Learning-Based Nonlinear Compensation Using Physics-Information Neural Networks for Data Access

Abstract

A system and computer-implemented method using a physical information neural network for data communication are provided. At a transmitter, the method is configured to obtain source data to be transmitted to encode the source data into code words based on a forward error correction code, map the code words to amplitude symbols, modify the mapped amplitude symbols into pre-equalized symbols including pre-distorted channel impairment symbols based on a predetermined physical model, and transmit the digital data of the pre-equalized symbols as channel data over a channel. At a receiver, the method is configured to receive and demodulate channel data from a channel to generate initial estimates of bits of the received channel data, mitigate channel noise and waveform distortion in the channel data based on the initial estimates, convert the channel data consisting of shaped non-uniform symbols as a non-uniform data sequence into an unshaped bit sequence, decode the unshaped bit sequence, correct residual errors in the converted channel data, and store the corrected channel data in a data sink.

Description

[Technical field]

[0001] The present invention relates generally to the field of waveform equalization for data communications, and more particularly to a method and system for learning-based nonlinear compensation based on an artificial neural network employing a physical model. [Background technology]

[0002] In digital communication and data storage systems, the transmitted signal waveform is often distorted by many factors, including hardware defects, channel media, and signal processing inadequacies. For example, optical fiber communication typically suffers from amplified spontaneous emission noise, shot noise, thermal noise, Kerr fiber nonlinearity, chromatic dispersion, polarization multiplexing, self-phase modulation, cross-phase modulation, Raman scattering, carrier offset, linewidth, timing jitter, etc. Many linear and nonlinear equalization or denoising methods have been proposed, such as decision feedback equalization, turbo equalization, and maximum likelihood sequence estimation. However, these methods rely heavily on simple signal models that are mathematically easy to handle, such as the linear dispersion model, autoregressive model, Volterra series, regular perturbation model, nonlinear Schrodinger equation, Maxwell equations, Helmholtz equation, and delay-Doppler-angular spectrum. Model-based methods have several drawbacks. These are: i) constructing accurate signal models is numerically intractable or complex; ii) model mismatch is inevitable; iii) the interactions of unknown physical phenomena cannot be adequately modeled; and iv) the models are highly sensitive to uncertain physical parameters.

[0003] In recent years, data-driven learning-based methods, mainly based on deep neural networks (DNNs), have emerged as a promising alternative to model-based methods. For example, convolutional neural networks, recurrent neural networks, and other network architectures have been shown to achieve superior performance for compensating signal distortions as nonlinear equalization. In general, DNNs consist of a very large number of trainable parameters, such as weights and biases for affine transformations, and these trainable parameters are optimized by stochastic gradient methods based on training data. Such over-parameterized DNNs have a higher ability to approximate almost any function with higher accuracy. However, data-driven learning-based methods using over-parameterized DNNs inherently have several drawbacks and challenges. Namely, i) a sufficiently large amount of high-quality training data is required, ii) large-scale data training requires high computational power, resources, and storage, iii) the interpretability and certifiability of the trained model are both low, and iv) the discovery of a suitable DNN architecture is not obvious.

[0004] Therefore, there is a need and demand to efficiently solve these problems regarding signal waveform equalization in data communication and storage systems. Summary of the Invention

[0005] Some embodiments of the present invention are based on the recognition that data communication and storage systems often suffer from signal distortions due to channel noise, device nonlinearities, and signal processing deficiencies. Our invention provides a method to improve data quality by using physical information neural networks. The present invention solves some of the challenges of existing methods in terms of efficient training and interpretability. In some embodiments, optical fiber physics, defined by the nonlinear Schrodinger equation, is modeled with Koopman operators, Volterra expansions, split-step Fourier methods, regular perturbations, and Runge-Kutta integrals. Model parameters such as Kerr nonlinearity, fiber attenuation, and dispersion are adaptively trained. To mitigate residual model mismatch errors, another embodiment uses additional DNNs in addition to the learned physical models. Another embodiment uses joint time / frequency / space domain physics. The imperfections are mitigated by designing transmitters and receivers with differentiable digital twin-based models, e.g., joint transmitter / receiver filtering, joint modulation / demodulation, and joint pre-distortion / post-distortion. The present invention allows for coupling designs at intermediate points, such as programmable amplifiers, repeaters, switches, multiplexers and routers.The physical information model includes implicit layers based on diffusion, equilibrium, quantum physics and molecular physics.

[0006] The present invention is based on the recognition that fully data-driven DNN models have difficulty interpreting learned parameters, and over-parameterized common DNN models require a large amount of high-quality training data. In addition, the present invention is based on the recognition that signal distortion for communication and storage systems is potentially governed by physical models of channel links (mediums), such as Maxwell's equations and nonlinear Schrodinger equations for electromagnetic, optical elements and electrical circuits. The present invention enables interpretable and efficiently trainable DNN models by integrating data-based methods and physical information models to compensate for signal waveform distortion in data communication and storage systems. Physical information DNN models can provide non-trivial benefits in terms of reducing the amount of redundant and unimportant parameters, and interpretably adjusting physically relevant parameters. At the same time, the present invention provides a method for adjusting the trade-off between model accuracy and complexity through the hybrid use of data-driven DNN models and physical information mathematical models.

[0007] In some embodiments, optical fiber physics defined by the nonlinear Schrodinger equation is merged into DNN models including, but not limited to, neural Koopman operators, neural Volterra expansions, neural split-step Fourier methods (SSFM), neural regular perturbations (RP), neural Runge-Kutta (RK) integrals, neural ordinary differential equations (ODE), neural diffusion models, neural equilibrium models, collocation models, exponential integrators, etc. In some embodiments, model parameters such as Kerr nonlinearity coefficients, fiber attenuation, and dispersion coefficients are trainable to adapt to a given training data set in either an online or offline manner. In addition, incremental parameters such as phase corrections, gain corrections, and phase shifter corrections are provided to generalize the physical model.

[0008] In some embodiments, multiple physics information models are used simultaneously, for example, RP and SSFM are merged. In addition, the residual DNN model is used to correct the residual error of model mismatch. Yet another embodiment uses joint time / frequency / space domain physics to perform the merging, for example, the time domain neural kernel and the frequency domain neural kernel are synthesized.

[0009] In some embodiments, signal impairments are mitigated by designing the transmitter and receiver ends together with a digital twin-based model that is discriminative to enable gradient-based optimization, e.g., joint transmitter / receiver filtering design, joint encoder / decoder design, joint modulation / demodulation design, joint pre-distortion / post-distortion design, etc. Some embodiments jointly design multiple intermediate points, e.g., programmable amplifiers, repeaters, switches, multiplexers, routers, etc., in addition to the transmitter / receiver ends, with discriminative computational graphs in the system.

[0010] In some embodiments, the physics-based model includes implicit layers based on diffusion, equilibrium, photon, quantum physics, and other differentiable computations with trainable parameters, such as molecular physics. In some embodiments, the integrated physics-based DNN model is applied to audio / visual signal processing, biosignal processing, data storage read / write, wireless communication, wired telecommunications, etc.

[0011] Thus, some embodiments disclose a method for reliably transferring data from a transmitter to a receiver over a communication channel. In some embodiments, the method and system include at least one computing processor, at least one memory bank, a transmitter having a source for receiving source data to be transferred, an encoder for encoding the source data, and a front-end interface for transmitting the encoded codewords as modulated signals to the communication channel. In some embodiments, the method and system further includes a communication channel comprising a wired medium channel, a wireless medium channel, an optical fiber channel, a vacuum free space channel, a data storage medium, or variations thereof. In some embodiments, the method and system further includes a receiver having a front-end interface for receiving an output signal from the communication channel, a decoder for decoding the output signal, and a sink for providing a decoded codeword. In yet other embodiments, the method and system includes a controller configured to specify (determine) parameters of the pre-shaping code, the post-shaping code, the shaping mapper, and the shaping demapper used in the encoder and the decoder depending on knowledge (type) of the channel link, the parameters further including codeword length, code rate, and overhead.

[0012] Thus, the method and system of the present invention uses an inverse operation at the receiver to recover the shaped bits and the original data by FEC decoding and a shaped demapper. Specifically, a computer-implemented method for recovering an encoded bit stream is executed by one or more computing processors at the receiver to perform the steps of accessing input data representing a noisy codeword, demapping the input data to generate a soft decision bit stream given noisy quadrature amplitude modulation symbols, decoding the bit stream according to a post-shaped error correction code, mapping the decoded bit labels to amplitude modulation symbols, demapping the amplitude symbols according to a shaped demapper, and decoding the unshaped bit stream according to a pre-shaped error correction code.

[0013] According to some embodiments of the present invention, there is provided a system for data communication over a channel link, the system being configured to include a communication module that accesses the channel link according to a set of specifications for data communication, and a controller including a physics-informed neural network (PINN), the controller determining parameters represented by the set of specifications of the communication module to compensate for signal waveform distortion in the data communication according to a physical model of the channel link.

[0014] Additionally, some embodiments may provide a method, sometimes referred to as a computer-implemented method, for data communication over a channel link, including accessing the channel link according to a communication method based on a set of specifications for data communication, and modifying the set of specifications for data communication according to a Physical Information Neural Network (PINN) based control method responsive to a physical model of the channel link. [Brief description of the drawings]

[0015] [Figure 1] 1 is a functional diagram illustrating a communication system and method for the transmission of digital data in accordance with an embodiment of the present invention. [Diagram 2] FIG. 1 illustrates a structure of an LPR with parallelizable branches for real-time high-throughput DNN in accordance with an embodiment of the present invention. [Diagram 3] FIG. 1 illustrates how to improve the RP model with trainable phase shifter correction terms (complex-valued scalars) C={CO, Cl,...,CN} at the end of each branch in accordance with an embodiment of the present invention. [Figure 4] FIG. 1 illustrates an embodiment using a cascaded LRP model controlled by multiple staggered DNNs. [Figure 5A] FIG. 13 illustrates another embodiment of using an external deep neural network (DNN) for residual model mismatch correction in additive aggregation, in accordance with an embodiment of the present invention. [Figure 5B] FIG. 13 illustrates another embodiment of using an external deep neural network (DNN) for residual model mismatch correction in multiplicative aggregation, in accordance with an embodiment of the present invention. [Figure 5C] FIG. 13 illustrates another embodiment of using an external deep neural network (DNN) for residual model mismatch correction in bilinear aggregation, in accordance with an embodiment of the present invention. [Figure 5D] FIG. 13 illustrates another embodiment using a parallel LRP model aggregated with an external deep neural network (DNN) for residual model mismatch correction, in accordance with an embodiment of the present invention. [Figure 5E] FIG. 13 illustrates another embodiment that uses multiple LRP models and a cascaded stack of multiple deep neural network (DNN) blocks for residual model mismatch correction, in accordance with an embodiment of the present invention. [Figure 5F] FIG. 13 illustrates another embodiment that uses multiple LRP models and cascaded stacks of multiple deep neural network (DNN) blocks with jumping paths for residual model mismatch correction, in accordance with an embodiment of the present invention. [Figure 6] FIG. 1 illustrates an exemplary performance gain for LRP-based PINN versus model-based SSFM digital backpropagation in an optical fiber channel plant in accordance with an embodiment of the present invention. [Figure 7] FIG. 1 is an exemplary block diagram illustrating a system suitable for implementing various components of a transceiver for performing encoding, decoding, equalization, and de-shaping in accordance with an embodiment of the present invention. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0016] Various embodiments of the present invention will now be described with reference to the figures. It should be noted that the figures are not drawn to scale, and that elements having similar structures or functions are represented by similar reference numerals throughout the figures. It should also be noted that the figures are intended only to facilitate the description of certain embodiments of the present invention. They are not intended as an exhaustive description of the present invention or as limitations on the scope of the present invention. In addition, aspects described in connection with a particular embodiment of the present invention are not necessarily limited to that embodiment, and may be implemented in any other embodiment of the present invention.

[0017] Some embodiments of the present invention are based on the recognition that signal equalization uses physical information neural networks (PINNs) that integrate physical models into neural networks to improve the quality of data communication and storage systems. For example, some embodiments enable improved machine learning methods and systems based on inverse regular perturbation (RP) models to represent the physical phenomena underlying light wave propagation through optical fiber communication. In particular, photonic physics for fibers is typically governed by the nonlinear Schrodinger equation (NLSE) and its variants, such as the Manakov equation. The RP model numerically solves the NLSE using finite difference approximations in a parallel structure. Some embodiments generalize parallel RP branches by jointly optimizing trainable parameters such as step size, gain, and phase rotation using a learned RP (LRP) model. DNN models based on LRP can potentially outperform the corresponding trained digital back-propagation (DBP) method based on split-step Fourier method (SSFM) in standard single-mode fiber links. Thus, LRP also allows for reduced complexity through fractional step-per-span (SPS) modeling while maintaining better performance than 1-SPS SSFM-DBP.

[0018] Physical models of the channel links to be embedded in the PINN include finite difference methods such as split-step Fourier methods, finite element methods such as regular perturbations, collocation models such as Runge-Kutta integrals, Volterra series expansions, Koopman operators, exponential integrators, ordinary differential equations, partial differential equations, nonlinear Schrödinger equations, finite impulse responses, infinite impulse responses, autoregressive models, diffusion processes, space-time-frequency dispersion models, Maxwell's equations, quantum dynamics, molecular dynamics, and variants thereof.

[0019] Over the past decades, rapid advances in digital signal processing (DSP) technology have enabled significant improvements in the throughput and reliability of coherent optical fiber communication systems. For example, linear impairments such as chromatic dispersion (CD), polarization mode dispersion (PMD) and phase noise have been extensively studied and addressed in the digital domain. Thus, Kerr fiber nonlinearities have become the primary cause limiting the achievable maximum information rate and effective signal-to-noise ratio (SNR) of optical communication systems.

[0020] The Manakoff equation has been used to model the propagation of dual-polarization (DP) signals through standard single mode fiber (SSMF). It can be solved numerically with the split-step Fourier method (SSFM), which sequentially calculates the nonlinear and linear operations with small finite difference steps. Higher model accuracy can be obtained by reducing the step size. Alternatively, several models have been investigated, including the symmetric perturbation (RP) method. Digital backpropagation (DBP) is the state-of-the-art method for nonlinearity compensation (NLC) by inverting the lightwave propagation with an inverse SSFM calculation. It offers significantly improved performance compared to most other methods, but at the cost of higher computational complexity. In addition, it is highly sensitive to model parameters such as the Kerr nonlinearity coefficient and the dispersion coefficient. Thus, the main challenge to realize practical NLC is to find the best balance between complexity and performance.

[0021] Recent advances in machine learning (ML) and deep neural networks (DNNs) have revolutionized a wide variety of scientific fields, including optical fiber communications and networking. For NLC and equalization, several physics-informed DNNs for DBP have been proposed. Meanwhile, ML techniques have proven effective in improving nonlinear equalization based on symbol triplet interactions, while perturbation theory has also been used to improve trained DBP.

[0022] In some embodiments, the ML-assisted NLC approach based on the inverse RP physical model employs a parallel branch structure with generalized phase shifter correction parameters that should be tuned using low-complexity ML techniques to significantly increase accuracy. Furthermore, LRP in the fractional step-per-span (SPS) case, where each RP branch models multiple spans, can maintain higher performance than 1-SPS SSFM-DBP while reducing computational complexity. PINN-assisted optical fiber communications

[0023] FIG. 1 illustrates a communication system for transmitting digital data from a transmitter 110 to a receiver 130 via a channel link 120 according to some embodiments. The channel link 120 is configured to provide data communication based on electromagnetic propagation, light wave propagation, acoustic propagation, molecular propagation, or variations thereof. For example, the communication channel 120 includes, but is not limited to, an air medium for wireless communication, a copper cable for wired communication, a solid state drive for data storage read / write, a vacuum for free space satellite communication, and a fiber cable for fiber optic communication. During communication to transfer digital data from the transmitter 110 to the receiver 130, the digital data may be corrupted by noise occurring across the channel and data processing path between the data source 111 and the data sink 135. For example, the noise includes, but is not limited to, thermal noise, spontaneous emission amplifier noise, reflection / refraction of inhomogeneous media, interference noise, and impulsive shot noise. The transmitter 110 uses a concatenated encoder 112, a shaped mapper 113, and a PINN-based pre-equalizer 114 to achieve reliable data transmission. The receiver 130 uses a concatenated PINN-based post-equalizer 132, a shaped demapper 133, and a decoder 134 to recover the original data from the noisy message transmitted by the transmitter 110. The encoder and decoder are based on forward error correction (FEC) codes such as low-density parity-check (LDPC), turbo, and polarity codes. Most communication systems use both the transmitter 110 and the receiver 130 as communication modules. Some communication systems use only the transmitter as a communication module, while others use only the receiver as a communication module.

[0024] In the transmitter 110, data to be transmitted is received from a source 111 configured to receive original data. The source may include, but is not limited to, a memory bank for reading data, an interface port for receiving data, and a device for generating data. For example, in some embodiments, the source includes a voice communication device that converts an input voice signal into digital data. The input data from the source 111 is encoded by an FEC encoder 112. In some cases, the encoded data is also referred to as a codeword. The pre-shaping FEC encoder adds a series of overhead parity bits to be robust against potential burst errors. The encoded data is mapped to a series of amplitude symbols according to a shaping mapper 113, which further adds redundancy to match the probability mass function (PMF) of the amplitude occurrence to a target distribution. The shaped amplitude symbols are labeled into a series of shaped bits and further transformed by a PINN pre-equalizer 114. The PINN pre-equalizer modifies the shaped amplitude to pre-distort the channel impairment based on a physical model such as RP. The pre-equalized symbols are fed to the channel medium via front-end circuitry 115, where various modulation formats are used, including but not limited to quadrature-amplitude modulation (QAM) with or without linear conversion such as orthogonal frequency-division multiplexing (OFDM). The front-end circuitry 115 may further include, but is not limited to, electro-optical circuits for optical communication and radio frequency circuits for wireless communication. For example, the front-end circuitry includes digital-to-analog converters, power amplifiers, and signal pre-processing, such as bandpass filters, pulse shaping, precoding, power loading, pilot insertion, and pre-distortion. For some embodiments, the transmitter uses wavelength-division multiplexing (WDM) and spatial multiplexing for multimode / multicore fiber.

[0025] The channel 120 distorts the transmitted signal, for example, by adding additive white Gaussian noise (AWGN), co-channel interference, deep fading, impulsive noise, inter-symbol interference, Kerr-induced nonlinear interference, polarization crosstalk, and linear chromatic dispersion, as well as residual hardware imperfections such as quantization error, clock jitter, overflow, laser linewidth, and carrier phase noise.

[0026] The receiver 130 first converts the channel output to an electrical received signal via a front-end circuit 131, which is typically complementary to the front-end 115 in the transmitter. For example, the front-end circuit includes, but is not limited to, an analog-to-digital converter, linear equalization, nonlinear equalization, adaptive filtering, channel estimation, carrier phase recovery, synchronization, and polarization recovery. The front-end circuit demodulates the received signal to mitigate channel noise and waveform distortion by generating initial estimates of the bits of the transmitted codeword through a PINN-based post-equalizer 132. For example, the demodulated signal represents a log-likelihood ratio (LLR) value as a soft input message to the FEC decoder.

[0027] The decoded message is then demapped by a shaping demapper 133, which converts the shaped non-uniform symbols into a uniform data sequence by removing the overhead introduced by the shaping mapper 113. To correct residual errors, the deshaped bit sequence is further decoded by an FEC decoder 134 before being provided to a sink 135. The data sink 135 includes, but is not limited to, a memory bank for writing data, an output interface port for transmitting data, and a device for receiving data. For example, in some embodiments, the sink includes a voice communication device that converts the decoded data into an audio signal.

[0028] Some embodiments are based on the recognition that the FEC code and shaping specifications are adaptively controlled by the controller 141 according to the side information 140 of the communication channel 120. For example, parameters such as code rate (or overhead), codeword length, shaping rate, and shaping length are adaptively selected based on channel knowledge, such as signal-to-noise ratio (SNR), power delay profile, channel transfer function, and nonlinear statistics including, but not limited to, mean, variance, kurtosis, and higher moments. The controller 141 is configured to determine transmitter and / or receiver specifications such as best FEC code and modulation selection given the channel knowledge for some embodiments using a look-up table or another PINN model. The specifications indicate the parameters of pre-shaping codes, post-shaping codes, shaping mappers, and shaping demappers to be used in the encoder and decoder depending on the knowledge (type) of the channel link. The parameters include codeword length, code rate, and overhead, and the method of best FEC code and modulation selection for the channel link is used in data communication for some embodiments. The set of specifications for the communication module further includes sets of parameters for the channel encoder, data compressor, constellation shaper, modulator, pulse shaping filter, multiplexer, pre-equalizer, amplifier, digital-to-analog converter, oscillator, antenna, mixer, analog-to-digital converter, equalizer, demultiplexer, demodulator, constellation de-shaper, data decompressor, channel decoder, etc. Thus, the set of parameters specified by the controller includes the generator matrix of the channel coding, the shaping rate, the shaping method, the modulation format, the pulse shaping filter length, the pre-equalizer method, the amplifier gain, the sampling rate and quantization bits of the digital-to-analog converter, the oscillator frequency, the antenna size, the sampling rate and quantization bits of the analog-to-digital converter, the pre-equalization method, the demodulation method, the constellation de-shaping method, the channel decoder method, etc.When controller 141 determines or modifies the specifications, the parameters are adaptively selected to best match the type of channel link used in the data communications system.

[0029] The transmitter 110 and / or the receiver 130 may be implemented with a hardware processor operably connected to a memory. Each of the transmitter 110 and the receiver 130 may include one or more processors. For example, the memory of the receiver 130 may store some information regarding one or a combination of the shaping mapper and demapper, the soft input and soft output of the decoder 133, the results and parameters of intermediate computations of encoding and decoding, and an adaptive coding policy. All components in the transmitter 110 and the receiver 130 may be implemented by hardware, one or more hardware processors, computer software (programs or program modules), or a combination of hardware and computer software. PR-based Physical Information Model

[0030]

number

[0031] This physical information PDE model based on Manakov NLSE is usually difficult to analytically derive a real solution at any position z and any initial condition. To solve the physical equations, typically finite difference or finite element methods are used to numerically identify the waveform propagation. For some embodiments, SSFM uses a finite difference step to subsequently solve the linear and nonlinear effects. The linear step takes into account the CD, fiber attenuation and PMD based on the Fourier domain to calculate the waveform transformation in an efficient way. The nonlinear step uses the time domain to modify its amplitude and phase according to the instantaneous lightwave power. ASE noise is also added to the amplifier.

[0032] For some embodiments, the RP model is used to numerically solve the Manakov NLSE by using finite difference steps to form parallel branches with linear and nonlinear operations. This is based on the realization that optical signal propagation through SSMF can be approximated with the RP model, which solves the Manakov equation in the linear domain and bundles the nonlinear components into small additive terms. Similar to SSFM-DBP, the direction of the RP model can be inverted by reversing the direction of f(z) (i.e., switching attenuation to amplification) and changing the signs of the beta2 and gamma parameters. Thus, the inverted RP takes the received signal u(L,t) to reconstruct u(0,t) (i.e., the backpropagation signal) for the NLC. For some embodiments, the gamma parameter is adjusted by a machine learning framework based on a gradient method to achieve better results.

[0033]

number

[0034]

number

[0035] Similar to SPS in SSFM-DBP, the accuracy of RP increases with the number of branches and decreases with the optical launch power, i.e., higher fiber nonlinearity. Note that the model is scaled to z such that the pre-linearity / post-linearity blocks cover the entire link length. n Note that we allow ={0, L}, which has been shown to improve performance when there are a small number of branches.

[0036] While the RP model can provide a reasonably accurate approximation with a large number of branches, it suffers from significant problems when the number of branches is small. As shown in Figure 3, the one-branch conventional RP model takes a linear signal 31 and adds an amplitude-dependent signal 32 that is shifted by 90 degrees with K nonlinear operations to provide a nonlinear phase rotation. Adding these two vectors can result in a phase shifter that tends towards the desired rotation 33. However, increasing nonlinearity necessarily leads to an expansion of the gain outside the unit power circle 34. Thus, the model accuracy drops significantly in higher power regimes.

[0037]

number

[0038] With this simple modification, we can collectively adjust the gain and rotation of each branch 35 to find a better phase shifter that is close to unit power 36. Therefore, the LRP model in the present invention can more accurately approximate the nonlinear effect and effectively mitigate the undesired gain expansion.

[0039] As an example implementation of the system, a single-carrier DP64-term quadrature amplitude modulated (QAM) signal with a symbol rate of 64 Gbaud is generated in the transmitter. The generated electrical signal is pulse shaped with a root-raised cosine (RRC) filter with a roll-off factor of 0.1. It is modulated onto an optical carrier with an in-phase / quadrature modulator. The carrier is provided by a laser with a center frequency of 1550 nm and zero frequency offset. The analog optical signal is upsampled to 8sa / sym and propagated through 10 spans of an 80 km SSMF link. Each span is followed by an erbium-doped OA with a gain of 16 dB and a noise figure of 4 dB. Nonlinear inter-channel interference can occur in a 5-channel, 50 GHz-spaced wavelength division multiplexed (WDM) system. SSMF has a gamma Kerr coefficient of 1.2 W / km, an alpha loss coefficient of 0.2 dB / km, and a beta dispersion coefficient of 17 ps / nm / km. The received optical signal is downsampled to 2sa / sym and passed through a PINN-based equalizer with linear CD compensation, SSFM-DBP or LRP-DBP model to mitigate the channel impairments. The output is then shaped with a matched RRC filter. The received signal is then normalized and de-rotated with a 1-tap equalizer and downsampled to Isa / sym.

[0040] For some embodiments, the LRP-based PINN equalizer is trained by an adaptive momentum (Adam) optimizer with a learning rate of 0.001 to find optimal values ​​for the phase shifter correction coefficient C and step size z. We use 200 randomly generated data frames split into 80% and 20% for training and testing, respectively. A single frame contains a snapshot of the coherently detected and digitally backpropagated signal as a target. Each snapshot has 8192 samples per polarization with an oversampling of 2sa / sym. The frames used for training are simulated (forward and backward propagated) with a 100-SPS Manakov SSFM. The forward propagated signal is downsampled to 2sa / sym. The raw nonlinear phase rotation is not compensated before training and is included in the optimization, while one-tap equalization is used after NLC. The signal is then passed to a 100-SPS DBP and the corresponding nonlinearity compensated output is saved as the second element of the snapshot. The detected signal is 2sa / sym, while the forward propagation in the analog domain is 8sa / sym. For some embodiments, we use a logarithmic step distribution across each fiber span, and asymmetric SSFM, i.e., the nonlinear steps are at the segment boundaries.

[0041]

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[0042] The complex value C is scaled by the gradient of the MSE loss function.

[0043] Proper initialization of trainable parameters is a key aspect for achieving higher performance and faster convergence. Some embodiments initialize z as a logarithmic step distribution over each span, while the phase correction C is initialized as a vector filled with ones on the real axis and zeros (0) on the imaginary axis. For some embodiments, the model yields better convergence when Delta is decoupled from the initial values ​​of z and f(z).

[0044] For some embodiments, the ML-based adjustment of trainable parameters in the physical model is realized by a deep neural network in an adaptive manner. Specifically, the method and system using the DNN uses additional trainable coefficients in addition to the RP model to correct the phase shifter. The DNN incorporates side information such as signal kurtosis and power of the instantaneous waveform at various positions z. Specifically, such a system uses various side information, including fiber plant information, to control the parameters of the LRP model via the DNN. For some embodiments, the PINN-based parallel equalizers are cascaded in a pipeline sequence to adjust the parallel and pipeline coefficients.

[0045] Figure 6 shows an example performance of the LRP-based PINN for a standard single-mode fiber plant. The PINN-based nonlinearity compensation outperforms the trained SSFM-based method. Compared with the 1-sps SSFM digital backpropagation, the 10-branch LRP achieves a 0.8 dB gain, but with comparable computational complexity. More importantly, the 5-branch LRP still performs better than the 1-sps SSFM by 0.25 dB, despite its lower computational complexity. This is the advantage of the trainable phase shifter correction. Compared with the 3-sps SSFM, the 30-branch LRP still shows a 0.6 dB gain.

[0046] Another embodiment is based on the cascaded use of physical models such as LRP controlled by additional DNNs. Figure 4 shows a cascaded model controlled by a staggered DNN. Here, the DNN uses intermediate signals to modify the parameters of LRP, such as step size and nonlinear constant. By using intermediate signals, the potential error caused by model mismatch can be reduced. In addition, the cascaded use of LPR can further reduce the model mismatch per LRP.

[0047] Yet another embodiment uses additional DNN modules to mitigate model mismatch errors. FIG. 5A illustrates an embodiment in which an external DNN such as a convolutional neural network (CNN) and a multilayer perceptron (MLP) is used to additively compensate for the residual error of the LRP. FIG. 5B illustrates an embodiment in which an additional DNN module is used multiplicatively. FIG. 5C illustrates an embodiment in which an additional DNN module is used together with a trainable physical model such as the LRP in a bilinear weighting manner. Here, the CNN is interpreted as a cascaded finite-impulse response (FIR) filter that generalizes the good linear behavior of the SSFM. For some embodiments, a recursive neural network (RNN) is used to represent the infinite impulse response (IIR) filter. The bilinear aggregation is further generalized by additional DNN blocks in some embodiments. The RP model has various variants, including log-domain RP and higher-order RP. Some embodiments use learnable parameterization according to a variant of the RP model.

[0048] FIG. 5D shows another embodiment in which a parallel physical model is aggregated by a DNN block. FIG. 5E shows another embodiment in which multiple LRPs and DNNs are stacked in a cascaded manner. FIG. 5F shows another embodiment using a jumping path for intermediate signals to be fed to a DNN block in a subsequent stage. The embodiments of FIG. 4 and FIG. 5A-FIG. 5F are only examples of potential integration of physical models into DNN architectures through combinations such as cascaded stacking, parallel aggregation, jumping aggregation, and staggered control.

[0049]

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[0050] Some embodiments use different parameterizations according to these different types of SSFM physical models. Agrawal SSFM requires an iterative loop to propagate the waveform. Some embodiments use an implicit neural network to solve this iterative process. For example, neural ordinary differential equation (NODE), deep equilibrium (DEQ), and neural diffusion models are used to realize the PINN-based equalizer. Linear operations are usually realized in the frequency domain, but time-domain convolution or filtering can also be used for some embodiments. In addition, both time-domain and frequency-domain operations are combined in the PINN architecture. For WDM and spatial multiplexing systems, spatial-domain processing is also merged into the PINN in some embodiments. Equalizers are also used as pre-equalizers or pre-distortion methods in the transmitter, which invert the signal distortion before transmission over the channel in a similar manner for post-equalization in the receiver. Some embodiments use both pre-equalizers and post-equalizers simultaneously.

[0051] Besides RP and SSFM, there are various methods for solving the NLSE. For example, the Runge-Kutta (RK) method, which is one of the collocation methods, can numerically solve the PDE with higher accuracy. Some embodiments use the RK model integrated in the DNN architecture to solve the inverse NLSE. Another example of a numerical method is the exponential integrator (EI), in which the Krylov subspace approximation is used for the matrix exponential operator. For example, the Marcuse method and the Blow-Wood method are used as EI frameworks. Some embodiments use the EI method as a PINN-based equalization. Nonlinear Fourier Transform (NFT)

[0052] There is another physical modeling framework known as the nonlinear Fourier transform (NFT) for solving the NLSE. The NFT transforms the waveform into an eigendomain where the NLSE dynamics act linearly. For example, the Koopman operator realizes the NFT. To realize the Koopman-based NFT, the exemplary PINN uses an autoencoder (AE) that can be used to find a coordinate transformation that transforms the nonlinear PDE into a linear PDE. Some embodiments of the present invention use a Koopman-AE DNN to model the behavior of the Manakoff equation applied to nonlinear optical fibers. For long sequence transformations with high oversampling rates, some embodiments split the symbol sequence into smaller chunks and finally splice them back together. To address chromatic dispersion-induced channel memory effects, chunk duplication and edge discarding are used during DNN recovery. Specifically, a sliding window is applied on the DNN input tensor and edge samples are discarded for each processed chunk. The PINN is trained to minimize a minimum squared error (MSE) function between the time-domain SSFM data and the generated predictions. In this case, the task of the Koopman DNN was to predict each step of a 100-step SSFM Manakov simulation over several spans of fiber. Other cost functions, such as cross-entropy loss, are alternatively used in some embodiments. Some embodiments use a reversible residual network to facilitate optimization of the forward and reverse NFTs. Digital Twin-based end-to-end design

[0053] PINN-based equalization can provide the added benefit of optimizing other system parameters such as pulse shaping filters, modulation schemes, constellation shaping, and FEC parameters through gradient propagation across the system diagram. For example, once an LRP-based PINN model is trained to model a channel for communication, the trained PINN model can be used to predict forward and reverse behavior as a digital twin model. In this manner, PINN enables end-to-end design of transmitter and receiver parameters. For some embodiments, these parameters are also trained by reinforcement learning to optimize without gradients. For some embodiments, the end-to-end design is extended to design intermediate subsystem components such as parametric amplifiers, repeaters, relays, switches, and routers. For example, gain control and routing optimization in amplifiers is achieved by the digital twin PINN model. Other features

[0054] Other embodiments include quantum communication, where quantum physics is modeled in a PINN-based equalization. Yet other embodiments include molecular communication, where molecular dynamics and chemical reaction physics are integrated as a PINN-based equalization. Some other embodiments use neuronal communication, where neurophysics is modeled in a PINN-based equalization.

[0055] 7 illustrates an example block diagram of a system suitable for implementing various components of a receiver for performing decoding, equalization, and unshaping according to some embodiments, and / or a transmitter for encoding, pre-equalizing, and shaping codewords according to some embodiments. The system 1700 may include one or a combination of a sensor 1710, an inertial measurement unit (IMU) 1730, a processor 1750, a memory 1760, a transceiver 1770, and a display / screen 1780, which may be operatively coupled to other components via a connection 1720. The connection 1720 may include a bus, a line, a fiber, a link, or a combination thereof.

[0056] The transceiver 1770 may include, for example, a transmitter capable of transmitting one or more signals over one or more types of communication networks and a receiver for receiving one or more signals transmitted over one or more types of communication networks. The transceiver 1770 may enable communication over wired or wireless networks based on various technologies, including, but not limited to, femtocells, Wi-Fi networks or wireless local area networks (WLANs) that may be based on the IEEE 802.11 family of standards, wireless personal area networks (WPANS) such as Bluetooth, near field communication (NFC), networks based on the IEEE 802.11 family of standards, and / or wireless wide area networks (WWANs) such as LTE, WiMAX, etc. The system may also include one or more ports for communicating over a wired network, such as optical fiber communication.

[0057] In some embodiments, the processor 1750 can also receive input from the IMU 1730. In other embodiments, the IMU 1730 can include a three-axis accelerometer, a three-axis gyroscope, and / or a magnetometer. The IMU 1730 can provide velocity, orientation, and / or other position-related information to the processor 1750. In some embodiments, the IMU 1730 can output measured information synchronized with the capture of each image frame by the sensor 1710. In some embodiments, the output of the IMU 1730 is used in part by the processor 1750 to fuse the sensor measurements and / or to further process the fused measurements.

[0058] The system 1700 may also include a screen or display and speakers 1780 for rendering images such as color and / or depth images. In some embodiments, the display and speakers 1780 may be used to display live images captured by the sensor 1710, fused images, augmented reality images, graphical user interfaces (GUIs), and other program output. In some embodiments, the display 1780 may include and / or house a touch screen that allows a user to input data through some combination of a virtual keyboard, icons, menus, or other GUIs, user gestures, and / or input devices, such as a stylus and other writing means. In some embodiments, the display 1780 may be implemented using a liquid crystal display or a light emitting diode (LED) display, such as an organic LED display, etc. In other embodiments, the display 1780 may be a wearable display.

[0059] The example system 1700 can also be modified in various ways in accordance with this disclosure, such as by adding, combining, or omitting one or more of the illustrated functional blocks. For example, in some configurations, the system 1700 does not include the IMU 1730 or the sensors 1770. In some embodiments, some portions of the system 1700 take the form of one or more chipsets.

[0060] The processor 1750 may be implemented using a combination of hardware, firmware, and software. The processor 1750 may represent one or more circuits configurable to execute at least a portion of the computational procedures or processes associated with sensor fusion and / or methods for further processing the fused measurements. The processor 1750 retrieves instructions and / or data from the memory 1760. The processor 1750 may be implemented using one or more application specific integrated circuits (ASICs), central processing units (CPUs), graphic processing units (GPUs), digital signal processors (DSPs), DSP devices, programmable logic devices (PLDs), field programmable gate arrays (FPGAs), controllers, microcontrollers, microprocessors, embedded processor cores, quantum computing processors, electronic devices, quantum processors, molecular processors, other electronic units designed to perform the functions described herein, or combinations thereof.

[0061] Memory 1760 may be implemented within processor 1750 and / or external to processor 1750. As used herein, the term "memory" refers to any type of long-term, short-term, volatile, non-volatile, or other memory and is not limited to any particular type or number of memories or the type of physical medium on which the memory is stored. In some embodiments, memory 1760 holds program code that facilitates equalization, pre-equalization, decoding, shaping, de-shaping, and encoding.

[0062] In some embodiments, additionally or alternatively, the processor 1750 can execute one or a combination of encoding, decoding, and shaping / deshaping applications 1755. For example, the output of the decoding can be used to decode a concatenated ECC formed from multiple component ECCs combined into a higher performance code. Another example is a system using iterative equalization and decoding, where the soft decision output from the decoder is sent back to a demodulator to iteratively refine the decoder input. Yet another example is to act on the decoded output, such as showing the output on a display and speaker 1780, storing the output in memory 1760, transmitting the output using a transceiver 1770, and / or performing an action based on the output and measurements of the sensor 1710.

[0063] Further, another embodiment of the present invention may provide an encoder circuit for generating encoded digital data from input digital data. The encoder / shaping circuit is configured to generate encoded / shaped digital data from a source bitstream as input digital data. The encoder / shaping circuit is configured to receive the source bitstream 111 and may include at least one memory bank configured to store a computer-executable encoding method (program / computer-executable instructions / or equivalent circuit) and one or more computing processors configured to generate codewords from the input digital data as encoded digital data by performing steps of the computer-executable encoding method. In this case, the computer-executable encoding method is configured to cause the processor to perform steps based on a code specification. In some embodiments, the generated digital data is modulated to be provided to the communication channel 120 via the front-end circuit 115.

[0064] All of the above-described embodiments of the present invention can be implemented in any of numerous ways. For example, the embodiments can be implemented using hardware, firmware, software, or a combination thereof. If implemented in software, the software code can be executed on any suitable processor or collection of processors, whether provided on a single computer or distributed among multiple computers. Such a processor can be implemented as an integrated circuit with one or more processors in an integrated circuit component. However, the processor can be implemented using circuitry in any suitable format.

[0065] Also, embodiments of the invention may be implemented as a method, an example of which has been provided above. The operations performed as part of the method may be ordered in any suitable manner. Thus, embodiments may be constructed in which operations are performed in an order different from that illustrated, which may include performing some operations simultaneously although shown as sequential operations in the exemplary embodiment.

[0066] The use of ordinal terms such as "first" and "second" in the claims to modify claim elements does not, in itself, imply a priority, precedence, or ordering of one claim element over another, nor does it imply a chronological order in which method operations are performed; it is merely used as a label to distinguish one claim element having a particular name from another element having the same name (but using ordinal terms) to distinguish among the claim elements.

[0067] Although the invention has been described by way of examples of preferred embodiments, it is to be understood that various other adaptations and modifications can be made within the spirit and scope of the invention.

[0068] Therefore, it is the object of the appended claims to cover all such variations and modifications as come within the true spirit and scope of the invention.

Claims

1. 1. A system for data communication over a channel link, comprising: a communication module for accessing said channel link in accordance with said set of specifications for data communication; a controller including a physics-based neural network; The controller determines parameters represented by the set of specifications for the communication module to compensate for signal waveform distortion in the data communication according to a physical model of the channel link.

2. The system of claim 1 , wherein the channel link is based on electromagnetic propagation, optical wave propagation, acoustic propagation, molecular propagation, or variations thereof for the data communication.

3. 2. The system of claim 1, wherein the set of specifications for the communications module includes sets of parameters for a channel encoder, a data compressor, a constellation shaper, a modulator, a pulse shaping filter, a multiplexer, a pre-equalizer, an amplifier, a digital-to-analog converter, an oscillator, an antenna, a mixer, an analog-to-digital converter, an equalizer, a demultiplexer, a demodulator, a constellation deshaper, a data decompressor, a channel decoder, and variations thereof.

4. 4. The system of claim 3, wherein the set of parameters includes a generator matrix for channel coding, a shaping rate, a shaping method, a modulation format, a pulse shaping filter length, a pre-equalizer method, an amplifier gain, a sampling rate and quantization bits of a digital-to-analog converter, an oscillator frequency, an antenna size, a sampling rate and quantization bits of an analog-to-digital converter, an equalizer method, a demodulator method, a constellation de-shaper method, a channel decoder method, and variations thereof.

5. 2. The system of claim 1, wherein the physical model of the channel link is based on finite difference methods such as split-step Fourier methods, finite element methods such as regular perturbations, collocation models such as Runge-Kutta integrals, Volterra series expansions, Koopman operations, exponential integrators, ordinary differential equations, partial differential equations, nonlinear Schrodinger equations, finite impulse responses, infinite impulse responses, autoregressive models, diffusion processes, space-time-frequency dispersion models, Maxwell's equations, quantum dynamics, molecular dynamics, and variations thereof.

6. 2. The system of claim 1, wherein the physical information neural network integrates the physical model of the channel link into a deep neural network having an additional set of trainable parameters to adjust for residual errors caused by model mismatch.

7. 7. The system of claim 6, wherein the deep neural network integrated into the physical information neural network is based on combinations of parallel aggregation, cascaded stacking, staggered control, jumping aggregation, and variations thereof.

8. 2. The system of claim 1, wherein the physical information neural network is based on a cascaded use of regular perturbation models with phase shifter correction trained by a deep neural network that uses statistical information of the channel link to adaptively modify the set of specifications for the communication module.

9. The system of claim 1 , wherein the communication module further modifies the set of specifications of the communication module, such as a pre-equalizer and an equalizer, using another physical information neural network modified by the controller.

10. The communication module includes: an encoder for encoding data into codewords based on the channel code; a shaped mapper for mapping the codewords to amplitude symbols; a pre-equalizer for modifying the mapped amplitude symbols into pre-equalized symbols based on the physical model using a trained physical information neural network; a digital-to-analog converter for transmitting digital data of the pre-equalized symbols as channel data via the channel link; an analog-to-digital converter receiving the channel data from the channel link and generating received symbols; a demodulator for modifying the received symbols as intermediate estimates of the digital data; a physical information neural network based post-equalizer for mitigating channel noise and waveform distortion caused by the channel link based on the intermediate estimate of the digital data to generate equalized data; a shaped demapper for modifying the equalized data into a shaped bit sequence as shaped data; a decoder for decoding the unshaped data using channel decoding to correct residual errors; and variations thereof.

11. 1. A method for data communication over a channel link, comprising: accessing said channel link according to a communication method based on said set of specifications for data communication; and modifying the set of specifications for the data communication according to a physical model of the channel link and according to a control method based on a physics-based neural network.

12. The method of claim 11 , wherein the channel link is based on electromagnetic propagation, optical wave propagation, acoustic propagation, molecular propagation, or variations thereof for the data communication.

13. 12. The method of claim 11, wherein the set of specifications for the data communication includes sets of parameters for channel coding, data compression, constellation shaping, modulation, pulse shaped filtering, multiplexing, pre-equalization, amplification, digital-to-analog conversion, oscillation, analog-to-digital conversion, equalization, demultiplexing, demodulation, constellation deshaping, data recovery, channel decoding, and variations thereof.

14. 14. The method of claim 13, wherein the set of parameters includes a generator matrix for channel coding, a shaping rate, a shaping method, a modulation format, a pulse shaping filtering length, a pre-equalization method, an amplification gain, a sampling rate and a quantization bit for digital-to-analog conversion, an oscillation frequency, a sampling rate and a quantization bit for analog-to-digital conversion, an equalization method, a demodulation method, a constellation de-shaping method, a channel decoding method, and variations thereof.

15. 12. The method of claim 11, wherein the physical model of the channel link is based on finite difference methods such as split-step Fourier methods, finite element methods such as regular perturbation, collocation models such as Runge-Kutta integrals, Volterra series expansions, Koopman operations, exponential integrators, ordinary differential equations, partial differential equations, nonlinear Schrodinger equations, finite impulse responses, infinite impulse responses, autoregressive models, diffusion processes, space-time-frequency dispersion models, Maxwell's equations, quantum dynamics, molecular dynamics, and variations thereof.

16. 12. The method of claim 11, wherein the physical information neural network integrates the physical model of the channel link into a deep neural network having an additional set of trainable parameters to adjust for residual errors caused by model mismatch.

17. 17. The method of claim 16, wherein the deep neural network integrated into the physics-information neural network is based on combinations of parallel aggregation, cascaded stacking, staggered control, jumping aggregation, and variations thereof.

18. 12. The method of claim 11, wherein the physical information neural network is based on a cascaded use of regular perturbation models with phase shifter correction trained by a deep neural network that uses statistical information of the channel link to adaptively modify the set of specifications for the communication module.

19. The method of claim 11, wherein the communication method further modifies the set of specifications of the communication method, such as pre-equalization and equalization, using another physical-information neural network modified by the control method.

20. The communication method includes: obtaining source data to be transmitted via the data interface port for data communication; encoding the source data into codewords based on a channel code; mapping said codewords to amplitude symbols; modifying the mapped amplitude symbols into pre-equalized symbols based on the physical model using a trained physical information neural network based pre-equalization; transmitting the digital data of the pre-equalized symbols as channel data over the channel link; receiving said channel data from said channel link to generate received symbols; demodulating the received symbols to generate an intermediate estimate of the digital data; mitigating channel noise and waveform distortion caused by the channel link based on the intermediate estimate of the digital data by using a physical information neural network based post-equalization to generate equalized data; converting the equalized data into an unshaped bit sequence as unshaped data by using shaped demapping; decoding the unshaped data using channel decoding to correct residual errors; storing the corrected channel data in a data sink; The method of claim 11 including combinations of these variations.

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