An Interpretable Operator-Learning Model for Electric Field Profile Reconstruction in Discharges Based on the EFISH Method

Abstract Machine learning (ML) models have recently been utilized for reconstructing the electric field distribution from a corresponding electric-field-induced second-harmonic (EFISH) signal profile—a task described in earlier works as the ‘inverse EFISH problem’. This approach is one of several that addresses the inaccuracy of a (line-of-sight) EFISH measurement caused by the Gouy phase shift present in focused laser beams. A key advantage of this approach is that the accuracy of the reconstructed, or ‘inverted’, profile (assuming that this solution is unique) can be readily evaluated through a ‘forward transform’ of the underlying EFISH equation. Motivated by this latest success, the present study introduces a novel ML model with significantly enhanced performance. This model is built on a more complex, but powerful operator-learning architecture, and moves beyond the simpler class of artificial and convolutional neural networks (i.e. ANNs and CNNs) employed in previous work. Termed here as Decoder-Deep Operator Network (or DDON), its main asset is the ability to learn function-to-function mappings, a feature essential for recovering electric field profiles of an 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. Another important aspect of this work is the use of ‘integrated gradients’, which helps identify input (signal) regions that most critically influence the accuracy of a reconstructed profile. This quantitative metric can in turn provide guidance on the optimal sampling region (or window) for acquiring EFISH data. 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.