Numerical Differentiation with Neural Operators in a Noisy Environment

This paper describes a neural operator-based approach to numerical differentiation of noisy data, which is a principal problem in image analysis and processing. The approach is based on the idea of modeling the differential operator with a special class of neural networks designed to build mappings between function spaces. Namely, a general-purpose neural operator is utilized, called DeepONet. The main advantage of such model compared to the classic approaches commonly used in applied mathematics is its ability to automatically perform noise suppression using the statistical dependencies in the data. Numerous modifications based on common deep learning practices are proposed to improve the training stability and the model’s generalization capacity. Preliminary experiments conducted on random one-dimensional data show the ability of the modified DeepONet to outperform the classic finite differences method and to generalize to certain unseen, out-of-distribution functions.