Physics-Informed and Data-Driven Forecasting of Chaotic Dynamics Across Lorenz and Rössler Systems

Abstract

Reliable finite-horizon forecasting of chaotic dynamics is challenging because small approximation errors grow rapidly during recursive prediction. This study presents a controlled comparison of data-driven and physics-regularized forecasting methods for the Lorenz and Rössler systems. The proposed Hybrid Physics-Informed Feedforward Neural Network (Hybrid PI-FNN) learns a discrete state-transition map from a ten-state observation window through a five-step recursive rollout. Unlike conventional continuous-coordinate physics-informed neural networks, physical consistency is imposed using fourth-order Runge–Kutta transition targets derived from the known governing equations. The physics weight is selected using chronological recursive validation and evaluated against an architecturally identical multi-step FNN with λ=0. Conventional FNN, LSTM, Echo State Network (ESN), Autoregressive AR(10), and Dynamic Mode Decomposition baselines are also evaluated using untouched test trajectories. For the 1000-step Lorenz test rollout, the ESN achieved the lowest mean squared error (MSE) of 0.1701, followed by the LSTM with 22.9962. The Hybrid PI-FNN produced an MSE of 95.8157, compared with 87.9260 for its λ=0 ablation; therefore, physics regularization did not improve Lorenz test MSE, although their forecast horizons at a 10% normalized-error threshold were similar (273 and 272 steps, respectively). For the Rössler system, the Hybrid PI-FNN reduced recursive MSE from 0.2202 for the λ=0 ablation to 0.0834, corresponding to a 62.11% reduction, while both models completed the maximum evaluated 1000-step forecast horizon. Nevertheless, the ESN again achieved the lowest Rössler MSE of approximately 8.04×10−5. Finite-horizon correlation-dimension analysis, exact governing-equation Lyapunov spectra, computational-cost comparisons, and five-seed paired experiments were additionally conducted. The exact spectra confirmed one positive, one approximately neutral, and one negative exponent for each system, indicating chaotic but not hyperchaotic behavior. The paired multi-seed analysis did not establish a statistically significant forecasting advantage from physics regularization. These findings show that higher-order physics consistency can benefit particular systems and configurations, but it does not guarantee universal superiority in recursive chaotic forecasting.