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Dual adaptive training of photonic neural networks

A preprint version of the article is available at arXiv.


Photonic neural networks (PNNs) are remarkable analogue artificial intelligence accelerators that compute using photons instead of electrons at low latency, high energy efficiency and high parallelism; however, the existing training approaches cannot address the extensive accumulation of systematic errors in large-scale PNNs, resulting in a considerable decrease in model performance in physical systems. Here we propose dual adaptive training (DAT), which allows the PNN model to adapt to substantial systematic errors and preserves its performance during deployment. By introducing the systematic error prediction networks with task-similarity joint optimization, DAT achieves high similarity mapping between the PNN numerical models and physical systems, as well as highly accurate gradient calculations during dual backpropagation training. We validated the effectiveness of DAT by using diffractive and interference-based PNNs on image classification tasks. Dual adaptive training successfully trained large-scale PNNs under major systematic errors and achieved high classification accuracies. The numerical and experimental results further demonstrated its superior performance over the state-of-the-art in situ training approaches. Dual adaptive training provides critical support for constructing large-scale PNNs to achieve advanced architectures and can be generalized to other types of artificial intelligence systems with analogue computing errors.

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Fig. 1: Training PNNs with DAT.
Fig. 2: Training DPNNs under three types of systematic errors for the MNIST classification.
Fig. 3: Physical experiments of in situ training of DPNN systems.
Fig. 4: Training MPNNs under two types of systematic errors for the MNIST and FMNIST classification.
Fig. 5: Training MPNN under joint systematic errors for MNIST and FMNIST classification.

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Data availability

The MNIST dataset is available at, whereas the FMNIST dataset is available at Other data needed to evaluate the conclusions are present in the main text or Supplementary Information. Source Data are provided with this paper.

Code availability

The code for training MPNN with DAT and the pre-trained models for replication are available at (ref. 42) (


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This work is supported by the National Key Research and Development Program of China through grant no. 2021ZD0109902 (to X.L.), and the National Natural Science Foundation of China through grant nos. 62275139 (to X.L.), 61932022 and 62250055 (to H.X.).

Author information

Authors and Affiliations



X.L. and H.X. initiated and supervised the project. X.L., Z.Z. and Z.D. conceived the research. X.L. designed the methods. Z.Z., Z.D. and H.C. implemented the algorithm and conducted experiments. Z.Z., Z.D., H.C., R.Y., S.G. and H.Z. processed the data. X.L., Z.Z., Z.D., H.C. and R.Y. analysed and interpreted the results. All authors prepared the manuscript and discussed the research.

Corresponding authors

Correspondence to Hongkai Xiong or Xing Lin.

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The authors declare no competing interests.

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Nature Machine Intelligence thanks the anonymous reviewers for their contribution to the peer review of this work. Primary Handling Editor: Mirko Pieropan, in collaboration with the Nature Machine Intelligence team.

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Extended data

Extended Data Fig. 1 Procedure of DAT with internal states for the DPNN in the nth PNN block.

The flow charts with blue and yellow backgrounds denote the forward inferences in the physical system and the numerical model, respectively. Four steps of DAT with internal states, labelled using dotted arrows with four different colours, are repeated over all training samples to minimize the loss functions until convergence for obtaining the numerical model and physical parameters for the system, that is, the phase modulation matrices \({\rm{M}}_{ni}\) for i = 1, 2, 3, 1≤ n ≤N. See Supplementary Section 2 for a detailed description.

Extended Data Fig. 2 Procedure of DAT without internal states for the MPNN.

The flow charts with blue and yellow backgrounds denote the forward inferences in the physical system and the numerical model, respectively. Four steps of DAT without internal states, labelled using dotted arrows with four different colours, are repeated over all training samples to minimize the loss functions and optimize the physical model, that is, the phase coefficients Θn and Φn of the nth photonic mesh for 1≤nN. See Supplementary Section 10 for a detailed description.

Extended Data Fig. 3 Architecture of the SEPN constructed with a complex-valued mini-UNet.

See Methods for a detailed description.

Extended Data Fig. 4 Training DPNN under joint systematic errors for the MNIST and FMNIST classification.

The performance of DAT is evaluated on the DPNN-S and DPNN-M architectures and compared with the PAT and direct deployment of in silico-trained model under different joint systematic error configurations, as shown in Table c. The first configuration of DPNN-M listed in Table c was selected for the visualization of the internal states with the example digit ‘7’, phase modulation layers, and confusion matrices on the MNIST classification in a, b, and d, respectively. See Supplementary Section 6 for a detailed description.

Extended Data Fig. 5 Training DPNNs under three types of systematic errors for the FMNIST classification.

The performances of DAT for DPNN-S (a) and DPNN-M (b) are compared with the PAT and direct deployment of in silico-trained models under different amounts of systematic errors. The DPNN-S is only trained without internal states (DAT w/o IS), while the DPNN-M is trained with (DAT w/ IS) and without internal states.

Source data

Extended Data Fig. 6 Convergence plots of DPNN-M evaluated with the blind test accuracy on the MNIST dataset.

a and b represent the results of training process under mild individual errors; c, d, and e are the results under the severe individual errors, and f is the result under the joint errors, with the error configurations shown above the subfigures. For the numerical comparisons of training DPNN-M, DAT uses the test accuracy at the tenth epoch, while PAT uses the highest test accuracy among the fifty epochs due to the different convergence speeds and stability. DAT outperforms PAT with a more robust and stable training process for optimizing the DPNN-M under both mild and severe errors.

Source data

Extended Data Fig. 7 Illustration of constantly changing systematic errors.

a, Experimental DPNN prototypes, where L1, L2 and L3 are relay lenses; POL1 and POL2 are polarizers; NPBS1 and NPBS2 are non-polarized beamsplitters. b, Measurement variation of three points at the CCD sensor using 30 successive frames for the physical DPNN-S. c, Variation of testing accuracies of in silico-trained physical DPNN-C within six days when directly deploying to the physical system with geometric calibrations. The fluctuations are caused by the recording errors and XY-Plane shift errors at the CCD sensor. The event ‘Calib.’ means that the CCD sensor is geometric calibrated with affine transformations. See Supplementary Section 7 for a detailed description.

Source data

Extended Data Fig. 8 Performance evaluation of SEPNs for a 3-block DPNN-C constructed by the physical system.

a, The internal states and final output with the input digit of ‘5’ from the MNIST test set (left), and the output differences between the numerical model and physical system of blocks 1, 2, and 3 with and without SEPNs by using pixel-level statistics (right). b, Output differences averaged over all samples from the MNIST test set with and without SEPNs. See Supplementary Section 8 for a detailed description.

Source data

Extended Data Fig. 9 Comparisons of DAT performances with all, partial, and without internal states for the 3-layer MPNN in the task of MNIST classification.

The DAT methods are implemented with each SEPN parameter of 9,648 in the unitary mode. The performance of DAT with all internal states P1, P2 (2 IS), one internal state P2 (1 IS), and without internal states are evaluated. The classification accuracy improves with more measurements of internal states, especially under severe systematic errors.

Source data

Extended Data Fig. 10 Computational complexities of different training methods.

The computational complexities, evaluated with FLOPs, of AT, PAT, and DAT with internal states (IS) for training DPNNs and MPNNs are compared under different input sizes (a) and PNN block numbers (b). The legend ‘AT’ represents adaptive training, and the postfix ‘(A)’ indicates the use of the angular spectrum method to implement diffractive weighted interconnections. The configurations used in the numerical or physical experiments are indicated by dotted vertical lines. The curves are plotted based on Supplementary Tables 1 and 2. See Supplementary Section 14 for a detailed description.

Supplementary information

Supplementary Information

Supplementary Sections 1–16, Algorithms 1 and 2, Figs. 1–4 and Tables 1 and 2.

Supplementary Data 1

Raw data obtained from the CCD detector in physical experiments for illustrating Fig. 3, Extended Data Fig. 8 and Supplementary Figs. 1 and 3.

Source data

Source Data Fig. 2

Statistical source data.

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Statistical source data.

Source Data Fig. 4

Statistical source data.

Source Data Extended Data Fig. 5

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Source Data Extended Data Fig. 6

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Source Data Extended Data Fig. 7

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Source Data Extended Data Fig. 8

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Source Data Extended Data Fig. 9

Statistical source data.

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Zheng, Z., Duan, Z., Chen, H. et al. Dual adaptive training of photonic neural networks. Nat Mach Intell 5, 1119–1129 (2023).

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