Neural Integral Operators for Inverse Problems: An Operator-Learning Framework for Small-Sample Spectroscopic Classification

Abstract

Learning maps between function spaces with a strong inductive bias is a central challenge in soft computing, especially when training data are scarce and standard deep architectures overfit. We introduce a \emph{neural integral operator} (NIO) framework based on integral equations of the first kind, in which the Urysohn kernel of the operator is parameterized by a feed-forward network~$G_{θ_G}$ and the latent function is produced by a convolutional encoder~$E_{ϕ_E}$, both trained jointly end-to-end via cross-entropy loss. The integral defining the learned operator is approximated by Monte Carlo sampling, which we argue acts as an implicit stochastic regularizer operating at the level of the integrand and complementing parameter-level regularizers such as weight decay and dropout. We benchmark the framework on three real-world spectroscopic classification tasks (FT-IR fruit purees, NIR meat, NIR textiles) of varying size and complexity, against traditional machine learning (decision tree, support vector machine, with and without UMAP) and modern deep learning baselines (FFNN, CNN+FFNN, shallow CNN, transformer). The proposed NIO is consistently among the top two performing models across all datasets and metrics, achieves the best results on the most challenging small-and-complex dataset (Textile), and yields lower performance variance than competing deep models in the small-data regime. The results suggest that operator-learning architectures with stochastic numerical integration are a viable soft-computing strategy for inverse problems in spectroscopy when conventional deep learning approaches are limited by data scarcity.