Approximation of 2D function with a jump discontinuity - the high noise case

This paper presents a novel two-stage neural network approach to approximate 2D functions with jump discontinuities from data across varying noise conditions, from noise-free to highly noisy scenarios. While traditional methods struggle particularly with discontinuity detection in high-noise environments, our methodology demonstrates consistent performance across a spectrum of noise levels in synthetic testing. Our approach combines a specialized U-Net architecture for discontinuity detection with a Dual UNet architecture for function reconstruction. The first stage uses a custom U-Net specifically designed to detect discontinuities in function data, demonstrating improved performance compared to general-purpose segmentation models on this task. The second stage introduces a Dual UNet architecture that processes probabilistic region classifications through multi-modal input fusion and detail-preserving reconstruction. Testing on synthetic data shows competitive performance compared to existing spline-based methods across the tested range of noise conditions. Our approach handles both open and closed curve discontinuities. Current scope and limitations: The method is designed for functions with a single jump discontinuity on gridded data. As a learning-based approach, it provides empirical performance without theoretical approximation guarantees. Within these constraints, the methodology demonstrates practical value for noisy environments and may have potential applications in domains such as medical imaging, signal processing, and computational fluid dynamics where robustness to noise is important.