Machine Learning-assisted spatiotemporal chaos forecasting

. Long-term forecasting of extreme events such as oceanic rogue waves, heat waves, floods, earthquakes, has always been a challenge due to their highly complex dynamics. Recently, machine learning methods have been used for model-free forecasting of physical systems. In this work, we investigated the ability of these methods to forecast the emergence of extreme events in a spatiotemporal chaotic passive ring cavity by detecting the precursors of high intensity pulses. To this end, we have implemented supervised sequence (precursors) to sequence (pulses) machine learning algorithms, corresponding to a local forecasting of when and where extreme events will appear.


Introduction
Chaos is a long-term aperiodic behavior in a deterministic system that is sensitive to initial conditions.Chaotic dynamics implies an exponential growth of any deviation in the initial conditions of deterministic systems.Therefore, long-term forecasting becomes a difficult task to achieve.Such as the prediction of extreme events which include hydrodynamic rogue waves [1], heat waves, large floods, earthquakes, etc.During the last decade, the efforts of the scientific community have been devoted to the identification of the mechanisms of formation of extreme events.Predictability and forecasting are open problems that are attracting increasing interest.The forecast of extreme events is of particular importance because it can allow to take safety measures, mitigating potentially large impacts.This work aims at treating the predictability of extreme events produced by chaotic dynamics in extended complex systems through the innovative approach of information theory combined with machine learning.

Complex Dynamics
Despite many years of intensive research to understand the complex dynamics of chaos, most of it is limited to theoretical investigations.Only a few experimental works have been reported due to the enormous precision required to gain knowledge about the initial conditions.Our study is based on a passive resonator consisting of a ring of optical fiber synchronously pumped near a cavity resonance [2].For simplicity, the ring was set to operate in a monostable regime, i.e., a region where the transmission function is single-valued for a given pump power.By pumping the cavity well beyond the cavity threshold, typically a few times, the continuous wave solution decomposes into a periodic wave train, which in turn experiences an oscillatory instability and then evolves into a chaotic regime.The dynamics of the light flowing in the cavity is accurately modeled by the driven and damped nonlinear Schrödinger equation, called the Lugiato-Lefever equation (LLE) [3]: here Ψ is the normalized slowly varying envelope of the electric field that circulates within the cavity, S is the pump, Δ is the frequency detuning, t corresponds to the slow evolution of Ψ over successive round-trips, τ accounts for the fast dynamics that describes how the electric field envelope changes along the fiber.

Complex Behaviour
Figure 1 shows the evolution of the output cavity field intensity |Ψ(t,τ)| 2 in the time domain, round trip to round trip.The description of the spatiotemporal chaos can be realized by analogy with hydrodynamics.Zooming in (from left to right panels) on the map in figure 1, we can see two regions, one with a turbulent evolution where the detection of precursors is very difficult, and another laminar region where we are able to detect a pattern consisting of a train of pulses.From this pattern, we can associate these pulses with precursors having a double peak shape and the pulses are located in the center of the double peaks.Using a moving window of two Gaussian double peaks separated by the average separation of the precursors, we perform a line-by-line convolution operation to detect these precursors.Once the precursors are detected, we perform a pulse-precursor association as shown in figure 2 (in this figure, pulses are found at specific horizon corresponding to a multiple of the position of the first precursor).Since the spatiotemporal chaos generated in the resonator is highly dimensional, the prediction of the fully developed turbulence in the fiber ring cavity is a great challenge.And since we cannot find exact solutions in this system, we try to understand the complex dynamics with some numerical tools.The prediction of high dimensional chaotic systems has taken a big step forward after the improvements in supervised machine learning algorithms.These studies have been done mainly using deep learning and recurrent neural networks.The ability of these neural networks to find hidden correlations in the data has allowed us to make predictions about complex spatiotemporal dynamics.By providing model-free processes, it is possible that the tools of chaos theory are no longer needed to deal with time series in general.

Results
So after choosing the horizon of our prediction time, we associate all pulses and precursors pairs and then we feed the neural network these pairs in order to predict the pulses after three round trips for a given precursors.We tried two types of neural networks, one which is the Long Short Term Memory (LSTM) and the other Gated Recurrent Unit (GRU), which are recurrent neural networks usually used for sequence to sequence forecasting.We can see the predicted pulses in figure 3. We compared the performance of each network using Pearson Correlation (PC) which is a measure of the linear correlation between measured and predicted data.And we did this study for different complexity by increasing the power.For the less chaotic system, we have a correlation of approximately 90% using the LSTM network and 81% using the GRU network as shown in the two graphs of figure 4.And we see that the correlation has decreased for the more chaotic system where we achieved a 75% correlation which is still considered a good correlation, knowing that a correlation higher than 50% is considered a good prediction.

Conclusion
Combining information theory and machine learning, we were able to predict when and where extreme events occur in the complex dynamics of a fiber ring cavity.

Fig. 1 .
Fig. 1.Shows the evolution of the output cavity field intensity |Ψ(t,τ)| 2 in the time domain, round trip to round trip.From the left to the right we have magnified the dynamics.

Fig. 2 .
Fig. 2. Shows the profile of (a) a precursors, (b) the pulse after associated with the precursor, (c) the pulse after three roundtrips.

Fig. 3 .
Fig. 3. Shows the profile of (a) the measured pulse, (b) and (c) the pulse predicted by the LSTM and GRU network respectively.

Fig. 4 .
Fig. 4. Graphs showing Pearson correlation for the less chaotic system.