or generalize poorly to systems with memory. Reservoir computing has proven to be a powerful tool for deterministic systems, but its direct application to stochastic systems leads to rapid divergence of forecasts. This work fills that gap by proposing a model that learns from data without knowing the equations.
The authors build a hybrid scheme: first, a reservoir computer is trained on a set of system trajectories, producing one-step predictions. Then, a normalizing flow estimates the distribution of the error between the prediction and the true value at each step. During forecasting, at each step, a random correction drawn from this distribution is added to the reservoir output. This approach corrects error accumulation and preserves the statistical properties of the process. The reservoir has 500–1000 nodes, and hyperparameters are tuned via Bayesian optimization. The normalizing flow uses rational-quadratic neural splines with an autoregressive structure.
The method was tested on seven systems, covering linear and nonlinear, Markovian and non-Markovian, stationary and non-stationary processes. For the Ornstein-Uhlenbeck process, RC-NF accurately reproduces the probability distribution over long intervals, unlike pure RC, which diverges quickly. For a double-well system, the method reproduces the transition rate between metastable states with an error below 0.001. For the stochastic Lorenz system, RC-NF yields a Wasserstein distance of 0.0266 versus 0.0479 for pure RC and better reproduces the maximal Lyapunov exponent (median 2.5622 versus 1.8525 for RC, with true value 2.5557). The method also successfully reproduces mixed-mode oscillations and relaxation oscillations.
This work shows that combining reservoir computing with a probabilistic error model overcomes a fundamental limitation of RC when dealing with noise. Unlike related works, which either treat noise as an external perturbation or require special trajectory selection, the proposed method directly handles the system's intrinsic noise and imposes no restrictions on the data. This opens the way to predicting complex stochastic systems without knowing their equations.
Possible directions include extending to multiplicative and non-Gaussian noise, exploring more complex dynamical regimes such as bifurcation transitions, and applying more advanced reservoir architectures. It is also important to investigate whether strong convergence of forecasts can be achieved, not just weak convergence (in the sense of distributions).
The method could be useful in climatology, neuroscience, financial mathematics, and other fields where predicting stochastic processes from data is needed.
Test the method on real data, such as climate records or biological signals, and extend it to systems with multiplicative noise.
This work directly addresses the problem of long-term prediction of stochastic systems and reproduction of rare events, such as transitions between metastable states, which remains an open problem in the theory of stochastic processes.
The RC-NF method reproduces the transition rate in a double-well system to the third decimal place, comparable to the accuracy of direct numerical simulation.