Simulate
Generate Lorenz-system trajectories and prepare time-series data for forecasting experiments, evaluation, and controlled comparison.
Chaotic System Forecasting
A scientific machine-learning project exploring long-horizon forecasting in chaotic dynamics using Echo State Networks, physics-informed reservoir computing, and systematic optimization.
The Problem
Chaotic systems are inherently difficult to forecast because even very small differences in initial conditions can cause trajectories to diverge rapidly over time. The challenge was to investigate whether reservoir computing, physics-informed constraints, and systematic experimentation could extend the useful forecasting horizon of the Lorenz system.
The Approach
The project combines dynamical-system simulation, reservoir computing, physical constraints, and optimization within one forecasting workflow.
Model
Use conventional Echo State Networks as a data-driven baseline and Physics-Informed ESNs to incorporate the governing dynamics.
Optimize
Explore initial-condition behavior and optimize important reservoir hyperparameters using Optuna to improve long-horizon forecasting performance.
Evaluate
Measure forecast quality in Lyapunov-time units and analyze when predicted trajectories begin to diverge from the true chaotic dynamics.
Technical Decisions
The experimental design emphasizes meaningful evaluation, reproducibility, and a clear distinction between referenced baselines and newly achieved results.
Lyapunov-Time Scale
Forecasting performance is evaluated using Lyapunov time rather than only raw simulation time, making the prediction horizon meaningful for a chaotic dynamical system.
Baseline Separation
Paper-reported ESN and PI-ESN baselines are kept explicitly separate from the achieved experimental forecasting horizon.
Forecasting Diagnostics
Evaluation focuses on trajectory behavior, prediction error, and predictability horizon rather than relying on a single black-box score.
Results
LONG-HORIZON CHAOTIC FORECASTING
Experimental forecasting horizon extended to 6 Lyapunov Times
Referenced results reported approximately 4.5 Lyapunov Times for a conventional ESN and 5.5 Lyapunov Times for a Physics-Informed ESN. Through initial-condition experimentation and hyperparameter optimization, the developed experimental approach reached a forecasting horizon of 6.0 Lyapunov Times.
Why It Matters
This work demonstrates how physics-informed constraints, reservoir computing, and systematic experimentation can be combined to extend useful forecasting horizons in highly sensitive nonlinear dynamical systems. The project also highlights the importance of evaluating chaotic forecasts using system-aware metrics rather than conventional prediction scores alone.
Experimental Evidence
Experimental results comparing trajectory forecasts and normalized prediction error across Lyapunov time, with the predictability threshold used to determine the useful forecasting horizon.
True Dynamics vs Forecasted Trajectories
Comparison of the true Lorenz-system dynamics with the conventional ESN and the optimized PI-ESN across the forecasting horizon.
Forecast Error Across Lyapunov Time
Normalized prediction error for the conventional ESN and optimized PI-ESN, evaluated against the predictability threshold to determine the useful forecasting horizon.
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