Fig. 1. (a) A communication system utilizing end-to-end (E2E) learning can be represented as an autoencoder, which consists of three main components: an encoder (transmitter), a decoder (receiver), and the actual transmission channel. The transmission channel can be various communication systems. (b) Conventional E2E learning method involves accurately modeling the transmission channel first, followed by performing the backpropagation (BP) algorithm in the digital domain. Channel modeling can be achieved using either physics-based approaches, known as “white-box” models, or pure data-driven methods, referred to as “black-box” models. (c) Proposed physics-guided learning: executing the forward pass of backpropagation on the actual transmission channel, while the backward pass estimates the gradient using a simplified white-box model.

Fig. 2. Schematic diagrams of methods adapted for DPD training. (a) Proposed physics-guided learning method. (b) Prior E2E learning methods. (b-i) Method 1: hybrid-domain learning with a data-driven model [18]. (b-ii) Method 2: digital-domain learning with a complicated physics-based model [16,19–24" target="_self" style="display: inline;">–24]. (b-iii) Method 3: digital-domain learning with a data-driven model (implemented by alternating training) [17,33,34].

Fig. 3. Experimental setup for our physics-guided learning method, configured in an either amplifier-less or 80-km transmission system. The physical transmission system and the Rx-DSP compose the actual channel. The simplified white-box digital channel employs a series of physical models that cover only several components. DPD, digital pre-distortion; AWG, arbitrary waveform generator; IQ-MOD, IQ modulator; VOA, variable optical attenuator; EDFA, erbium-doped fiber amplifier; SSMF, standard single-mode fiber; LO, local oscillator; OSC, oscilloscope; LPF, low-pass filter.

Fig. 4. Training process comparisons between our method and prior Method 1 (hybrid-domain data-driven method) and Method 3 (digital-domain data-driven method, implemented by alternating training). (a) Training loss versus training iteration in experiments, under the conditions of 5 dB link loss and 600 mV Vpp. (b) Validation MSE versus training iteration in simulations. (b-ii) Zoom-in of (b-i) to compare the required iteration numbers for different methods when reaching the same MSE.

Fig. 5. Performance comparison in impairments mitigation. (a) Calculated SNR versus Vpp of DPDs trained through different methods. (b) Calculated BER versus Vpp, followed by (b-i) and (b-ii) showing the received constellations without and with DPD at their respective optimal Vpp values. (c) Comparison of transmitted signal spectra with and without DPD.

Fig. 6. (a) BER versus optical link loss for DPD modules from different E2E learning methods. All DPDs were trained at a fixed 5 dB link loss. (b) Noise resilience investigation in comparison with ILA. (b-i) Calculated BER versus link loss of DPDs trained at fixed 3 dB, 5 dB, and 8 dB link loss values, respectively. (b-ii) Zoom-in of (b-i).

Fig. 7. Applying the DPD module trained for 32-QAM to other formats without retraining. SNR versus Vpp for 16-QAM and 64-QAM. The insets (i)–(iv) show the received constellations at the optimal Vpp.

Fig. 8. Performance comparison in the 80-km transmission system. DPD modules trained in the amplifier-less system by different learning methods are tested without retraining.

Fig. 9. (a) Schematic of E2E learning for a communication system. (b) Proposed physics-guided learning: gradient estimation with a physics-based (white-box) model. (c) Gradient estimation with a data-driven (black-box) model.

Fig. 10. Simulation setup of the amplifier-less coherent system. The inset shows the NN structure of the DPD module. Tx, transmitter; Rx, receiver.

Fig. 11. Training process comparison between training DPD alone and joint learning with an NN-based post-equalizer.