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

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.

DefTransNet: A Transformer-based Method for Non-Rigid Point Cloud Registration in the Simulation of Soft Tissue Deformation

Soft-tissue surgeries, such as tumor resections, are complicated by tissue deformations that can obscure the accurate location and shape of tissues. By representing tissue surfaces as point clouds and applying non-rigid point cloud registration (PCR) methods, surgeons can better understand tissue deformations before, during, and after surgery. Existing non-rigid PCR methods, such as feature-based approaches, struggle with robustness against challenges like noise, outliers, partial data, and large deformations, making accurate point correspondence difficult. Although learning-based PCR methods, particularly Transformer-based approaches, have recently shown promise due to their attention mechanisms for capturing interactions, their robustness remains limited in challenging scenarios. In this paper, we present DefTransNet, a novel end-to-end Transformer-based architecture for non-rigid PCR. DefTransNet is designed to address the key challenges of deformable registration, including large deformations, outliers, noise, and partial data, by inputting source and target point clouds and outputting displacement vector fields. The proposed method incorporates a learnable transformation matrix to enhance robustness to affine transformations, integrates global and local geometric information, and captures long-range dependencies among points using Transformers. We validate our approach on four datasets: ModelNet, SynBench, 4DMatch, and DeformedTissue, using both synthetic and real-world data to demonstrate the generalization of our proposed method. Experimental results demonstrate that DefTransNet outperforms current state-of-the-art registration networks across various challenging conditions. Our code and data are publicly available.