Typically, nonlinearity is considered to be problematic and sometimes can lead to dire consequences. However, the nonlinearity in a Duffing oscillator array can enhance its ability to be used as a reservoir computer. Machine learning and artificial neural networks, inspired from the biological computing framework, have shown their immense potential, especially in real-time temporal data processing. Here, the efficacy of a Duffing oscillator array is explored as a reservoir computer by using information theory. To do this, a reservoir computer model is studied numerically, which exploits the dynamics of the array. In this system, the complex dynamics stem from the Duffing term in each of the identical oscillators. The effects of various system parameters of the array on the information processing ability is discussed from the perspective of information theory. By varying these parameters, the information metric was found to be topologically mixed. Additionally, the importance of asynchrony in the oscillator array is also discussed in terms of the information metric. Since such nonlinear oscillators are used to model many different physical systems, this research provides insight into how physical nonlinear oscillatory systems can be used for dynamic computation, without significantly modifying or controlling the underlying dynamical system. To the authors' knowledge, this is the first use of Shannon's information rate for quantifying a reservoir computer of this kind, as well as the first comparison between synchronization phenomena and the computing ability of a reservoir.
This paper explores the stochastic dynamics of a Hopf adaptive frequency oscillator when driven by noise. Adaptive oscillators are nonlinear oscillators that store information via plastic states. As noise is ubiquitous in physical systems, it is important to gain an understanding of the stochastic effects on adaptive oscillators. Previously, it has been shown that a simplified analysis of the Fokker–Planck equation results in affecting the plastic frequency state of these oscillators. However, when the full Fokker–Planck equation is considered, new behaviors are observed due to changes in oscillation amplitudes in addition to frequencies. The plastic frequency state of these oscillators may benefit from enhanced learning due to small amplitudes of noise, converge to incorrect values for medium amplitudes of noise, and even collapse to zero in the limit of large amplitudes of noise. Interestingly, not all averaged states collapse equally, which leads a two dimensional limit cycle to collapse into single dimensional oscillations when considering the averaged dynamics. These behaviors are compared analytically through the Fokker–Planck equation, numerically using the Euler–Maruyama simulations, and finally validated experimentally using an analog, electronic circuit. These results show that noise can enhance, mislead, or even reduce the dimensionality of the averaged adaptive Hopf oscillator.
Physical reservoir computing utilizes a physical system as a computational resource. This nontraditional computing technique can be computationally powerful, without the need of costly training. Here, a Hopf oscillator is implemented as a reservoir computer by using a node-based architecture; however, this implementation does not use delayed feedback lines. This reservoir computer is still powerful, but it is considerably simpler and cheaper to implement as a physical Hopf oscillator. A non-periodic stochastic masking procedure is applied for this reservoir computer following the time multiplexing method. Due to the presence of noise, the Euler–Maruyama method is used to simulate the resulting stochastic differential equations that represent this reservoir computer. An analog electrical circuit is built to implement this Hopf oscillator reservoir computer experimentally. The information processing capability was tested numerically and experimentally by performing logical tasks, emulation tasks, and time series prediction tasks. This reservoir computer has several attractive features, including a simple design that is easy to implement, noise robustness, and a high computational ability for many different benchmark tasks. Since limit cycle oscillators model many physical systems, this architecture could be relatively easily applied in many contexts.
Adaptive oscillators (AOs) are nonlinear oscillators with plastic states that encode information. Here, an analog implementation of a four-state adaptive oscillator, including design, fabrication, and verification through hardware measurement, is presented. The result is an oscillator that can learn the frequency and amplitude of an external stimulus over a large range. Notably, the adaptive oscillator learns parameters of external stimuli through its ability to completely synchronize without using any pre- or post-processing methods. Previously, Hopf oscillators have been built as two-state (a regular Hopf oscillator) and three-state (a Hopf oscillator with adaptive frequency) systems via VLSI and FPGA designs. Building on these important implementations, a continuous-time, analog circuit implementation of a Hopf oscillator with adaptive frequency and amplitude is achieved. The hardware measurements and SPICE simulation show good agreement. To demonstrate some of its functionality, the circuit’s response to several complex waveforms, including the response of a square wave, a sawtooth wave, strain gauge data of an impact of a nonlinear beam, and audio data of a noisy microphone recording, are reported. By learning both the frequency and amplitude, this circuit could be used to enhance applications of AOs for robotic gait, clock oscillators, analog frequency analyzers, and energy harvesting.
Adaptive oscillators are a subset of nonlinear oscillators that can learn and encode information in dynamic states. By appending additional states onto a classical Hopf oscillator, a four-state adaptive oscillator is created that can learn both the frequency and amplitude of an external forcing frequency. Analog circuit implementations of nonlinear differential systems are usually achieved by using operational amplifier-based integrator networks, in which redesign procedures of the system topology is time consuming. Here, an analog implementation of a four-state adaptive oscillator is presented for the first time as a field-programmable analog array (FPAA) circuit. The FPAA diagram is described, and the hardware performance is presented. This simple FPAA-based oscillator can be used as an analog frequency analyzer, as its frequency state will evolve to match the external forcing frequency. Notably, this is done without any analog-to-digital conversion or pre-processing, making it an ideal frequency analyzer for low-power and low-memory applications.
The Hopf oscillator is a nonlinear oscillator that exhibits limit cycle motion. This reservoir computer utilizes the vibratory nature of the oscillator, which makes it an ideal candidate for reconfigurable sound recognition tasks. In this paper, the capabilities of the Hopf reservoir computer performing sound recognition are systematically demonstrated. This work shows that the Hopf reservoir computer can offer superior sound recognition accuracy compared to legacy approaches (e.g., a Mel spectrum + machine learning approach). More importantly, the Hopf reservoir computer operating as a sound recognition system does not require audio preprocessing and has a very simple setup while still offering a high degree of reconfigurability. These features pave the way of applying physical reservoir computing for sound recognition in low power edge devices.
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