Photoionization by an eXtreme UltraViolet (XUV) attosecond pulse train (APT) in the presence of an infrared pulse (RABBITT method) conveys information about the atomic photoionization delay. By taking the difference of the spectral delays between pairs of rare gases (Ar,He), (Kr, He) and (Ne,He) it is possible to eliminate in each case the larger group delay ('attochirp') associated with the APT itself and obtain the Ar, Kr and Ne Wigner delays referenced to model calculations of the He delay. In this work we measure how the delays vary as a function of XUV photon energy but we cannot determine the absolute delay difference between atoms due to lack of precise knowledge of the initial conditions. The extracted delays are compared with several theoretical predictions and the results are consistent within 30 as over the energy range from 10 to 50 eV. An 'effective' Wigner delay over all emission angles is found to be more consistent with our angle-integrated measurement near the Cooper minimum in Ar. We observe a few irregular features in the delay that may be signatures of resonances.
Reservoir computing is a neural network approach for processing time-dependent signals that has seen rapid development in recent years. Physical implementations of the technique using optical reservoirs have demonstrated remarkable accuracy and processing speed at benchmark tasks. However, these approaches require an electronic output layer to maintain high performance, which limits their use in tasks such as time-series prediction, where the output is fed back into the reservoir. We present here a reservoir computing scheme that has rapid processing speed both by the reservoir and the output layer. The reservoir is realized by an autonomous, time-delay, Boolean network configured on a field-programmable gate array. We investigate the dynamical properties of the network and observe the fading memory property that is critical for successful reservoir computing. We demonstrate the utility of the technique by training a reservoir to learn the short-and long-term behavior of a chaotic system. We find accuracy comparable to state-of-the-art software approaches of similar network size, but with a superior real-time prediction rate up to 160 MHz.Reservoir computers are well-suited for machine learning tasks that involve processing timevarying signals such as those generated by human speech, communication systems, chaotic systems, weather systems, and autonomous vehicles. Compared to other neural network techniques, reservoir computers can be trained using less data and in much less time. They also possess a large network component, called the reservoir, that can be re-used for different tasks. These advantages have motivated searches for physical implementations of reservoir computers that achieve high-speed and real-time information processing, including opto-electronic and electronic devices. Here, we develop an electronic approach using an autonomous, time-delay, Boolean network configured on a field-programmable gate array (FPGA). These devices allow for complex networks consisting of 1,000's of nodes with arbitrary network topology. Time-delays can be incorporated along network links, thereby allowing for extremely high-dimension reservoirs. The characteristic time scale of a network node is less than a nanosecond, allowing for information processing in the GHz regime. Further, because the reservoir state is Boolean rather than realvalued, calculation of an output from the reservoir state can be done rapidly with synchronous FPGA logic. We use such a reservoir computer for the challenging task of forecasting the dynamics of a chaotic system. This work paves the way for low-cost, compact reservoir computers that can be embedded in various commercial and industrial systems for real-time information processing.
We develop and test machine learning techniques for successfully using past state time series data and knowledge of a time-dependent system parameter to predict the evolution of the “climate” associated with the long-term behavior of a non-stationary dynamical system, where the non-stationary dynamical system is itself unknown. By the term climate, we mean the statistical properties of orbits rather than their precise trajectories in time. By the term non-stationary, we refer to systems that are, themselves, varying with time. We show that our methods perform well on test systems predicting both continuous gradual climate evolution as well as relatively sudden climate changes (which we refer to as “regime transitions”). We consider not only noiseless (i.e., deterministic) non-stationary dynamical systems, but also climate prediction for non-stationary dynamical systems subject to stochastic forcing (i.e., dynamical noise), and we develop a method for handling this latter case. The main conclusion of this paper is that machine learning has great promise as a new and highly effective approach to accomplishing data driven prediction of non-stationary systems.
We propose and demonstrate a nonlinear control method that can be applied to unknown, complex systems where the controller is based on a type of artificial neural network known as a reservoir computer. In contrast to many modern neural-network-based control techniques, which are robust to system uncertainties but require a model nonetheless, our technique requires no prior knowledge of the system and is thus model-free. Further, our approach does not require an initial system identification step, resulting in a relatively simple and efficient learning process. Reservoir computers are well-suited to the control problem because they require small training data sets and remarkably low training times. By iteratively training and adding layers of reservoir computers to the controller, a precise and efficient control law is identified quickly. With examples on both numerical and high-speed experimental systems, we demonstrate that our approach is capable of controlling highly complex dynamical systems that display deterministic chaos to nontrivial target trajectories.
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