Taming Chaos: Tensor Networks Predict Turbulent Systems

Author: Denis Avetisyan

A new approach leverages tensor networks to accurately forecast the long-term behavior of complex, chaotic dynamics.

A tensor network model efficiently encodes non-Markovian correlations within chaotic dynamics by hierarchically mapping sequences of past states—represented as input vectors with dimension $dd$ at each time step—to predicted future states through successive tensor contractions performed by rank-4 weight tensors.

This review details a tensor network framework capable of stable, compact forecasting of nonlinear systems like the Lorenz and Rössler attractors, capturing non-Markovian correlations over multiple Lyapunov times.

Predicting the long-term behavior of complex systems remains a fundamental challenge due to inherent sensitivity to initial conditions and non-Markovian dynamics. This is addressed in ‘Tensor Network Framework for Forecasting Nonlinear and Chaotic Dynamics’, which introduces a novel approach leveraging tensor networks to model and forecast chaotic systems. The study demonstrates that this framework accurately reconstructs trajectories and extends predictive horizons beyond several Lyapunov times, achieving stable forecasts with a compact, interpretable representation. Could this paradigm unlock robust, data-driven modeling capabilities for diverse complex systems, from climate prediction to hybrid quantum-classical simulations?

Decoding Chaos: The Limits of Traditional Forecasting

Traditional forecasting methods falter when applied to chaotic systems like the Lorenz system, as they rely on assumptions of linearity and stationarity that are fundamentally violated by the inherent unpredictability of chaos. Even minor initial errors can rapidly diverge into drastically different outcomes. The Lorenz system, defined by its three coupled differential equations, exemplifies these limitations, its sensitivity compounded by complex, non-Markovian temporal correlations. Accurate long-term prediction requires capturing the multiscale, hierarchical structure embedded within the chaotic attractor, a challenge recent work addresses using tensor network models (TNM), leveraging their ability to represent high-dimensional data efficiently.

Utilizing a bond dimension of 8 and the Adam optimizer with a learning rate of 0.001, the inhomogeneous parametrization of the tensor network model (TNM) demonstrates faster convergence and lower final errors, as evidenced by training and validation losses, while accurately reproducing the Lorenz system's dynamics. — Utilizing a bond dimension of 8 and the Adam optimizer with a learning rate of 0.001, the inhomogeneous parametrization of the tensor network model (TNM) demonstrates faster convergence and lower final errors, as evidenced by training and validation losses, while accurately reproducing the Lorenz system’s dynamics.

The patterns within chaos reveal their secrets only to meticulous observation.

Tensor Networks: A New Lens for Chaotic Dynamics

The Tensor Network Model (TNM) offers a compact, low-rank representation of chaotic dynamics, inspired by techniques from quantum many-body physics. This approach efficiently encodes complex correlations, potentially enabling long-term prediction. The model’s strength lies in its ability to approximate high-dimensional functions with significantly fewer parameters. Hierarchical tensor contractions drive the TNM’s computational efficiency, capturing long-range dependencies without exponential cost. Enhancements, such as inhomogeneous weight tensors, accelerate convergence and improve robustness. This results in accurate predictions, as demonstrated by low cumulative root mean squared error (CRMSE) when forecasting the Lorenz system’s behavior on unseen data.

When forecasting the Lorenz system's behavior on unseen test data with a bond dimension of 8 and the Adam optimizer at a learning rate of 0.001, the TNM, utilizing an inhomogeneous parametrization, achieves accurate predictions with a low cumulative root mean squared error (CRMSE). — When forecasting the Lorenz system’s behavior on unseen test data with a bond dimension of 8 and the Adam optimizer at a learning rate of 0.001, the TNM, utilizing an inhomogeneous parametrization, achieves accurate predictions with a low cumulative root mean squared error (CRMSE).

Optimizing Expressivity: Balancing Detail and Efficiency

A tensor network model (TNM)’s capacity to represent complex systems is directly linked to its bond dimension, controlling the amount of entanglement it can capture and relating to the system’s Lyapunov exponent. Increasing the bond dimension allows the TNM to represent intricate correlations, but increases computational cost. Evaluations on the Lorenz and Rössler systems, using Root Mean Squared Error (RMSE), demonstrate an RMSE of 0.70 with the inhomogeneous TNM on the Lorenz system—an improvement over the homogeneous model’s RMSE of 0.79. Long-term predictive capability was assessed using Cumulative Root Mean Squared Error (CRMSE), revealing the inhomogeneous TNM maintained a CRMSE below 2.1 for approximately 5.0 Lyapunov times, significantly exceeding the homogeneous TNM, which reached the same threshold after 3.6 Lyapunov times.