# Smooth Piecewise Cutting for Neural Operator to Handle Discontinuities and Sharp Transitions

arXiv: https://arxiv.org/abs/2605.19823 · HTML: https://mlsanity.com/paper/2605.19823/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 34 references that are in the map.
- First posted on arXiv 19 May 2026.

## Builds on (most cited first, 21 of 21)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [DeepOKAN: Deep Operator Network Based on Kolmogorov Arnold Networks for Mechanics Problems](https://mlsanity.com/paper/2405.19143/) (2024)
- [NOMAD: Nonlinear Manifold Decoders for Operator Learning](https://mlsanity.com/paper/2206.03551/) (2022)
- [Residual-based attention in physics-informed neural networks](https://mlsanity.com/paper/2307.00379/) (2023)
- [Derivative-Informed Neural Operator: An Efficient Framework for High-Dimensional Parametric Derivative Learning](https://mlsanity.com/paper/2206.10745/) (2022)
- [SVD Perspectives for Augmenting DeepONet Flexibility and Interpretability](https://mlsanity.com/paper/2204.12670/) (2022)
- [Nonlinear Reconstruction for Operator Learning of PDEs with Discontinuities](https://mlsanity.com/paper/2210.01074/) (2022)
- [HyperDeepONet: learning operator with complex target function space using the limited resources via hypernetwork](https://mlsanity.com/paper/2312.15949/) (2023)
- [A Discontinuity Capturing Shallow Neural Network for Elliptic Interface Problems](https://mlsanity.com/paper/2106.05587/) (2021)
- [Enhancing training of physics-informed neural networks using domain-decomposition based preconditioning strategies](https://mlsanity.com/paper/2306.17648/) (2023)
- [Derivative-enhanced Deep Operator Network](https://mlsanity.com/paper/2402.19242/) (2024)
- [Fusion-DeepONet: A Data-Efficient Neural Operator for Geometry-Dependent Hypersonic and Supersonic Flows](https://mlsanity.com/paper/2501.01934/) (2025)
- [A Neural Network based Shock Detection and Localization Approach for Discontinuous Galerkin Methods](https://mlsanity.com/paper/2001.08201/) (2020)
- [A Variational Framework for Residual-Based Adaptivity in Neural PDE Solvers and Operator Learning](https://mlsanity.com/paper/2509.14198/) (2025)
- [Neural operators struggle to learn complex PDEs in pedestrian mobility: Hughes model case study](https://mlsanity.com/paper/2504.18267/) (2025)
- [Lifting the Winding Number: Precise Discontinuities in Neural Fields for Physics Simulation](https://mlsanity.com/paper/2502.00626/) (2025)
- [Walsh-Hadamard Neural Operators for Solving PDEs with Discontinuous Coefficients](https://mlsanity.com/paper/2511.07347/) (2025)
- [Shock-Aware Physics-Guided Fusion-DeepONet Operator for Rarefied Micro-Nozzle Flows](https://mlsanity.com/paper/2510.17887/) (2025)
- [Discontinuous Galerkin finite element operator network for solving non-smooth PDEs](https://mlsanity.com/paper/2601.03668/) (2026)
- [φ-DeepONet: A Discontinuity Capturing Neural Operator](https://mlsanity.com/paper/2604.08076/) (2026)

## Abstract (arXiv)

Neural operators have achieved strong performance in learning solution operators of partial differential equations (PDEs), but their inherently continuous representations struggle to capture discontinuities and sharp transitions. Existing approaches typically approximate such features within continuous function spaces, often requiring increased model capacity and high-resolution data. In this work, we propose Cut-DeepONet, a two-stage training framework that explicitly models discontinuities while reducing learning complexity. Our approach reformulates the problem via a lifting strategy, partitioning the domain into smooth subregions while representing discontinuities as boundaries in a higher-dimensional space. This separation aligns the operator learning task with the inductive bias of neural networks and avoids directly approximating discontinuities. An additional network predicts input-dependent discontinuity locations for unseen inputs, which are then used to guide the neural operator in generating smooth components within each region. Experiments on benchmark PDEs show that Cut-DeepONet outperforms state-of-the-art methods, even when trained on low-resolution datasets. The method excels on problems with discontinuities and sharp transitions, while using fewer trainable parameters. Our results highlight the benefits of changing the representation of operator learning rather than increasing model complexity.