[0001] This invention was made with Government support under contract FA9550-17-1-0002 awarded by the Air Force Office of Scientific Research. The Government has certain rights in the invention. CROSS-REFERENCE TO RELATED APPLICATIONS

[0002] The present application claims the benefit of and priority to U.S. Provisional Patent Ap- plication No. 62/669,899 entitled”Training of Photonic Neural Networks Through in Situ Back- propagation”,filed May 10, 2018 and to U.S. Provisional Patent Application No. 62/783,992 entitled”Training of Photonic Neural Networks Through in Situ Backpropagation”,filed Decem- ber 21, 2018. The disclosure of U.S. Provisional Patent Application Serial Nos. 62/669,899 and 62/783,992 are herein incorporated by reference in its entirety. FIELD OF THE INVENTION

[0003] The present invention generally relates to photonic neural networks and more specifically relates to training of photonic neural networks through in situ backpropagation. BACKGROUND

[0004] Recently, integrated optics has gained interest as a hardware platform for implementing machine learning algorithms, including artificial neural networks (ANNs), which rely heavily on matrix-vector multiplications that may be done efficiently in photonic circuits. Artificial neural net- works, and machine learning in general, are becoming ubiquitous for an impressively large number of applications. This has brought ANNs into the focus of research in not only computer science, but also electrical engineering, with hardware specifically suited to perform neural network op- erations actively being developed. There are significant efforts in constructing artificial neural network architectures using various electronic solid-state platforms, but ever since the conception of ANNs, a hardware implementation using optical signals has also been considered. Photonic implementations benefit from the fact that, due to the non-interacting nature of photons, linear op- erations– like the repeated matrix multiplications found in every neural network algorithm– can be performed in parallel, and at a lower energy cost, when using light as opposed to electrons.

[0005] Many implementations of photonic neural networks are trained using a model of the sys- tem simulated on a regular computer, but this can be inefficient for two reasons. First, this strategy depends entirely on the accuracy of the model representation of the physical system. Second, unless one is interested in deploying a large number of identical,fixed copies of the ANN, any ad- vantage in speed or energy associated with using the photonic circuit is lost if the training must be done on a regular computer. Alternatively, training using a brute force, in situ computation of the gradient of the objective function has been proposed. However, this strategy involves sequentially perturbing each individual parameter of the circuit, which is highly inefficient for large systems. SUMMARY OF THE INVENTION

[0006] Systems and methods for training photonic neural networks in accordance with embodi- ments of the invention are illustrated. One embodiment includes a method for training a set of one or more optical interference units (OIUs) of a photonic artificial neural network (ANN), wherein the method includes calculating a loss for an original input to the photonic ANN, computing an adjoint input based on the calculated loss, measuring intensities for a set of one or more phase shifters in the set of OIUs when the computed adjoint input and the original input are interfered with each other within the set of OIUs, computing a gradient from the measured intensities, and tuning phase shifters of the OIU based on the computed gradient.

[0007] In a further embodiment, computing the adjoint input includes sending the calculated loss through output ports of the ANN.

[0008] In still another embodiment, the intensities for each phase shifter of the set of phase shifters is measured after the phase shifter.

[0009] In a still further embodiment, the set of OIUs includes a mesh of controllable Mach- Zehnder interferometers (MZIs) integrated in a silicon photonic circuit.

[0010] In yet another embodiment, computing the adjoint input includes sending the calculated loss through output ports of the ANN.

[0011] In a yet further embodiment, the loss is the result of a mean squared cost function.

[0012] In another additional embodiment, the ANN is a feed-forward ANN.

[0013] In a further additional embodiment, the ANN is a recurrent neural network (RNN).

[0014] In another embodiment again, the ANN further includes a set of one or more activation units, wherein each OIU performs a linear operation and each activation unit performs a non-linear function on the input.

[0015] In a further embodiment again, computing the adjoint input includes linearizing at least one activation unit of the set of activation units prior to sending an input through the ANN.

[0016] In still yet another embodiment, the method further includes steps for performing dropout during training by shutting off channels in the activation units.

[0017] In a still yet further embodiment, non-linear functions of the ANN are performed using an electronic circuit.

[0018] In still another additional embodiment, each OIU of the set of OIUs includes a same num- ber of input ports and output ports.

[0019] Additional embodiments and features are set forth in part in the description that follows, and in part will become apparent to those skilled in the art upon examination of the specification or may be learned by the practice of the invention. A further understanding of the nature and advantages of the present invention may be realized by reference to the remaining portions of the specification and the drawings, which forms a part of this disclosure. BRIEF DESCRIPTION OF THE DRAWINGS

[0020] The description and claims will be more fully understood with reference to the follow- ingfigures and data graphs, which are presented as exemplary embodiments of the invention and should not be construed as a complete recitation of the scope of the invention.

[0021] Figure 1 illustrates an example of a schematic for an artificial neural network (ANN).

[0022] Figure 2 illustrates an example of operations in an ANN.

[0023] Figure 3 illustrates a schematic of a process for experimental measurement of gradient information.

[0024] Figure 4 conceptually illustrates a process for experimental measurement of gradient in- formation.

[0025] Figure 5 illustrates a numerical demonstration of a time-reversal procedure.

[0026] Figure 6 illustrates how a time-reversal interference technique could be performed for a layer embedded in a network.

[0027] Figure 7 conceptually illustrates a process for using a time-reversal inference method to measure sensitivities.

[0028] Figure 8 illustrates how a time-reversal inference technique can be applied without inter- nal coherent detection and preparation.

[0029] Figure 9 illustrates a schematic of a recurrent neural network (RNN).

DETAILED DESCRIPTION

[0030] Turning now to the drawings, systems and methods in accordance with certain embodi- ments of the invention can be used to train photonic neural networks. In some embodiments, meth- ods can compute the gradient of the cost function of a photonic artificial neural network (ANN) by use of only in situ intensity measurements. Processes in accordance with several embodiments of the invention physically implement the adjoint variable method (AVM). Furthermore, methods in accordance with a number of embodiments of the invention scale in constant time with respect to the number of parameters, which allows for backpropagation to be efficiently implemented in a hybrid opto-electronic network. Although many of the examples described herein are described with reference to a particular hardware implementation of a photonic ANN, one skilled in the art will recognize that methods and systems can be readily applied to other photonic platforms without departing from the heart of the invention.

[0031] Currently, there is no efficient protocol for the training of photonic neural networks, which is a crucial step for any machine learning application, and should ideally be performed on the same platform. Methods in accordance with a number of embodiments of the invention enable highly ef- ficient, in situ training of a photonic neural network. In a variety of embodiments, adjoint methods may be used to derive the photonic analogue of the backpropagation algorithm, which is the stan- dard method for computing gradients of conventional neural networks. Gradients in accordance with a number of embodiments of the invention may be obtained exactly by performing intensity measurements within the device.

[0032] Training protocols in accordance with many embodiments of the invention can greatly simplify the implementation of backpropagation. Beyond the training of photonic machine learn- ing implementations, methods in accordance with some embodiments of the invention may also be of broader interest to experimental sensitivity analysis of photonic systems and the optimization of reconfigurable optics platforms, among other applications. Photonic Neural Networks [0033] In its most general case, a feed-forward ANN maps an input vector to an output vector via an alternating sequence of linear operations and element-wise nonlinear functions of the vectors, also called‘activations’. A cost function, L, is defined over the outputs of the ANN and the ma- trix elements involved in the linear operations are tuned to minimize L over a number of training examples via gradient-based optimization. The‘backpropagation algorithm’ is typically used to compute these gradients analytically by sequentially utilizing the chain rule from the output layer backwards to the input layer.

[0034] A photonic hardware platform implemented in accordance with certain embodiments of the invention is illustrated in Figure 1. The boxed regions correspond to optical interference units (OIUs) 105 that perform a linear operation represented by the matrix Wˆ _l . Each OIU can include a number of integrated phase shifters (e.g., 110) illustrated as rounded shapes within each OIU. In many embodiments, integrated phase shifters can be used to control an OIU and train a network. Photonic hardware platform 100 also includes nonlinear activations 115 represented as f _l (·).

[0035] In this example, photonic element 100 performs linear operations using optical interfer- ence units (OIUs). OIUs in accordance with several embodiments of the invention are meshes of controllable Mach-Zehnder interferometers (MZIs) integrated in a silicon photonic circuit. By tuning the phase shifters integrated in the MZIs, any unitary N ×N operation on the input can be implemented, whichfinds applications both in classical and quantum photonics. In photonic ANNs in accordance with some embodiments of the invention, OIUs can be used for each linear matrix-vector multiplication. In certain embodiments, nonlinear activations can be performed us- ing an electronic circuit, which involves measuring the optical state before activation, performing the nonlinear activation function on an electronic circuit such as a digital computer, and preparing the resulting optical state to be injected to the next stage of the ANN.

[0036] In the description of this example, the OIU is described by a number, N, of single-mode waveguide input ports coupled to the same number of single-mode output ports through a linear and lossless device. In certain embodiments, the device may also be extended to operate on differ- ing numbers of inputs and outputs. OIUs in accordance with some embodiments of the invention implement directional propagation such that all powerflows exclusively from the input ports to the output ports. In its most general form, devices implement the linear operation

where X _in and Z _out are the modal amplitudes at the input and output ports, respectively, and Wˆ , or the transfer matrix, is the off-diagonal block of the system’s full scattering matrix,

[0037] The diagonal blocks are zero because forward-only propagation is assumed, while the off-diagonal blocks are the transpose of each other because a reciprocal system is assumed. Z _in and X _out correspond to the input and output modal amplitudes, respectively, if the device were run in reverse, i.e., sending a signal in from the output ports.

Operation and training with backpropagation [0038] A key requirement for the utility of any ANN platform is the ability to train the net- work using algorithms such as error backpropagation. Such training typically demands significant computational time and resources and it is generally desirable for error backpropagation to be im- plemented on the same platform.

[0039] The operation and gradient computation in an ANN in accordance with an embodiment of the invention is illustrated in Figure 2. In this example, propagation through a square cell (e.g., 215) corresponds to matrix multiplication, while propagation through a rounded region (e.g., 220) corresponds to activation. The ^ (e.g., 225) indicates element-wise vector multiplication.

[0040] The top row 205 corresponds to the forward propagation steps in the operation and train- ing of an ANN. For the forward propagation step, processes in accordance with numerous embodi- ments of the invention begin with an initial input to the system, X ₀ , and perform a linear operation on this input using an OIU represented by the matrix Wˆ ₁ . In several embodiments, processes can apply an element-wise nonlinear activation, f ₁ (·), on the outputs, giving the input to the next layer. This process can be repeated for each layer l until the output layer, L. Written compactly, for l = 1... L

[0041] Once forward propagation is completed, a cost function is computed to train the network. Cost function L is an explicit function of the outputs from the last layer . To train

the network, cost functions can be minimized with respect to the linear operators, Wˆ _l , which may be adjusted by tuning the integrated phase shifters within the OIUs in accordance with some em- bodiments of the invention. In a variety of embodiments, training methods can operate without resorting to an external model of the system, while allowing for the tuning of each parameter to be done in parallel, therefore scaling significantly better with respect to the number of parameters when compared to a brute force gradient computation method.

[0042] Once a cost (or loss) function is computed, backward propagation is performed to adjust the model based on the computed loss. Bottom row 210 of Figure 2 illustrates the backward propa- gation steps. In a number of embodiments, backpropagation processes can derive an expression for the gradient of the cost function with respect to the permittivities of the phase shifters in the OIUs. In the following, e _l is the permittivity of a single phase shifter in layer l, as the same derivation holds for each of the phase shifters present in that layer. Note that Wˆ _l has an explicit dependence on e _l , but allfield components in the subsequent layers also depend implicitly on e _l .

[0043] As a demonstration, a mean squared cost function is calculated

where T is a complex-valued target vector corresponding to the desired output of a system given input X ₀ .

[0044] Starting from the last layer in the circuit, the derivatives of the cost function with respect to the permittivity of the phase shifters in the last layer e _L are given by

where ^ is element-wise vector multiplication, defined such that, for vectors a and b, the i-th

¢

element of the vector a^b is given by a _i b _i . R{·} gives the real part, f _l (·) is the derivative of the

¢ lth layer activation function with respect to its (complex) argument. The vector d _L º G _L ^ f _L is defined in terms of the error vecto

[0045] For any layer l < L, the chain rule can be used to perform a recursive calculation of the gradients,

[0046] This process is illustrated in the backward propagation of the second row 210 of Figure 2, which computes the d _l vectors sequentially from the output layer to the input layer. The com- putation of d _l requires performing the operation which corresponds physically to

sending d _l+1 into the output end of the OIU in layer l + 1. In this way, processes in accordance with many embodiments of the invention‘backpropagate’ the vectors d _l and G _l physically through the entire circuit.

[0047] In some embodiments, training a photonic ANN relies on the ability to create arbitrary complex inputs. Processes in accordance with several embodiments of the invention require an integrated intensity detection scheme to occur in parallel and with virtually no loss. In numerous embodiments, this can be implemented by integrated, transparent photo-detectors.

[0048] The problem of overfitting is one that can be addressed by‘regularization’ in any practi- cal realization of a neural network. Photonic ANNs in accordance with various embodiments of the invention provide a convenient alternative approach to regularization based on‘dropout’. In various embodiments, in a dropout procedure, certain nodes can be probabilistically and temporar- ily‘deleted’ from the network during train time, which has the effect of forcing the network to find alternative paths to solve the problem at hand. This has a strong regularization effect and has become popular in conventional ANNs. Dropout in accordance with some embodiments of the invention can be implemented in the photonic ANN by‘shutting off’ channels in the activation functions during training. Specifically, at each time step and for each layer l and element i, one may set f _l (Z _i ) = 0 with somefixed probability.

[0049] Specific processes for training photonic neural networks in accordance with embodiments of the invention are described above; however, one skilled in the art will recognize that any number of processes can be utilized as appropriate to the requirements of specific applications in accor- dance with embodiments of the invention.

[0050] For example, the discussion above assumes that the functions f _l (·) are holomorphic. For each element of input Z _l , labeled z, this means that the derivative of f _l (z) with respect to its com- plex argument is well defined. In other words, the derivative

does not depend on the direction that Dz approaches 0 in the complex plane.

[0051] In numerous embodiments, the backpropagation derivation can be extended to non-holomorphic activation functions. In the backpropagation algorithm, the change in the mean-squared loss func- tion with respect to the permittivity of a phase shifter in the last layer OIU as written in Eq. (5) is

Where the error vector was defined as for simplicity and ) is the output of thefinal layer.

[0052] To evaluate this expression for non-holomorphic activation functions, f _L (Z) and its argu- ment can be split into their real and imaginary parts

where i is the imaginary unit and a and b are the real and imaginary parts of Z _L , respectively. [0053] Evaluating gives the following via the chain rule

where the layer index has been dropped for simplicity. Here, terms of the form correspond

to element-wise differentiation of the vector x with respect to the vector y. For example, the i-th element of the vector is given by

[0054] Now, inserting into Eq. (12):

[0055] The real and imaginary parts of G _L are defined as G _R and G _I , respectively. Inserting the definitions of a and b in terms of Wˆ _L and X _L-1 and doing some algebra:

[0056] Finally, the expression simplifies to

[0057] As a check, if the conditions for f _L (Z) to be holomorphic are set, namely

Eq. (20) simplifies to

as before.

[0058] This derivation may be similarly extended to any layer l in the network. For holomorphic activation functions, whereas the d vectors were defined as

for non-holomorphic activation functions, the respective definition is

( ) ( )

where G _R and G _I are the respective real and imaginary parts of G _l , u and v are the real and imagi- nary parts of f _l (·), and a and b are the real and imaginary parts of Z _l , respectively. [0059] This can be written more simply as

[0060] In polar coordinates where Z = rexp(if) and f = f(r,f), this equation becomes

where all operations are element-wise.

Gradient computation [0061] Computing gradient terms of the form which contain derivatives with

respect to permittivity of the phase shifters in the OIUs, can be a crucial step in training an ANN. In certain embodiments, gradients can be expressed as the solution to an electromagnetic adjoint problem.

[0062] OIUs used to implement the matrix Wˆ _l , relating the complex mode amplitudes of input and output ports, can be described usingfirst-principles electrodynamics. This can allow for the gradient to be computed with respect to each e _l , as these are the physically adjustable parameters in the system in accordance with some embodiments of the invention. Assuming a source at frequency w, at steady state, Maxwell’s equations take the form

which can be written more succinctly as Here, eˆ _r describes the spatial distribution of the relative permittivity (e _r ), k ₀ = w ² /c ² is the free- space wavenumber, e is the electricfield distribution, j is the electric current density, and Aˆ = Aˆ ^T due to Lorentz reciprocity. Eq. (34) is the starting point of thefinite-difference frequency-domain (FDFD) simulation technique, where it is discretized on a spatial grid, and the electricfield e is solved given a particular permittivity distribution, e _r , and source, b.

[0063] To relate this formulation to the transfer matrix Wˆ , source terms b _i , i Î 1...2N can be defined, which correspond to a source placed in one of the input or output ports. In this example, it is assumed that there are a total of N input and N output waveguides. The spatial distribution of the source term, b _i , matches the mode of the i-th single-mode waveguide. Thus, the electricfield amplitude in port i is given by b ^T

i e, and a relationship can be established between e and X _in , as

for i = 1 ... N over the input port indices, where is the i-th component of X _in . Or more compactly,

Similarly,

for i+N = (N +1) ...2N over the output port indices, or,

and, with this notation, Eq. (1) becomes

[0064] Based on the above, the cost function gradient in Eq. (10) can be evaluated. In particular, with Eqs. (34) and (39),

Here b _x,l-1 is the modal source profile that creates the inputfield amplitudes X _l-1 at the input ports.

[0065] The key insight of the adjoint variable method is that the expression can be interpreted as an operation involving thefield solutions of two electromagnetic simulations, which can be referred to as the‘original’ (og) and the‘adjoint’ (aj)

using the symmetric property of

[0066] Eq. (40) can now be expressed in a compact form as

[0067] If it is assumed that this phase shifter spans a set of points, rf in the system, then, from

where dˆ _r,r ¢ is the Kronecker delta.

[0068] Inserting this into Eq. (43), the gradient is given by the overlap of the twofields over the phase-shifter positions

[0069] A schematic illustration of methods in accordance with many embodiments of the inven- tion for experimental measurement of gradient information is illustrated in three stages 305-315 in Figure 3. The box region 320 represents the OIU. The ovals (e.g., 350) represent tunable phase shifters. Computation of the gradient is illustrated with respect to phase shifters 360 and 365.

[0070] In thefirst stage 305, the original set of amplitudes X _l is sent through the OIU 32. The constant intensity terms is measured at each phase shifter. The second stage 310 shows that the adjoint mode amplitudes, given by d _l , are sent through the output side of the OIU. is recorded

from the opposite side, as well as in each phase-shifter. In the third stage, X _l + X _{T R} is sent

through the OIU, interfering e _og and inside the device and recovering the gradient information

for all phase shifters simultaneously.

[0071] Methods in accordance with numerous embodiments of the invention compute the gra- dient from the previous section through in situ intensity measurements. Specifically, an intensity pattern is generated with the form matching that of Eq. (45). Interfering

directly in the system results in the intensity pattern:

the last term of which matches Eq. (45). Thus, in many embodiments, the gradient can be com- puted purely through intensity measurements if thefield can be generated in the OIU.

[0072] The adjointfield as defined in Eq. (42), is sourced by meaning that it physi-

cally corresponds to a mode sent into the system from the output ports. As complex conjugation in the frequency domain corresponds to time-reversal of thefields, is expected to be sent in from

the input ports. Formally, to generate a set of input source amplitudes, X _{T R} , is found such that

the output port source amplitudes, , are equal to the complex conjugate of the adjoint

amplitudes, or Using the unitarity property of transfer matrix Wˆ _l for a lossless system, along with the fact that output modes, the input mode amplitudes for the time-reversed adjoint can be computed as

As discussed earlier is the transfer matrix from output ports to input ports. Thus, X _{T R} can be

experimentally determined by sending into the device output ports, measuring the output at the input ports, and taking the complex conjugate of the result.

[0073] A process 400 for experimentally measuring a gradient of an OIU layer in an ANN with respect to the permittivities of the OIU layer’s integrated phase shifters is conceptually illustrated in Figure 4. Process 400 sends (405) originalfield amplitudes X _l-1 through the OIU layer and measures the intensities at each phase shifter. Process 400 sends (410) d _l into the output ports of the OIU layer and measures the intensities at each phase shifter. Process 400 computes (415) the time-reversed adjoint inputfield amplitudes. In numerous embodiments, time-reversed adjoint inputfield amplitudes are calculated as in Eq. (47). Process 400 interferes (420) the original and time-reversed adjointfields in the device and measures the resulting intensities at each phase shifter. Process 400 computes (425) a gradient from the measured intensities. In a number of embodiments, processes compute gradients by subtracting constant intensity terms measured from the originalfield amplitudes and d _l (e.g., at steps 405 and 410 of process 400) and multiply by k ²

0 to recover the gradient, as in Eq. (45).

[0074] In many embodiments, the isolated forward and adjoint steps are performed separately, storing the intensities at each phase shifter for each step, and then subtracting this information from thefinal interference intensity. In a variety of embodiments, rather than storing these constant in- tensities, processes can introduce a low-frequency modulation on top of one of the two interfering fields, such that the product term of Eq. (46) can be directly measured from the low-frequency signal.

[0075] Specific processes for experimentally measuring a gradient of an OIU layer in an ANN in accordance with embodiments of the invention are described above; however, one skilled in the art will recognize that any number of processes can be utilized as appropriate to the requirements of specific applications in accordance with embodiments of the invention.

[0076] A numerical demonstration of a time-reversal procedure is illustrated infive panels 505- 530 of Figure 5. In this example, the procedure is performed with a series of FDFD simulations of an OIU implementing a 3× 3 unitary matrix. These simulations are intended to represent the gradient computation corresponding to one OIU in a single layer, l, of a neural network with input X _l-1 and delta vector d _l . In these simulations, absorbing boundary conditions are used on the outer edges of the system to eliminate back-reflections.

[0077] Thefirst panel 505 shows a relative permittivity distribution for three MZIs arranged to perform a 3x3 linear operation. Boxes (e.g., 550) represent where variable phase shifters could be placed in this system. As an example, the gradient information for a layer wi

with unit amplitude and d _l = [0 1 0] ^T is computed, corresponding to the bottom left and middle right port, respectively.

[0078] The second panel 510 illustrates the real part of a simulated electricfield E _z corresponding to injection from the bottom left port. Specifically, stage 510 shows the real part of e _og , correspond- ing to the original, forwardfield.

[0079] In the third panel 515, the real part of the adjoint E _z is shown, corresponding to injection from the middle right port. The third panel 515 shows the real part of the adjointfield, e _aj , corre- sponding to the cost function

[0080] The fourth panel 520 shows a time-reversed adjointfield in accordance with some em- bodiments of the invention that can be fed in through all three ports on the left. Panel 520 shows the real part of the time-reversed copy of e _aj as computed by the method described in the previous section, in which X ^*

T _R is sent in through the input ports. There is excellent agreement, up to a constant, between the complex conjugate of thefield pattern of panel 515 and thefield pattern of panel 520.

[0081] Thefifth panel 525 shows gradient information as obtained directly by the adjoint

method, normalized by its maximum absolute value. Panel 525 shows the gradient of the objective function with respect to the permittivity of each point of space in the system, as computed with the adjoint method, described in Eq. (45).

[0082] In the sixth panel 530, the gradient information as obtained by methods in accordance with a variety of embodiments of the invention is shown, normalized by its maximum absolute value. Namely, thefield pattern from panel 510 is interfered with the time-reversed adjointfield of panel 520 and the constant intensity terms are subtracted from the resulting intensity pattern. In certain embodiments, the results are then multiplied by an appropriate set of constants. Panels 525 and 530 match with high precision.

[0083] In a realistic system, the gradient must be constant for any stretch of waveguide between waveguide couplers because the interferingfields are at the same frequency and are traveling in the same direction. Thus, there should be no distance dependence in the corresponding intensity distribution. This is largely observed in this simulation, although smallfluctuations are visible because of the proximity of the waveguides and the sharp bends, which were needed to make the structure compact enough for simulation within a reasonable time. In practice, the importance of this constant intensity is that it can be detected after each phase shifter, instead of inside of it. Intensity measurements in accordance with some embodiments of the invention can occur in the waveguide regions directly after the phase shifters, which eliminates the need for phase shifter and photo-detector components at the same location.

[0084] Numerically generated systems in accordance with many embodiments of the invention experience a power transmission of only 59% due to radiative losses and backscattering caused by very sharp bends and stair-casing of the structure in the simulation. Nevertheless, the time-reversal interference procedure still reconstructs the adjoint sensitivity with very goodfidelity. Further- more, a reasonable amount of this loss is non-uniform due to the asymmetry of the structure. Choice and implementation of activation functions [0085] A major drawback of saturable absorption is that it is fundamentally lossy. Depending on the threshold power and waveguide implementation, an attenuation per layer of at least 1 dB can be expected. In a large scale photonic ANN with many layers, the compounding attenuation from each layer can bring the signal levels below the optical noisefloor. Moreover, this scheme may require lowered activation power thresholds for successively deeper layers, which can be challeng- ing to achieve for afixed hardware platform and saturable absorption mechanism.

[0086] It is therefore of substantial interest to develop activation functions that are not subjected to the above limitations. Additionally, for simple implementation of the backpropagation algo- rithm described in this work, activation functions in accordance with various embodiments of the invention can have derivatives that allow the operation d _l = G _l ^ f ^¢ (Z _l ) to be performed simply in the photonic circuit.

[0087] A possible alternative to saturable absorption is the rectified linear unit (ReLU) activation, which is a common activation function in conventional real-valued ANNs with several complex- valued variants. For example, the complex ReLU (c-ReLU) variant returns the output only if it is above a power threshold. This function is convenient because it is holomorphic (away from the discontinuity) and its derivative is simply 0 below the power threshold and 1 above the power threshold. Therefore, forward and backward propagation steps can be performed on the same hard- ware. For forward propagation, one wouldfirst measure the power of the waveguide mode with an electro-optic circuit and close the channel if it is below a threshold. For backward propagation, simply leave this channel either closed or open, depending on the forward pass.

[0088] The mod-ReLU variant similarlyfirst checks whether the input is above a power thresh- old. If not, 0 is returned. Otherwise, it returns the input but with the power threshold subtracted from its amplitude.

[0089] Specific activation functions in accordance with embodiments of the invention are de- scribed above; however, one skilled in the art will recognize that any number of activation func- tions can be utilized as appropriate to the requirements of specific applications in accordance with embodiments of the invention. Avoiding internal coherent detection and reconstruction [0090] As presented thus far, the time-reversal interference method requires coherent detection and state preparation inside of the device. This introduces potential technical challenges and the need for additional components within the device. To mitigate this issue, methods in accordance with some embodiments of the invention (also referred to as a‘linearization methods’) can recover gradient information without needing coherent detection within the device. In a variety of embod- iments, methods can allow one to work entirely on the outsides of the ANN by selectively turning off or‘linearizing’ the activation functions between layers in a controlled fashion. However, such methods can require an activation that may implement backpropagation through some protocol that, itself, does not require internal coherent measurement and state preparation.

[0091] A time-reversal interference method for a single linear OIU embedded in the network is described above. As described, time-reversal interference methods can require coherent detec- tion to obtain X ^*

T _R , external computation of X _TR , and subsequent coherent preparation of X _TR in the device. Thus, implementing this in a multi-layered network could require additional elements between each layer, complicating the experimental setup. An example of such a time-reversal in- terference technique for a layer embedded in a network is illustrated in Figure 6. In this example, coherent detection, external computation, and state preparation is required between layers 610 and 615, which can be undesirable.

[0092] To overcome this issue, methods in accordance with various embodiments of the invention can obtain the sensitivity of the internal layers without needing coherent detection or preparation within the network. The strategy relies on the ability to‘linearize’ the activations, such that they become represented as simple transmission for both forward and backward passes. For a linearized activation given by f _l (Z _l ) = Z _l , Z _l = X _l and d _l = G _l , which greatly simplifies both forward and backward propagation.

[0093] A process for tuning a linear layer Wˆ _l is conceptually illustrated in Figure 7. Process 700 linearizes (705) all activations after Wˆ _l and performs a forward pass with original input X ₀ . Pro- cess 700 records (710) intensities within Wˆ _l and measures theoutput. Process 700 linearizes (715) all activations in network and sends the complex conjugate of the measured output into the output end of network and measures output b. Sending b ^* into the input end of the network recreates the desired X _l-1 at the input of Wˆ _l . Process 700 linearizes (720) all activations in network before Wˆ _l and performs backpropagation, sending G _L into the output end of network. Process 700 measures (725) intensities in Wˆ _l for subtraction and the output c. Sending c ^* into the input end of the network recreates the desired X _TR at the input end of Wˆ _l and d* at the output end of Wˆ _l . Process 700 inputs (730) b ^* + c ^* into completely linearized network, which reproduces the desired interference term X ₂ +X _TR at the input end of Wˆ _l . Process 700 measures (735) the intensities in Wˆ _l and computes sensitivity. Methods for computing the sensitivity based on the measured intensities are described in greater detail above.

[0094] By linearizing the activation functions, large sections of the network can be made unitary. This allows the encoding of information about the internalfields in the externally measured states a, b, c. These states can be used to recreate the desired interference terms needed to obtain gradi- ents as intensity measurements.

[0095] In numerous embodiments, methods in accordance with several embodiments of the in- vention can be used in a general network. In order to compute gradients for layer l of L, each OIU can be described by a matrix Wˆ _l and each activation can be described by a Fˆ _l operator in the forward propagation and Bˆ _l in the backward propagation. Note that even though the activations are nonlinear, depending on the result of a forward pass through the network, they can be set and then may resemble linear operations.

[0096] For the time-reversal interference method to work, an input to the system can be produced such that the mode X _l +X _TR is created right before Wˆ _l . Equivalently, the state can be pro- duced directly after Wˆ _l . Using the notation introduced, these states can be explicitly described as

[0097] A stage diagram of a procedure for using the time-reversal interference method to measure sensitivities in the OIUs without internal coherent detection or preparation is illustrated in six stages 805-830 of Figure 8. In thefirst stage 805, the gradient measurement with respect to the phase shifters is shown in the l = 3 layer, which requires creating an input state to recreate X ₂ + X _TR at the input Empty ovals 850 correspond to’linearized’ activations, which can be represented as an identity matrix. Thefirst stage 805 shows all of the channels following layer l are linearized. The output of the system when sending in the original input, X ₀ , is computed, which is labeled a.

[0098] The second stage 810 shows that all of the activation channels are linearized. Then, a ^* is sent into the output end of the network and the output is measured, which is labeled b.

The complex conjugate of b is given by

[0099] The third stage 815 shows that sending b ^* into the input end of the network recreates the desired X _l-1 at the input of Wˆ _l . In the fourth stage 820, only the activation channels before layer l are linearized and G _L is input into the output end of the system, measuring c.

[0100] Thefifth stage 825 shows that sending c ^* into the input end of the network recreates the desired X _TR at the input end of Wˆ _l and d*

l at the output end of Wˆ _l . The sixth stage 830 shows that inputting b ^* + c ^* will output directly after layer l, which will be sufficient for training via

the time-reversal intensity measurement technique. To make the derivation more clear, these two terms are split up and the total output is defined as o º o ₁ +o ₂ where

Inserting the form of b ^* from Eq. (55) into the expression for o ₁ ,

as desired. Similarly, inserting the form of c ^* from Eq. (57) into the expression for o ₂ ,

also as desired.

[0101] Thus, inputing b ^* + c ^* will give o = Z _l + d*

l at the output of layer l. Equivalently, this same input reproduces X _l-1 +X _TR at the input end of later l, which is what we need to do time- reversal sensitivity measurements. The derivation presented here holds for an arbitrary choice of activation function, even those not representable by a linear operator. Other Applications [0102] In addition to the feed-forward ANNs discussed in this work, methods in accordance with some embodiments of the invention can be used to train recurrent neural networks (RNNs), which are commonly used to analyze sequential data, such natural language or time series inputs. Re- current networks, as diagrammed in Fig. 9, have a single linear layer and activation function. A schematic of a recurrent neural network (RNN) is illustrated in Figure 9 with a single OIU and activation function. A time series of inputs to the system are provided from the left. The output of this layer may be read out with a portion routed back into the input end of the network. In many embodiments, this splitting and routing may be done simply in the photonic platform using an integrated beam splitter. RNNs may also be trained using the same backpropagation method as described above.

[0103] Methods in accordance with several embodiments of the invention can be used to train Convolutional Neural Networks (CNNs). CNNs are another popular class of neural network that is commonly used to analyze image data, where there are many inputs with a large amount of corre- lation between adjacent elements. In CNNs, the matrix multiplications are replaced by convolution operations between the input values and a trainable kernel. This convolution may be performed on a mesh of MZIs similar to those presented here, but with far fewer MZI units necessary. In various embodiments, backpropagation for CNNs may be performed using methods as described above.

[0104] The training scheme presented here can also be used to tune a single OIU without non- linearity, which mayfind a variety of applications. Specifically, the linear unit can be trained to map a set of input vectors {X _i } to a corresponding set of output vectors {Y _i }. One example for an objective function for this problem is

where j is a vector containing all the degrees of freedom. Other choices that also maximize the overlap between X _i and Y _i for every i are, of course, also possible.

[0105] One application of this training is a mode sorter for sorting out a number of arbitrary, or- thogonal modes. In particular, for an N ×N linear system, N orthogonal inputs X _i can be chosen. Define Y _i = B _i as the unit vector for the i-th output port. While this specific problem may also be solved sequentially by existing methods, gradient-based implementations in accordance with some embodiments of the invention may have benefits in speed and scaling with number of parameters in the system, especially when the degrees of freedom can be updated in parallel.

[0106] Training protocols in accordance with numerous embodiments of the invention may also be used for optimizing a single (potentially random) input into a desired output, i.e.

This type of problem arises for example when considering optical control systems for tuning the power delivery system for dielectric laser accelerators. In this application, a series of waveguides carry power to a central accelerator structure. However, it is likely that these signals will initially consist of randomly distributed powers and phases due to the coupling, splitting, and transport stages earlier in the process. Thus, the OIUs can be used as a phase and amplitude sorting element, where now X is an initially random amplitude and phase in each waveguide, and d is a vector of target amplitudes and phases for optimal delivery to the dielectric laser accelerator. The adjoint field is directly given by the radiation from an electron beam, so the target vector may be generated physically by sending in a test electron beam. In several embodiments, a similar system can be used for electron beam focusing and bunching applications.

[0107] In numerous embodiments, OIUs can also be used to implement reconfigurable optical quantum computations, with various applications in quantum information processing. In such sys- tems, linear training with classical coherent light can be used to configure the quantum gates, e.g., by setting up a specific matrix Wˆ described by complete, orthonormal sets {X _i } and {Y _i }. After the tuning, systems in accordance with numerous embodiments of the invention can be run in the quantum photonic regime.

[0108] Methods in accordance with numerous embodiments of the invention work by physically propagating an adjointfield and interfering its time-reversed copy with the originalfield. In a number of embodiments, the gradient information can then be directly obtained out as an in-situ intensity measurement. While processes are described in the context of ANNs, one skilled in the art will recognize that the processes are broadly applicable to any reconfigurable photonic system. Such a setup can be used to tune phased arrays, optical delivery systems for dielectric laser accel- erators, or other systems that rely on large meshes of integrated optical phase shifters. Methods in accordance with some embodiments of the invention can implement the nonlinear elements of the ANN in the optical domain.

[0109] Although the present invention has been described in certain specific aspects, many ad- ditional modifications and variations would be apparent to those skilled in the art. It is therefore to be understood that the present invention may be practiced otherwise than specifically described. Thus, embodiments of the present invention should be considered in all respects as illustrative and not restrictive.