| This thesis investigates whether nonlocal neural operator models, particularly neural integral operators, provide a useful inductive bias for learning functional mappings in scientific data, with a central focus on capturing nonlocal structure in brain dynamics and a complementary case study in spectroscopic classification. Scientific measurements are often more naturally understood as functions over spectral, spatial, or temporal domains than as fixed-length feature vectors, motivating an operator-learning perspective that maps between function spaces rather than vectors. The thesis develops this perspective across two studies of increasing complexity. In spectroscopy, a Neural Integral Operator formulates classification as a first-kind integral equation with a learned Urysohn kernel, using Monte Carlo integration both as a numerical scheme and as an implicit stochastic regularizer. Evaluated on three spectroscopic benchmarks of varying size and difficulty, the model achieves consistent top-tier performance relative to classical and deep-learning baselines, with the clearest advantage on the hardest, most spectrally overlapping dataset. In fMRI, an Attentional Neural Integral Equation formulates encoding and decoding as a second-kind, fixed-point equation operating in a latent spatiotemporal representation. The model supports both directions—decoding stimulus information from brain activity and encoding predicted brain activity from stimuli—and experiments show that increased temporal and spatial context improves performance and yields more separable learned representations, with decoding results providing the stronger evidence. Together, these results support a calibrated claim: nonlocal operator learning is a viable and adaptable inductive bias for scientific data with distributed functional structure, most beneficial where relevant information cannot be captured locally. Keywords: neural operators, neural integral equations, operator learning, inverse problems, spectroscopy, fMRI, nonlocal modeling |