Deep-learning inverse techniques have attracted significant attention in recent years. Among them, the neural adjoint (NA) method, which employs a neural network surrogate simulator, has demonstrated impressive performance in the design tasks of artificial electromagnetic materials (AEM). However, the impact of the surrogate simulators' accuracy on the solutions in the NA method remains uncertain. Furthermore, achieving sufficient optimization becomes challenging in this method when the surrogate simulator is large, and computational resources are limited. Additionally, the behavior under constraints has not been studied, despite its importance from the engineering perspective. In this study, we investigated the impact of surrogate simulators' accuracy on the solutions and discovered that the more accurate the surrogate simulator is, the better the solutions become. We then developed an extension of the NA method, named Neural Lagrangian (NeuLag) method, capable of efficiently optimizing a sufficient number of solution candidates. We then demonstrated that the NeuLag method can find optimal solutions even when handling sufficient candidates is difficult due to the use of a large and accurate surrogate simulator. The resimulation errors of the NeuLag method were approximately 1/50 compared to previous methods for three AEM tasks. Finally, we performed optimization under constraint using NA and NeuLag, and confirmed their potential in optimization with soft or hard constraints. We believe our method holds potential in areas that require large and accurate surrogate simulators.

翻译：深度学习逆技术近年来引起了广泛关注。其中，神经伴随（NA）方法通过采用神经网络代理模拟器，在人工电磁材料（AEM）设计任务中展现出卓越性能。然而，代理模拟器精度对NA方法解的影响仍不明确。此外，当代理模拟器规模较大且计算资源有限时，该方法难以实现充分优化。尽管从工程视角看约束条件下的行为至关重要，但相关研究尚属空白。本研究探究了代理模拟器精度对解的影响，发现代理模拟器越精确，所得解越优。进而，我们开发了NA方法的扩展版本——神经拉格朗日（NeuLag）方法，该方法能高效优化足够数量的候选解。实验表明，即使在使用大规模高精度代理模拟器导致候选解处理困难的情况下，NeuLag方法仍能寻得最优解。在三个AEM任务中，NeuLag方法的再模拟误差约为传统方法的1/50。最后，我们分别采用NA和NeuLag方法进行了约束条件下的优化，验证了其在软约束与硬约束优化中的潜力。我们认为，该方法在大规模高精度代理模拟器需求领域具有应用前景。