Forecasting complex systems such as climate or turbulence often stumbles on our ignorance of the exact equations. The classical route is to fit parametric models, but their predictive power depends on the choice of model and fitting algorithm. An alternative is nonparametric methods, which build forecasts directly from data. This work sits alongside studies that use local linearization of shift maps, but it goes a step further: instead of predicting only mean values, it recovers the entire evolution of the probability distribution, which is critical for uncertainty quantification.
The key idea is to represent the time-shift operator in a basis of smooth functions obtained by diffusion maps with variable kernel width. This basis minimizes the approximation error of the stochastic integral arising in the semigroup estimation. In practice, from a time series one constructs a transition matrix between values of basis functions at successive steps, then uses it to evolve the coefficients of the initial density. The method does not require knowledge of the vector field or diffusion tensor and works at any discretization step.
On a stochastic equation on a torus with non-gradient drift and anisotropic diffusion, the diffusion forecast accurately reproduced the evolution of the first two moments, matching an ensemble forecast of 50,000 trajectories. For the Lorenz-63 model, the method showed significant improvement over locally linear methods: at a step of 0.1, iterative local forecasts accumulated bias and overestimated variance, whereas the diffusion forecast remained consistent with the ensemble reference. On real El Niño 3.4 index data, the 14-month forecast achieved a correlation of 0.64 and RMSE of 0.60, better than a previously published result (correlation 0.4, RMSE 1.4).
This work shows that one can build probabilistic forecasts for systems with unknown dynamics without explicitly estimating equation coefficients. Unlike neighboring works that either estimate the vector field in the ambient space or use local linear models, here the semigroup is directly approximated on the manifold, which reduces data requirements and improves forecast stability. This extends the applicability of nonparametric methods to problems with substantial uncertainty.
It may be possible to combine local linear models through a diffusion basis to build a global model that respects the invariant measure. Also promising is separating observational and dynamical noise for a more accurate estimate of the attractor structure. The theoretical question of approximating fractal attractors by smooth manifolds remains open and could become a basis for further research.
The method could be useful in climatology, oceanography, and other fields where long time series are available and probabilistic forecasts are needed.
A next step could be applying the method to more complex climate data and comparing it with operational forecast models.
In the field of nonparametric forecasting, the problem of uncertainty quantification for chaotic systems with positive Lyapunov exponents remains. This work offers one path through Markov approximation, but the question of the accuracy of such approximation for fractal attractors requires further study.
The diffusion maps method underlying the forecast was originally developed for analyzing high-dimensional data, such as images, and only later adapted for dynamical systems.