Short-wavelength Reverberant Wave Systems for Enhanced Reservoir Computing

10 Machine learning (ML) has found widespread application over a broad range of important tasks. 11 To enhance ML performance, researchers have investigated computational architectures whose 12 physical implementations promise compactness, high-speed execution, physical robustness, and 13 low energy cost. Here, we experimentally demonstrate an approach that uses the high sensitivity 14 of reverberant short wavelength waves for physical realization and enhancement of computational 15 power of a type of ML known as reservoir computing (RC). The potential computation power 16 of RC systems increases with their eﬀective size. We here exploit the intrinsic property of short 17 wavelength reverberant wave sensitivity to perturbations to expand the eﬀective size of the RC 18 system by means of spatial and spectral perturbations. Working in the microwave regime, this 19 scheme is tested on diﬀerent ML tasks. Our results indicate the general applicability of reverberant 20 wave-based implementations of RC and of our eﬀective reservoir size expansion techniques. 21


22
Machine learning (ML) algorithms have demonstrated the capability to perform a variety 23 of tasks without being constructed with specific knowledge of the rules governing the task 24 [1, 2]. Important ML performance metrics, such as speed and energy efficiency, depend on 25 the computing platform on which the ML algorithm operates. Accordingly, researchers have 26 been motivated to find platforms and associated algorithms that optimize these metrics. In 27 this regard, reservoir computing (RC) [3-6], a type of ML that we describe in Sec. II, has 28 attracted attention because it can be realized in a variety of physical forms [7][8][9][10][11][12][13][14][15][16][17][18][19]. 29 Based on the proceeding motivation, we present in Sec. III an implementation of reservoir 30 computing [3, 5, 12, 20-22] that utilizes the complex response of short wavelength modes 31 in a reverberant cavity as the reservoir. When the wavelength of the fields in a cavity is 32 much smaller than the size of the cavity, the wave field has effectively a high degree of 33 freedom. Equivalently, in this 'short wavelength' regime the field can be thought of as a 34 superposition of many modes: the number of which is determined by the bandwidth of the 35 time dependent signals to be produced and the spectral mode density of the cavity. This Reservoir computing is a general type of ML whose structure, in the case of continuous 50 time operation, can be specified as follows. Input variables, in the form of a time (t) 51 dependent vector u(t), drive the evolution of a reservoir stater(t). The reservoir stater 52 is typically a high dimensional vector, and the input u is a much lower dimensional signal 53 vector to whichr responds. (In our case,r represents the field within the wave confining 54 structure, e.g., the microwave cavity shown in Fig. 1a.)

55
The reservoir state evolves according to a reservoir dynamical system f , In Eq. 1, it is assumed that f satisfies the 'echo-state' property [3, 5], which requires that 57 for any input time series u(t),r(t) becomes independent of the initial conditionr(0) as t 58 becomes large. A time series of output vectors r(t) of dimension N r is derived from the 59 reservoir stater(t) via a function g, 60 r(t) = g(r(t)).  Figure 1 shows the proof-of-principle experimental microwave reverberant wave-based

81
RC system considered in this paper. In Fig. 1a  The input information is transferred from a lab computer to the AWG and injected into the chaotic enclosure through an electric dipole antenna. Several diode-loaded antennas are used to probe the EM field, whose voltage signals are measured by the oscilloscope and further transferred to the lab computer and stored. The cavity shown in panel a is thin in the vertical, z-direction, and has a shape in the (x, y) directions in which the bottom and left walls are straight lines and the upper and right walls are circular arcs. This leads to a purely vertical electrical field E z (x, y) whose complex two-dimensional spatial distribution is shown in panel a via the blue-to-red color coding within the cavity. b, Schematic of the diode-loaded port. c, The dynamics of diodeport voltage (red) under single frequency input wave at 4 GHz (black). d, The Fourier transform (FT) of the diode-port signal shown in c.
into analog waveforms by an AWG (arbitrary waveform generator) and stored for both the Correspondingly, we write f as so that Thus, each of ther (i) evolves independent of the others, except from the mutual dependence 122 on the input stream u(t). Comparing Eq. 5 with Eq. 1, we consider each of the components 123r i ofr to be the wave field distribution in a hypothetical cavity described by f i . The c A plot at N r = 90 (N 0 = 3, N b = 30) of the percent deviation of the Rössler observer task NMSE, n |s(t) − s ′ (t)| 2 / n |s(t)| 2 , from the NMSE with optimal parameters (T osc , T decay ) = (250, 700) as a function of the input duration T osc and the system decay time T decay . d The normalized mean square error in s versus N r the dimension of r, which is varied, e.g., for the experimental (solid) curves, by starting at a maximum value of N r = 630 (corresponding to N 0 = 3, N b = 30, N f = 7), and then randomly removing virtual outputs to successively lower N r .

157
We have experimentally investigated the effectiveness of our reverberant wave approach 158 to implementing RC on several different tasks. Ensembles of new reservoirs are created by 159 translating a metallic perturber and/or changing the oscillation period of the input signal 160 from the AWG. In our experiment, a combined RC of size N r is given by where N 0 , N b , and N f represent the number of measurement channels, the number of applications of the boundary condition perturbation RET, and the number of applications of the 163 frequency stirring RET, respectively.

164
For the experiments described below we used the bowtie cavity with 3 output ports.

165
Without using RETs we find that, for all of the 5 tasks tested, the RC system fails to give 166 useful results. However, with RET implemented, the performance of all 5 tasks improves, 167 becoming better as the effective RC system size N r increases (Figs. 3d, 4b, 4d, and 4f). For

168
the 5 examples tested we found that using RET to sufficiently boost N r resulted in excellent 169 performance. In the rest of this section we give results of our tests. test are shown in Fig. 4c and d.

219
Example 4: The function simulator task. For the function simulator task, the RC is 220 expected to output any periodic waveform that is desired. For this purpose, we take the 221 input to be a sinusoidal waveform with the period of the desired waveform. In our test 222 example we take the desired waveform to be the cube of the sine wave, and we train the Observer tasks for dynamical systems. For the observer tasks, the input is a subset 287 of a multi-variable dynamical system states and the output is the inferred set of unobserved 288 variables of the system. We have applied the observer task for two systems with chaotic 289 attractors: the chaotic Rössler system, governed by Eqns.

304
The NARMA-10 test. For the NARMA-10 task, the input is a random bit series 305 u(n) drawn from a uniform distribution over the interval [0, 0.5]. The target output is 306 computed from the following 10th-order nonlinear relationship: y(n + 1) = 0.3y(n) + 307 0.05y(n) 9 i=0 y(n − i) + 1.5u(n − 9) · u(n) + 0.1. Its complex behavior and a 10-state 308 memory requirement make the NARMA-10 task a popular benchmark test for both soft- The authors declare no competing interests.