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.

翻译：神经算子在求解偏微分方程解算子方面取得了优异性能，但其固有的连续表示难以捕捉不连续与剧烈过渡现象。现有方法通常将此类特征在连续函数空间内近似逼近，往往需要增加模型容量和高分辨率数据。本研究提出Cut-DeepONet双阶段训练框架，在显式建模不连续性的同时降低学习复杂度。该方法通过提升策略重新构建问题，将定义域划分为光滑子区域，同时将不连续性表示为高维空间中的边界。这种分离使算子学习任务与神经网络的归纳偏置对齐，避免直接逼近不连续特征。额外网络预测未知输入中依赖输入的不连续位置，进而引导神经算子在各区域内生成光滑分量。在基准偏微分方程上的实验表明，即便使用低分辨率数据集进行训练，Cut-DeepONet仍优于现有最优方法。该方法在处理含不连续与剧烈过渡的问题时表现卓越，同时使用的可训练参数更少。研究结果凸显了改变算子学习表征方式而非增加模型复杂度的优势。