A Distribution-to-Distribution Neural Probabilistic Forecasting Framework for Dynamical Systems
Authors: Tianlin Yang, Hailiang Du, Louis Aslett
Organizations: Department of Mathematical Sciences, Durham University, Durham DH1 3LE, United Kingdom · School of Mathematics, East China University of Science and Technology, Shanghai 200237, China · Data Science Institute, The London School of Economics and Political Science, London WC2A 2AE, United Kingdom
Probabilistic forecasts provide a principled representation of forecast uncertainty through predictive distributions. Directly propagating predictive distributions with conventional physical models requires solving distributional evolution equations, a task well-known to be computationally prohibitive for high-dimensional nonlinear systems. Consequently, multi-step predictive distributions are commonly approximated using ensembles or sampled trajectories. A novel distribution-to-distribution (D2D) neural probabilistic forecasting framework is proposed with compatible distributional encoding and decoding interfaces around a replaceable neural forecasting module. Kernel mean embeddings represent input distributions, while mixture density networks parameterise outputs, enabling recursive multi-step forecasting in distributional form. The framework is demonstrated on the Lorenz63 system. Results show that the D2D model captures nontrivial distributional evolution, produces skillful probabilistic forecasts without explicit ensemble simulation, and remains competitive with a naively initialised perfect-model benchmark. This provides proof-of-concept evidence that predictive distributions can be recursively propagated through a neural forecasting model and evaluated in terms of probabilistic forecast performance.
Figures & tables
Figure 1: Schematic illustration of the proposed D2D neural network architecture.
Figure 2: An example of the evolution of the probability density function (PDF) of the x -variable. Panel (a) shows the reference distributional evolution under the underlying true dynamical system, approximated empirically by propagating a large ensemble of initial conditions under the numerically simulated Lorenz63 system. Panel (b) shows the corresponding PDF evolution generated by the proposed D2D model trained with the iterative strategy. Red dots indicate the corresponding observed values at each lead time.
Figure 3: Logarithmic score skill relative to climatology as a function of lead time for iterative D2D forecasts and perfect model forecasts. Panels (a)-(d) correspond to observational noise levels of 0.01, 0.02, 0.04, and 0.08, respectively. The horizontal black dashed line represents the zero skill climatological forecast. The red curve denotes the perfect model forecast based on a 128-member ensemble propagated under the numerically simulated Lorenz63 system and converted to a predictive distribution using Gaussian kernel dressing [ 42 ] . The remaining curves correspond to iterative D2D models trained with maximum curriculum lead times of 16Δt , 32Δt , 64Δt , and 128Δt . Shaded bands indicate 95% bootstrap resampling intervals computed on the test set.
Figure 4: Evolution of the predictive PDF for the x -variable generated by the 32Δt iterative D2D model, using the same initial condition as in Fig. 2 . Panel (a) corresponds to observational noise level 0.01 , and panel (b) corresponds to observational noise level 0.08 .
Figure 5: Logarithmic score skill relative to climatology for the direct and iterative D2D strategies under different observational noise levels. Panels (a)–(d) correspond to the same observational noise levels as in Fig. 3 . The blue curve denotes the iterative D2D model trained with maximum curriculum lead time 128Δt . The red curve denotes the direct-forecast strategy, constructed from multiple direct D2D models trained at fixed lead times 1Δt , 2Δt , 4Δt , … , 128Δt and composed through a hierarchical temporal aggregation strategy to produce forecasts at arbitrary lead times.
Faculty of Economics and Business, KU Leuven, Belgium · Department of Electrical Engineering, Shanghai Jiao Tong University, China · Dyson School of Design Engineering, Imperial College London, United Kingdom +3