We present an approach for solving optimal Dirichlet boundary control problems of nonlinear optics by using deep learning. For computing high resolution approximations of the solution to the nonlinear wave model, we propose higher order space-time finite element methods in combination with collocation techniques. Thereby, $C^{l}$-regularity in time of the global discrete is ensured. The resulting simulation data is used to train solution operators that effectively leverage the higher regularity of the training data. The solution operator is represented by Fourier Neural Operators and Gated Recurrent Units and can be used as the forward solver in the optimal Dirichlet boundary control problem. The proposed algorithm is implemented and tested on modern high-performance computing platforms, with a focus on efficiency and scalability. The effectiveness of the approach is demonstrated on the problem of generating Terahertz radiation in periodically poled Lithium Niobate, where the neural network is used as the solver in the optimal control setting to optimize the parametrization of the optical input pulse and maximize the yield of $0.3\,$THz-frequency radiation. We exploit the periodic layering of the crystal to design the neural networks. The networks are trained to learn the propagation through one period of the layers. The recursive application of the network onto itself yields an approximation to the full problem. Our results indicate that the proposed method can achieve a significant speedup in computation time compared to classical methods. A comparison of our results to experimental data shows the potential to revolutionize the way we approach optimization problems in nonlinear optics.

翻译：我们提出一种利用深度学习求解非线性光学中最优狄利克雷边界控制问题的方法。为获得非线性波动模型解的高分辨率近似，我们提出结合配点技术的高阶时空有限元方法，从而保证了全局离散解在时间上具有$C^{l}$-正则性。基于此生成的仿真数据用于训练解算子，这些算子能够有效利用训练数据的高正则性特征。解算子由傅里叶神经算子和门控循环单元表示，可作为最优狄利克雷边界控制问题中的正向求解器。所提出的算法在现代化高性能计算平台上实现并测试，重点关注效率与可扩展性。该方法的效果在周期性极化铌酸锂中产生太赫兹辐射的问题中得到验证：在该问题中，神经网络被用作最优控制框架下的求解器，以优化光输入脉冲的参数化，并使$0.3\,$THz频率辐射的产额最大化。我们利用晶体的周期性层状结构设计神经网络，训练网络学习光在单个周期层内的传播规律。通过将网络递归应用于自身，即可得到完整问题的近似解。结果表明，与经典方法相比，所提方法可在计算时间上实现显著加速。将我们的结果与实验数据对比表明，该方法有望彻底改变非线性光学优化问题的解决路径。