Abstract
We show that coherent laser networks (CLNs) exhibit emergent neural computing capabilities. The proposed scheme is built on harnessing the collective behavior of laser networks for storing a number of phase patterns as stable fixed points of the governing dynamical equations and retrieving such patterns through proper excitation conditions, thus exhibiting an associative memory property. It is discussed that despite the large storage capacity of the network, the large overlap between fixedpoint patterns effectively limits pattern retrieval to only two images. Next, we show that this restriction can be uplifted by using nonreciprocal coupling between lasers and this allows for utilizing a large storage capacity. This work opens new possibilities for neural computation with coherent laser networks as novel analog processors. In addition, the underlying dynamical model discussed here suggests a novel energybased recurrent neural network that handles continuous data as opposed to Hopfield networks and Boltzmann machines that are intrinsically binary systems.
1 Introduction
In the recent years, there has been a growing interest in developing new platforms for generalpurpose or applicationspecific computing that offer an advantage over classical processors in terms of computational time, energy efficiency and scalability [1]. Although quantum computing is widely considered as a promising route, it appears that the classical nonlinear systems exhibit a largely underexplored computational capacity that is not properly utilized in conventional digital computers [2]. In this regard, there is great interest in developing alternative hardware platforms, which subsequently demand for compatible new algorithms.
Inspired by the biological brain, an interesting computational platform seems to be a network of nonlinear units, i.e., neurons, with a complex architecture that allows dense longrange interactions [3]. In such systems, computing is an emergent nonlinear dynamical behavior of the network, and, in principle, can be much more efficient for certain tasks in comparison with the wellestablished sequential architecture. Interestingly, in the physics community interest in the subject of neural computation was raised at an early stage by the introduction of Hopfield networks [4, 5]. In these contexts, mainly influenced by spin systems in statistical mechanics, computing is viewed as finding states that minimize a global network energy function. Analog physical implementations of Hopfield networks with optoelectronics [6] and CMOS circuits [7, 8] were demonstrated for a small number of neurons at early stages. More importantly, such networks inspired unconventional methods for solving combinatorial optimization problems [9] as well as energybased models for machine learning [10]. On the other hand, interest in physical implementation of unconventional computing with densely connected architectures has recently regained interest in photonics [11–14]. In fact, energyefficiency and the possibility of implementing longrange interactions make photonics an attractive candidate for neural computation. Accordingly, there is interest in developing novel methods and algorithms that allow for taking advantage of the existing photonics systems for unconventional computing.
Here, we show that coherent laser networks (CLNs) exhibit collective neural computing capabilities, and devise the fundamental requirements for realizing an associative memory for continuous patterns. What facilitates this work is recent experimental progress in creating large networks of coherently coupled photonic oscillators [11, 12, 15], [16], [17]. These activities have been primarily centered on solving computationallyhard problems by optical simulation of classical spin models. In particular, coherent laser networks have been used for solving nonconvex optimization problems of the form of the classical XY Hamiltonian [18], while numerical simulation of the governing dynamical models have been shown to be an efficient optimization method [19]. Here, it is shown that coherent laser networks hold a great potential as a physical energybased neural computing platform.
The present work is timely due to two important recent realizations that make coupled laser systems an attractive choice as a physical neural network. First, is the possibility of implementing dissipative interaction among laser networks which ensures the presence of fixed point attractors for such dynamical systems [15, 20]. The presence of dissipative coupling is shown to shift the dynamical model governing laser networks toward a class of reactiondiffusion systems that are known to be the host of exotic phenomena most notably pattern formation, which is the core of the present work [21]. In contrast, driven by device applications, traditionally the general trend has been centered on dispersive interaction among laser arrays to avoid power loss, which in turn could result in unstable and chaotic behavior. Second, several recent works show the possibility of creating coupling through complex graph topologies, which is essential for implementing and training a recurrent neural network based on laser networks with desired wiring [15, 22]. In contrast, in the past the emphasis has been on lattice geometries with nearest neighbor couplings. It is because of this latter that we opt to call the system a laser network rather than a laser array.
Figure 1 illustrates the concept of the proposed neural network based on coherent laser networks. Considering two coupled lasers, as shown schematically in Figure 1(a), by utilizing dissipative coupling, the two lasers can reach an equilibrium frequencylocked regime which can be described through the lasers phase degrees of freedom interacting with the coupling rate w (Figure 1(b)). The associated energy landscape function governing the equilibrium phases of the two lasers is shown in Figure 1(c). It can be shown that for w > 0 (attractive coupling) the energy function is minimum at ϕ _{2} = ϕ _{1}, while for w < 0 (repulsive coupling) the energy function is minimum for ϕ _{2} − ϕ _{1} = ±π [20, 21, 23]. Therefore, the dynamic interaction between these two lasers can promote the stabilization of the system into a given equilibrium phase distribution, in this case either inphase or outofphase depending on the sign of the coupling coefficient, which forms the basis of utilizing larger laser networks for storing highdimensional patterns.
Figure 1(d) schematically depicts a network of n lasers that are coherently coupled through diffraction engineering. This coherent laser network can be considered as a complex network represented with a graph as shown in Figure 1(b). Here, each graph node, represents an artificial neuron associated with a laser that is described by its amplitude and phase, a _{ i }(t) = a _{ i }(t) exp(iϕ _{ i }(t)), as two dynamical variables. In addition, two representative neurons i and j, interact dynamically through rates (w _{ ij }, w _{ ji }), which could in general be nonreciprocal, i.e., w _{ ij } ≠ w _{ ji }. Assuming that all lasers are identical, starting from an initial condition, under proper conditions the network can reach a phaselocking state where the amplitudes are nearly equal and the phases have a fixed pattern [20]. In this regime, the system can be viewed as a network of phase oscillators that are governed by an ndimensional energy landscape function as shown schematically in Figure 1(c). The equilibrium phase patterns of the laser network are associated with the local minima of this energy landscape function. Thus, the laser network can be viewed as an energybased neural network. The use of such an energybased model can be best demonstrated through associative memory functionality. In such a system, by properly choosing the weight matrix, one can suitably engineer the landscape function such that desired patterns are located at its local minima as illustrated in Figure 1(d). In this manner, the network memorizes a given pattern which can be retrieved when it is suitably initialized.
In this work, first, it is shown that the conservative reciprocal coupling allows for the formation of binary patterns. We show that by using the Hebbian learning desired patterns can be memorized by the network, although the storage capacity is limited to only two images. Next, it is shown that these restrictions can be uplifted by considering nonreciprocal coupling that allows for treating continuous patterns, while increasing the storage capacity. A simple learning rule for training such coherent laser networks is introduced, which is based on simultaneously embedding a number of patterns as fixed point solutions of the dynamical models governing laser networks. These results are justified by numerical simulation of the dynamical equations governing laser networks.
2 Formulation
2.1 A single laser
Given the importance of a single laser oscillator as an artificial neuron and a building block of the coherent laser network, first we discuss it in the following. Here, laser oscillations is modeled through a secondorder nonlinear oscillator as: [24, 25]
where, a is the complex modal amplitude of the electric field in laser cavity, g _{0} is the small signal gain, b represents the complex amplitude of a drive laser for seeding or optical injection [26–28], and ξ represents fluctuations. Here, the oscillation frequency ω _{0} is gauged out for simplicity, the laser is assumed to be frequencylocked with the drive, and the time is normalized to the photon lifetime, 1/γ, where γ is the passive cavity decay rate. This model, which is similar to the singlesideband Van der Pol [29] or the socalled Stuart–Landau oscillator [30], represents a classA laser, in which the field decay rate is much less than the decay rates of the atomic degrees of freedom, i.e., atomic polarization and population inversion [31]. The analysis presented in this work is based on this minimal model which facilitates integrability. However, it is later discussed that the results are applicable to a more general class of laser systems.
In the absence of seeding, i.e., b = 0, Eq. (1) admits a stable fixed point at
2.2 Laser networks
The extension of the dynamical model to the case of n coupled laser oscillators is straightforward. Considering n identical oscillators, the evolution equations can be written as:
In this relation, ‘⋅’ shows entrywise product,
The symmetry of the coupling matrix allows for writing the dynamical model in terms of the gradient of a Lyapunov function, i.e.,
It is straightforward to show that along the trajectories of Eq. (2) the time derivative of F is negative. This guarantees that starting from a given set of initial conditions, the evolution of the dynamical system (2) is toward the local minima of the multivariate cost function
It is important to note that the governing cost function F can be greatly simplified in the strong pump regime, where the amplitudes tend to become uniform and the phase degrees of freedom become the key players in the phase space [20]. This can be seen from Eq. (3), which shows the pump parameter g
_{0} as a penalty for intensity inhomogeneity across the laser network. By directly enforcing the condition of equal equilibrium intensity, i.e.,
It is worth recalling that ϕ _{ i } (i = 1, …, n) represent the phases of the lasers as dynamical variables that describe the phase space of the system, while φ _{ i } (i = 1, …, n) are constants that represent the phases of the drives. In the following, the attention is focused on the case of the large gain limit, which concerns only the phase degrees of freedom. In addition, for simplicity, the drive term is not considered.
3 Associative memory
The cost function of Eq. (4) is in general a nonconvex function, thus, a coherent laser network with a given weight matrix W could have numerous local minima with different basins of attractions in the phase space. In this case, if the initial point in the phase space is located within the attractor basin of a local minimum, the network will evolve toward the associate stationary state, say
The cases of binary and continuous patterns are to be treated separately. First, the case of binary pattern, e.g.,
3.1 Binary patterns
As mentioned earlier, the goal of the training is to find the coupling matrix W that results in the presence of local minima of the energy landscape function f (Eq. (4)) at desired points. To draw this connection, it is easier to start with identifying the stationary points of the energy landscape function f. Enforcing the condition of stationary solutions ∇f ≡ 0, results in the following stationary phase relations for the fixed points:
Clearly, the stationary state condition is satisfied for any binary pattern
for i ≠ j. This weight matrix clearly respects the reciprocity condition, i.e., w
_{
ij
} = w
_{
ji
}, while the energy conservation can be enforced by choosing the diagonal elements as w
_{
ii
} = ∑
_{
j
}w
_{
ij
}. It can be shown that for the weight matrix given by Eq. (6), the desired pattern is a local minimum. This can be shown by using this weight matrix in the XY Hamiltonian of Eq. (6), which results in
Figure (3) depicts the reconstruction of a binary pattern in a coherent laser network trained according to Eq. (6). Here, a binary 64 × 64 pixel image is considered (Figure 3(a)). Accordingly, we consider a network of n = 4096 lasers with the coupling coefficients of Eq. (6) based on the desired pattern shown in Figure 3(a). A corrupted version of the image is considered as the initial phases of the oscillators (Figure 3(b)). By numerically integrating the dynamics of Eq. (2), it is observed that the network successfully retrieves the original image after reaching equilibrium (Figure 3(c)). It is worth noting that in practice, the initial phases might not be controllable, while instead seeding can be used to suitably drive the network toward the memorized pattern.
It is worth mentioning the similarity of the laser network with the Hopfield network in case of binary patterns. For binary values with π contrast, the XY Hamiltonian of Eq. (4) becomes equivalent with the Ising Hamiltonian ∑
_{
i,j
}
w
_{
ij
}
s
_{
i
}
s
_{
j
} (s
_{
i
} = ±1), which forms the basis of the Hopfield network. Similarly, the weight matrix given by Eq. (6) becomes equivalent to the Hebbian learning rule of the Hopfield network, i.e.,
The Hebbian learning of Eq. (6) can be readily generalized to store more than one pattern. In this case, for k given patterns {Θ^{(1)}, …, Θ^{(k)}}, where,
It is worth noting that in case of nonbinary patterns the weight matrix of Eq. (6) does not guarantee that a desired continuous pattern is a stationary point. However, it guarantees local convexity of the landscape function at that point (see Methods). Accordingly, a network trained with relation (6) can evolve into a nearby local minimum, which, given the highly nonconvex nature of the landscape function could be close to the desired pattern. The exact reconstruction of continuous patterns is possible by utilizing complex coupling as discussed next.
3.2 Continuous patterns
The challenge with embedding a continuous pattern as a stable local minimum of the XY Hamiltonian of Eq. (4) can be resolved by making a simple change in the form of the Hamiltonian as suggested in Ref. [36]. This is done by considering the training parameters as a phase factor in the sinusoidal function according to f = ∑
_{
i,j
} cos(ϕ
_{
i
} − ϕ
_{
j
} − ψ
_{
ij
}), where the network can be simply trained to exhibit a stable local minimum at the desired continuous pattern
Inspired by the clock model proposed in Ref. [36], here the following modification of the XY Hamiltonian is suggested:
where, w _{ ij } = w _{ ij } exp(iψ _{ ij }) are complex weights. This energy function contains additional parameters, i.e., the amplitudes and phases of the weight matrix elements, which can be trained to store multiple patterns. In the following, it is shown that this phase cost function can be effectively mapped onto a coherent laser network by uplifting the physical limitations of the coupling matrix.
Considering a given continuous pattern
where, again, C is an arbitrary matrix and I is the identity matrix. A convenient choice is C = I which results in W = I − AA ^{+}.
Assuming that the target k patterns are linearly independent vectors, the weight matrix W is of rank k. Therefore, its physical implementation requires n × k independent matrix elements. In addition, similar to the case of a single pattern, it is straightforward to show that this weight matrix is generally complex but Hermitian, i.e., W ^{†} = W.
It is important to note that the presence of nonreciprocal coupling (w
_{
ij
} ≠ w
_{
ji
}) does not generally rule out the possibility of phase locking of the network [38]. In fact, the Hermiticity of the weight matrix allows the system to admit a Lyapunov function, which guarantees asymptotic stability of the laser network. In this case, due to the Hermiticty of the coupling matrix,
The proposed learning is tested with a dataset of k = 64 continuous patterns of n = 64 × 64 pixels, shown in Figure 4(a). These grayscale images are selected from a collection of dog faces from the downsampled ImageNet dataset [39]. The amplitude and phase of the complex weight matrix of Eq. (8) are plotted in Figures 4(b). Here, the network successfully stores and retrieves all the 64 training patterns. For demonstration purposes, the reconstruction of two exemplary images from their corrupted versions is depicted in Figure 4(c and d).
4 Discussion
4.1 Frequency locking
It is worth mentioning that the results presented above were built on the idealistic assumption of identical oscillators, while in practice, individual laser cavities can have deviations in their resonance frequencies and linewidths. However, simulation results show that the system exhibits selforganizing behavior and can reach phaselocking in presence of tolerable perturbations. To explore this aspect, the network of Figure 4 is simulated under the presence of random frequency and linewidth detunings of the individual network elements. The effect of detuning is considering by changing the first term of Eq. (2) according to −a(t) → −(1+ δγ +i δω ) ⋅a(t), thus
where,
4.2 Gain dynamics
The results presented in this work were based on the socalled classA laser model, where the gain can be considered a constant, while many practical lasers fall in the category of classB lasers, where the gain evolves dynamically [31]. The simplified model used here admits a Lyapunov function, which allows for an analytical treatment of the laser network and finding a training method. However, it should be noted that the proposed training method concerns solely the stationary behavior of the network through the coupling matrix. Therefore, the dynamics of the gain is not expected to violate the associative memory functionality, so long as the stability of the fixed points is guaranteed. As shown recently, a large gain lifetime, compared to the photon lifetime, can give rise to destabilization of shallow local minima or metastable states such as vortex states in a lattice of coupled lasers [21]. In this case, however, numerical simulations indicate that the patterns embedded through the learning rule of Eq. (8) remain stable even for large gain lifetimes. This is justified by repeating the simulations of Figure 4 with a classB laser model. In this model, the gain of a laser oscillator is driven at a constant pump rate, while it decays linearly for small field intensities and nonlinearly when the field intensity grows. The normalized rate equations can then be written as:
Here,
To investigate the effect of the dynamic gain, Eq. (10) are simulated for the network described in Figure 4. The results show that the associative memory functionality is preserved. Figure 6 exemplifies the dynamics for reconstructing one of the stored patterns.
5 Conclusions
In summary, in this paper the potential of using coherent laser networks for neural computing was proposed. The coherent laser network is governed by a nonconvex energy landscape function that can contain a large number of fixed point attractors. The use of the coherent laser network as an energybased neural network model was demonstrated through an associative memory functionality. It was shown that using nonreciprocal coupling between lasers allows for going beyond binary data and adding the capability of handling continuous patterns. This work outlines the great potential of coherent laser networks for optical neural computing. In addition, the proposed dynamical model could have applications as a novel continuoustime neural network for conventional digital computing.
The present work was focused on the associative memory functionality as a generic task for energybased models, while it remains to examine the full capacity of coherent laser networks as energybased models in different network architectures and for various machine learning functionalities [40]. Likewise, the proposed pseudo inverse learning is a simplistic approach, which is suitable for experimental realization given that it requires a lowrank weight matrix. However, of interest would be to develop advanced training algorithms that allow for harnessing the full capacity of the coherent laser networks for machine learning. Finally, it is worth stressing that the proposed system that builds on simulating the dynamics of laser networks can pave the way for developing novel energybased models for handling continuous patterns in applications such as pattern recognition and feature extraction. In addition, while requiring large numbers of laser oscillators, the proposed neural computing framework can be implemented with the existing photonic technology. In particular this system can be realized through solid state lasers in selfimaging cavities with high spatial mode degeneracy [15, 18] and by harnessing diffraction engineering for creating an arbitrary complex coupling matrix. In addition, the proposed laser network can be realized in timemultiplexed oscillators in fiber loops and by utilizing electronic feedback for realizing an arbitrary complex coupling matrix [13, 14].
6 Methods
6.1 Numerical simulations
The coherent laser as described by Eq. (2) is in essence a continuoustime energybased recurrent neural network. Considering the potential importance of the proposed model for unconventional computing through simulations of the underlying model with digital computers, in the following, numerical simulations are briefly discussed. The numerical simulations of Eq. (2) are performed with a forwarddifference Euler method according to:
For the simulations discussed in this paper, the network converges rapidly (after
6.2 The Hessian matrix
The Lyapunov function of Eq. (3) is a function of 2n variables, which can be cast in a vector as
where, ∇F is the gradient vector and H is a 2n × 2n Hessian matrix. In this representation, stationary states are points associated with
where, H _{ d } = (g _{0} − 1)I − 2g _{0}diag(a*·a) − W, and H _{ o } = −g _{0}diag(a·a).
For the phase cost function of Eq. (4), the Hessian matrix is an n × n matrix with elements h _{ ij } = ∂ ^{2} f/∂ϕ _{ i } ∂ϕ _{ j }, which are found to be:
For the choice of the weight matrix of Eq. (6) for a given pattern
Funding source: Air Force Office of Scientific Research
Award Identifier / Grant number: FA95502210189
Funding source: U.S. Air Force
Award Identifier / Grant number: Unassigned

Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

Research funding: This project was supported by the U.S. Air Force Office of Scientific Research (AFOSR) Young Investigator Program (YIP) Grant No. FA95502210189. VMM was supported by the Army Research Office through Grant No. W911NF2210091.

Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Data availability: The data that support the findings of this study are available from the corresponding author upon reasonable request.
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