We propose a new machine-learning approach for fiber-optic communication systems whose signal propagation is governed by the nonlinear Schrödinger equation (NLSE). Our main observation is that the popular split-step method (SSM) for numerically solving the NLSE has essentially the same functional form as a deep multi-layer neural network; in both cases, one alternates linear steps and pointwise nonlinearities. We exploit this connection by parameterizing the SSM and viewing the linear steps as general linear functions, similar to the weight matrices in a neural network. The resulting physics-based machine-learning model has several advantages over “black-box” function approximators. For example, it allows us to examine and interpret the learned solutions in order to understand why they perform well. As an application, low-complexity nonlinear equalization is considered, where the task is to efficiently invert the NLSE. This is commonly referred to as digital backpropagation (DBP). Rather than employing neural networks, the proposed algorithm, dubbed learned DBP (LDBP), uses the physics-based model with trainable filters in each step and its complexity is reduced by progressively pruning filter taps during gradient descent. Our main finding is that the filters can be pruned to remarkably short lengths—as few as 3 taps/step—without sacrificing performance. As a result, the complexity can be reduced by orders of magnitude in comparison to prior work. By inspecting the filter responses, an additional theoretical justification for the learned parameter configurations is provided. Our work illustrates that combining data-driven optimization with existing domain knowledge can generate new insights into old communications problems.

中文翻译：

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用于光纤通信系统的基于物理的深度学习

我们为光纤通信系统提出了一种新的机器学习方法，其信号传播由非线性薛定谔方程 (NLSE) 控制。我们的主要观察结果是，用于数值求解 NLSE 的流行分步法 (SSM) 与深层多层神经网络具有基本相同的功能形式；在这两种情况下，一种交替线性步骤和逐点非线性。我们通过参数化 SSM 并将线性步长视为一般线性函数来利用这种联系，类似于神经网络中的权重矩阵。与“黑盒”函数逼近器相比，由此产生的基于物理的机器学习模型有几个优点。例如，它允许我们检查和解释学习的解决方案，以了解为什么它们表现良好。作为应用程序，考虑了低复杂度非线性均衡，其中任务是有效地反转 NLSE。这通常称为数字反向传播 (DBP)。所提出的算法被称为学习 DBP (LDBP)，而不是使用神经网络，它在每个步骤中使用基于物理的模型和可训练的滤波器，并通过在梯度下降过程中逐步修剪滤波器抽头来降低其复杂性。我们的主要发现是，可以在不牺牲性能的情况下将滤波器修剪为非常短的长度——少至 3 抽头/步。因此，与之前的工作相比，复杂性可以降低几个数量级。通过检查滤波器响应，为学习的参数配置提供了额外的理论依据。