Tensor Network Framework for Forecasting Nonlinear and Chaotic Dynamics

TL;DR

This work introduces a tensor network model (TNM) for forecasting chaotic dynamics by encoding non-Markovian memory and multiscale structure through hierarchical contractions. By applying the TNM to the Lorenz and Rössler attractors, the authors demonstrate accurate short-term trajectory reconstruction and predictive horizons extending across several Lyapunov times, with an inhomogeneous parametrization of weight tensors yielding faster convergence and greater robustness than a homogeneous counterpart. Bond-dimension scaling reveals that modest $D$ suffices due to the low intrinsic dimensionality of chaotic attractors, while the approach offers a physically interpretable control on model capacity. The TNM provides a compact, versatile framework for data-driven chaotic forecasting with potential applications in climate modeling and hybrid quantum-classical simulations, where memory effects and multiscale correlations are essential.

Abstract

We present a tensor network model (TNM) for forecasting nonlinear and chaotic dynamics, bridging quantum many-body methods with classical complex systems. The TNM leverages hierarchical tensor contractions to encode non-Markovian temporal correlations and multiscale structures, enabling compact and interpretable representations of chaotic flows. Using the Lorenz and Rössler systems as benchmarks, we show that the TNM accurately reconstructs short-term trajectories and faithfully captures the attractor geometry. The model enables robust short-term forecasting beyond several Lyapunov times, offering a meaningful horizon for data-driven prediction under chaos. Inhomogeneous parametrization of weight tensors improves convergence and robustness compared to homogeneous parametrization, while scaling with bond dimension reveals saturation beyond modest values, consistent with the low intrinsic dimensionality of the chaotic attractor. This work establishes tensor networks as a universal paradigm for data-driven modeling of complex dynamical systems, offering physically motivated control of model expressivity and opening pathways toward applications in climate systems and hybrid quantum-classical simulations.

Figures (8)

• Figure 1: Schematic of the tensor network model (TNM), which maps a sequence of past states to the next state via hierarchical tensor contractions, efficiently encoding non-Markovian correlations in chaotic dynamics. The bottom layer ($v_0^0, v_1^0, \dots, v_6^0$) represents input vectors at successive time steps, each with feature dimension $d$. Hidden layers consist of rank-4 weight tensors performing tensor contractions to extract hierarchical correlations. The output node at the top encodes the predicted future state.
• Figure 2: Forecasting performance of the TNM on the Lorenz system with homogeneous (a),(c) and inhomogeneous (b),(d) parametrizations of weight tensors. Panels (a),(b) show the evolution of $x(t)$: gray solid curves denote the true trajectory, blue dashed lines are training predictions, and orange dash-dot lines are validation predictions. Both models reproduce the dynamics, but the inhomogeneous case adheres more closely to the ground truth. Panels (c),(d) display training and validation losses on a logarithmic scale, showing faster convergence and lower final errors for the inhomogeneous model. The bond dimension is fixed at $D=8$, and training is performed by the Adam optimizer with a learning rate of 0.001, using 80 epochs for the homogeneous and 60 epochs for the inhomogeneous model.
• Figure 3: Prediction performance of TNM on the Lorenz attractor. (a) Parity plot for the homogeneous model, showing predicted versus true coordinates $(x, y, z)$ with RMSE = 0.79. (b) Parity plot for the inhomogeneous model, achieving improved accuracy with RMSE = 0.70. (c) Histogram and cumulative distribution function (CDF) of pointwise Euclidean prediction error for the homogeneous model. 90.7% of predictions lie within a distance of 1.0 from the ground truth. (d) Same as (c) but for the inhomogeneous model, where 92.1% of predictions fall below the error threshold of 1.0. Together, these results demonstrate the superior predictive accuracy and robustness of the inhomogeneous TNM.
• Figure 4: Forecasting performance of the TNM on unseen test data using homogeneous (a) and inhomogeneous (b) parametrizations. Gray soild curves show the ground-truth trajectory $x(t)$, green dashed lines indicate TNM predictions, and orange lines (right axis) represent cumulative RMSE (CRMSE). Bond dimension is set to $D=8$, and models are trained with Adam optimizer at learning rate 0.001.
• Figure 5: Dependence of TNM performance on the bond dimension $D$, comparing homogeneous (dashed lines) and inhomogeneous (solid lines) parametrizations. Training (triangles) and validation (circles) losses are shown at epoch 200. Increasing $D$ from 2 to 5 reduces the losses for both cases, but further gains plateau for $D>5$. At all bond dimensions, the inhomogeneous parametrization achieves consistently lower losses than the homogeneous one.