LSTM Surrogate Model Accelerates Second‑Order Nonlinear Optics Simulations by 252× to Millisecond Scale

Second‑order nonlinear optics underpins quantum information, integrated photonics, biomedical imaging and high‑power laser systems, but conventional split‑step Fourier method (SSFM) simulations demand massive compute resources and struggle to incorporate experimental imperfections.

High‑Fidelity Dataset Construction

The authors generated a dataset using the SLAC LCLS‑II laser‑cathode chain model. Randomly sampling second‑order dispersion, third‑order dispersion and spectral‑shaping parameters produced 10,000 pulse‑shaping configurations (≥400 with phase‑only shaping). Each configuration was simulated with SSFM, yielding 100 propagation slices and three coupled fields (SHG1, SFG, SHG2) sampled at 32,768 points.

Pre‑processing involved three steps: (1) frequency‑domain truncation and down‑sampling (SFG → 348 complex values, each SHG → 1,892 complex values); (2) concatenating real and imaginary parts into an 8,264‑element real vector; (3) normalizing all elements to [0,1] using global extrema. The final split comprised 890,000 training, 10,000 validation and 90,000 test samples.

LSTM Surrogate Architecture

The surrogate adopts a sequence‑to‑sequence design, treating each propagation slice as a time step. It contains 2,048 hidden units followed by three fully‑connected layers with dimensions (2048→4096), (4096→4096) and (4096→8264), using ReLU, Tanh and Sigmoid activations respectively. Training employs the Adam optimizer and a weighted mean‑square‑error (wMSE) loss.

During training, sequences of 10 slices form the input; a sliding window over the 100‑slice simulations generates 100 input‑output pairs per configuration. Tensor shapes are (batch, 10, 8264) → (batch, 8264).

Autoregressive Inference

Inference runs autoregressively: the initial slice is repeated ten times, the model predicts the next slice, which replaces the oldest slice in the window, and the process repeats until all 100 slices are generated, reconstructing the full field evolution.

Accuracy Evaluation

A composite error metric combines (1) cosine similarity of normalized waveforms, (2) energy‑scaled NMSE, and (3) Wasserstein distance between intensity distributions. After ~180 epochs on an NVIDIA A10G GPU (≈160 h), training loss reached 2.05 × 10⁻⁵ and validation loss 2.03 × 10⁻⁵.

Representative test cases yielded composite errors of 0.012 and 0.003. The LSTM accurately reconstructed SFG and SHG1 in both time and frequency domains; only under large spectral‑amplitude modulation did SHG1 show minor local deviations, confirming robust generalization across diverse shaping conditions.

Efficiency Gains

Baseline SSFM on a single‑core CPU required 1.98 s total (≈1.875 s for the nonlinear step). The LSTM on the same CPU showed comparable latency due to batch overhead, but on an NVIDIA A100 GPU with batch size 200 inference time dropped to 7.43 ms per sample, delivering a 252× speedup.

Conclusion

The LSTM surrogate eliminates repeated time‑frequency transforms, enabling real‑time prediction of second‑order nonlinear optical processes and extending to other χ⁽²⁾ scenarios. This work demonstrates how machine‑learning models can bridge numerical simulation and experimental practice, offering a scalable, efficient paradigm for photonic system design.