Neural Integral Operators for Inverse problems in Spectroscopy

TL;DR

The paper tackles the challenge of scarce data in spectroscopic inverse problems by introducing neural integral operators (NIE) that learn kernels of integral equations of the first kind, enabling stable classification from spectra. The core idea is to parameterize the kernel with a neural network $G_\theta$ and solve the first-kind integral equation $T_\theta(u)(\sigma) = f$ using an encoder $E_\theta$ to map spectra to a reduced representation $u_\theta$, with Monte Carlo integration providing regularization and data augmentation. Experiments on three real-world datasets show that NIE outperforms traditional ML methods like DT and SVM and remains competitive with deep learning models on small datasets, addressing the overfitting issues common in spectroscopy. This approach offers a data-efficient deep learning paradigm for inverse spectroscopy, with potential impact in analytical chemistry, food quality control, forensic science, and materials research, where small sample sizes are common. The method also highlights the growing role of neural operators in learning inverse mappings directly from spectral data, leveraging integral equation formulations to improve generalization.

Abstract

Deep learning has shown high performance on spectroscopic inverse problems when sufficient data is available. However, it is often the case that data in spectroscopy is scarce, and this usually causes severe overfitting problems with deep learning methods. Traditional machine learning methods are viable when datasets are smaller, but the accuracy and applicability of these methods is generally more limited. We introduce a deep learning method for classification of molecular spectra based on learning integral operators via integral equations of the first kind, which results in an algorithm that is less affected by overfitting issues on small datasets, compared to other deep learning models. The problem formulation of the deep learning approach is based on inverse problems, which have traditionally found important applications in spectroscopy. We perform experiments on real world data to showcase our algorithm. It is seen that the model outperforms traditional machine learning approaches such as decision tree and support vector machine, and for small datasets it outperforms other deep learning models. Therefore, our methodology leverages the power of deep learning, still maintaining the performance when the available data is very limited, which is one of the main issues that deep learning faces in spectroscopy, where datasets are often times of small size.

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