Universal replication of chaotic characteristics by classical and quantum machine learning

TL;DR

This work proposes that a ML model, trained to predict the state one-step-ahead from several latest historic states, can accurately replicate the bifurcation diagram and the Lyapunov exponents of discrete dynamic systems.

Abstract

Replicating chaotic characteristics of non-linear dynamics by machine learning (ML) has recently drawn wide attentions. In this work, we propose that a ML model, trained to predict the state one-step-ahead from several latest historic states, can accurately replicate the bifurcation diagram and the Lyapunov exponents of discrete dynamic systems. The characteristics for different values of the hyper-parameters are captured universally by a single ML model, while the previous works considered training the ML model independently by fixing the hyper-parameters to be specific values. Our benchmarks on the one- and two-dimensional Logistic maps show that variational quantum circuit can reproduce the long-term characteristics with higher accuracy than the long short-term memory (a well-recognized classical ML model). Our work reveals an essential difference between the ML for the chaotic characteristics and that for standard tasks, from the perspective of the relation between performance and model complexity. Our results suggest that quantum circuit model exhibits potential advantages on mitigating over-fitting, achieving higher accuracy and stability.

Figures (4)

• Figure 1: (Color online) The illustration on the main process of time-series prediction by machine learning. The top-left panel shows the one-dimensional Logistic map [Eq. (\ref{['1D logistic']})] as an example. By implementing the $\mu$-tuned pre-processing map [see Eqs. (\ref{['eq-preprocess']}) and (\ref{['eq-featuremap_mu']})], a sample (the data of several states) is mapped to a series of vectors as the input of the ML model such as automatically-differentiable quantum circuit (ADQC) illustrated in the bottom-left panel. The ML model is optimized by minimizing the loss function that is taken as the root mean-square error of the one-step-ahead predictions [see the bottom-right panel and Eq. (\ref{['eq-MSE']}).
• Figure 2: (Color online) (a) The bifurcation diagrams obtained by ADQC, LSTM, and Logistic map (see the inset). The dense of the data points is indicated by the darkness of the colors. (b) The semi-logarithmic plot of the relative $\varepsilon_R$ versus time $t$ in different regions of the bifurcation diagram. We take $\mu= 2.2, 3.2, 3.4$, and $3.92$, which give three negative and one positive values for the Lyapunov exponent. The exponential growth of $\varepsilon_R$ for $\mu=3.92$ is fitted by Eq. (\ref{['eq-LEexp']}) with the exponential index $\eta = 0.44$ (see the black solid line). The table in (b) gives the $\eta$ for different values of $\mu$ in the chaotic region.
• Figure 3: (Color online) The LE's of the (a) 1D and (b) 2D Logistic maps and that of the corresponding ML models (LSTM and ADQC). Each point is the average of five independent simulations. At the top of (a), the two horizontal stripes show whether the LSTM and ADQC correctly (green) or incorrectly (red) give the sign of LE. In (b), the first two stripes at the top show the accuracy on giving the sign of the two LE's by LSTM, and the last two stripes give that by ADQC.
• Figure 4: (Color online) The errors of predicting the state one-step-ahead and those of replicating LE for the 1D and 2D Logistic maps. These two errors are characterized by $L$ [Eq. (\ref{['eq-MSE']})] and $L_{\text{LE}}$, respectively. In (a), we show the $L$ and $L_{\text{LE}}$ obtained by LSTM versus its hidden dimension $d_{\text{h}}$, and in (b) we show those by ADQC versus its number of layers $N_{\text{L}}$. Each data point is the average on five independent simulations, with the standard deviations demonstrated by the error bars and colored shadows.