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.

Solving nonlinear partial differential equations (PDEs) using kernel methods offers a compelling alternative to traditional numerical solvers. However, the performance of these methods strongly depends on the choice of kernel. In this work, as the available information is inherently multifidelity, we propose a kernel learning approach based on cokriging, leveraging empirical information from multifidelity simulations. In the first step, we fit a differentiable non-stationary kernel to an empirical kernel obtained from low-fidelity simulations. In the second step, we derive a high-fidelity kernel with estimated hyperparameters, and construct a corresponding high-fidelity mean using the multifidelity framework. These components can then be used within a Gaussian process framework for solving PDEs. Finally, we demonstrate the performance of the proposed physics-informed method on the Burgers' equation.

Global Feature Extractor As mentioned in the paper, the global feature extractor GFE()is composed of 3 modules: GFE(Mn) = GAP(Conv(Sample(Mn))). The sampling module Sample()is implemented by the built-in interpolation interface in Firedrake[Rathgeber et al., 2016], with sampling density 32x32. The convolutional block contains 4 convolutional layers. The SELU [Klambauer et al., 2017] activation function is used to increase the representation capability of the model. The output tensor from the convolutional block is then fed into a Global Average Pooling (GAP) layer to get a mesh resolution invariant global feature embedding En.

We will prove by the induction. Let's suppose that the formula holds for k up to n. We will prove that this formula also holds for k = n+1. By the definition in Eq. 4 and the chain rule, we can get that: Ns,n+1(t) = t τs A.2 Spline representation In this section, we give error bounds for spline representation. For simplicity, we consider 1D scenario and assume the target function u: [0,1] R is periodic and defined on the unit interval Ω = [0,1].

Parallel-in-time (PinT) techniques have been proposed to solve systems of time-dependent differential equations by parallelizing the temporal domain.

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