  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.
