Chemical lasers, as a significant class of high-energy lasers, have long relied on numerical analysis for optimal design. However, the underlying processes in chemical lasers are complex and involve chemical reactions, supersonic flows, and resonant laser amplification within a cavity. Owing to their inherent complexity and strong nonlinearity, traditional numerical methods often incur high computational costs and face challenges in achieving convergence. This study focuses on the interaction between laser gain and intracavity laser fields. The forward problem involves predicting the complex laser field distribution from a given gain profile, while the inverse problem involves reconstructing the gain profile from the complex laser field distribution. Our objective is to employ deep learning techniques to address both the forward and inverse problems.

This paper proposes a substantial numerical methodology by introducing physics-informed neural networks (PINNs) to address both the forward and inverse problems in chemical lasers. The performance of the PINNs is evaluated using a paraxial wave equation, which is derived from the Helmholtz equation under the paraxial approximation and homogeneous medium assumption. Because traditional neural network optimizers cannot directly compute the gradients of complex variables, we decompose the complex-valued laser and corresponding complex-valued partial differential equations into their real and imaginary components. The network architecture consists of a 12-layer neural network, including 1 input layer, 1 output layer, and 10 hidden layers, each with 100 neurons. The Swish activation function is used in the hidden layers, and the output layer has no activation function. The Adam optimizer is employed to achieve convergence. The database contains approximately 4×10⁵ data points for each spatial coordinate and variable. During training, the mini-batch size is set to 1×10⁴, with a total of 10⁵ iterations. A dynamic learning rate is adopted during training. The optimizer iteratively adjusts the network parameters and minimizes the total loss function towards the target value.

The dataset contains 7 planes perpendicular to the laser propagation direction (z-axis). Each plane includes 241×241 uniformly distributed sampling points, which constituted a dataset of 406567 sampling points. These data are obtained through numerical calculations in MATLAB, where the fast Fourier transform algorithm is used to simulate laser propagation within the resonant cavity. However, residuals exist between the numerically computed data and governing equation, with an average residual of 13.71% for the real part and 14.32% for the imaginary part. In the forward problem, the intracavity gain distribution and initial laser distribution (the first plane along the laser propagation direction) are known, and the laser field distribution on subsequent planes is predicted using PINNs. The PINN-based approach achieves relative errors of 2.82% and 6.76% for the real and imaginary components of the complex laser field, respectively. Moreover, PINNs significantly outperform traditional numerical methods in terms of computational efficiency, reducing the inference time from 17.6 s to 0.43 s. In the inverse problem, a complex laser field distribution across the 7 planes is given, and PINNs are used to reconstruct the intracavity gain distribution. The average relative error between the PINN-predicted and numerically-obtained gain distributions is 17.98%. By incorporating the boundary gain as a label, the relative error of the predicted gain decreases to 6.78%. Additionally, if the homogeneous medium assumption is applied to the active resonant cavity model, it is essential that the data be sampled from the equiphase surface to obtain accurate results when using the PINNs to solve both the forward and inverse problems. This reflects an inherent property of PINNs: the governing equations in the loss function must accurately represent the physical laws governing the system without the inclusion of empirical terms or approximations. However, in engineering practice, it is often challenging to obtain complete control equations and precise coefficients, which limits the broader applicability of PINNs in real-world engineering. Therefore, future research should focus on developing new methods for the numerical simulation of chemical lasers that integrate both physical laws and data-driven approaches, particularly in cases where some detailed physics is unavailable.

This paper presents a numerical study of chemical lasers using PINNs. Focusing on the active paraxial wave equation, the forward and inverse problems of intracavity complex-valued laser fields are addressed using PINNs. Although this study specifically targets chemical lasers, the methodology is generalizable to calculating the gain distribution and complex laser fields in any laser cavity. During the training of neural networks, optimizers cannot differentiate between complex numbers. Therefore, the complex laser field and associated partial differential equations are decomposed into their real and imaginary components. The results demonstrate that, in the forward problem, the PINNs accurately predict the laser field distribution and significantly enhance the computational efficiency. In the inverse problem, PINNs successfully reconstruct the intracavity gain distribution, a task that traditional numerical methods cannot achieve, and further improve the accuracy by incorporating known boundary conditions. The accurate and efficient solutions provided by PINNs for both the forward and inverse problems offer new approaches for the design and optimization of lasers.