Machine learning (ML) models have recently been used to reconstruct electric field distributions from EFISH signal profiles-the 'inverse EFISH problem'. This addresses the line-of-sight EFISH inaccuracy caused by the Gouy phase shift in focused beams. A key benefit of this approach is that the accuracy of the reconstructed profile can be directly checked via a 'forward transform' of the EFISH equation. Motivated by this latest success, the present study introduces a novel ML model with markedly improved performance. Based on a more powerful operator-learning architecture, it goes beyond the ANNs and CNNs employed previously. Termed Decoder-DeepONet (DDON), its main strength is learning function-to-function mappings, essential for recovering electric field profiles of unknown shape. The superior performance of DDON is exemplified via a comparison with our published CNN model and the feasibility of a classical mathematical method, as well as its application to both discharge simulations and experimental EFISH data from a nanosecond pulsed discharge. In almost all cases, the DDON model exhibits better generalizability, higher prediction accuracy, and wider applicability. Furthermore, the intrinsic nature of this operator-learning architecture renders it less sensitive to the exact location(s) of the acquired data, enabling electric field reconstruction even with seemingly 'incomplete' input profiles--an issue often accompanying poor signal sensitivity. We also employ Integrated Gradients (IG) to identify the signal regions most critical to reconstruction accuracy, providing guidance on the optimal sampling window for EFISH acquisition. Overall, we believe that the DDON model is a robust and comprehensive model which can be readily applied to reconstruct 'bell-shaped' electric field profiles with an existing axis of symmetry, especially in non-equilibrium plasmas.

翻译：机器学习模型最近被用于从EFISH信号剖面重构电场分布——即"EFISH逆问题"。这解决了聚焦光束中古依相移导致的视线EFISH测量不准确问题。该方法的一个关键优势在于，重构剖面的精度可以通过EFISH方程的"前向变换"直接验证。受此最新进展启发，本研究提出了一种性能显著提升的新型机器学习模型。基于更强大的算子学习架构，它超越了先前使用的ANN和CNN。该模型称为解码器深度算子网络，其主要优势在于学习函数到函数的映射，这对于恢复未知形状的电场剖面至关重要。通过与已发表的CNN模型、经典数学方法的可行性比较，以及在纳秒脉冲放电模拟和实验EFISH数据中的应用，展示了DDON的优越性能。在几乎所有案例中，DDON模型都表现出更好的泛化能力、更高的预测精度和更广的适用性。此外，这种算子学习架构的内在特性使其对数据采集点的精确位置不敏感，即使面对看似"不完整"的输入剖面也能实现电场重构——这通常是信号灵敏度不足导致的常见问题。我们还采用积分梯度方法识别对重构精度最关键的信号区域，为EFISH采集的最佳采样窗口提供指导。总体而言，我们认为DDON模型是一个鲁棒且全面的解决方案，特别适用于重构具有现有对称轴的"钟形"电场剖面，在非平衡等离子体研究中具有直接应用价值。