Solving dynamical systems in complex and diverse geometries remains challenging for both numerical solvers and learning-based surrogate models. We propose Neural Operator Splitting, a framework for le...
The paper introduces a kernel‑based operator learning method that preserves key physical properties—such as incompressibility, periodicity, and turbulence—of the incompressible Navier–Stokes equations. By mapping input functions to expansion coefficients in a property‑preserving kernel basis, the method guarantees that predicted velocity fields analytically maintain these properties. The authors provide theoretical convergence guarantees, develop efficient computational techniques for large‑scale training, and demonstrate significant accuracy and speed improvements over neural operators on 2D and 3D flow benchmarks.
arXiv:2605. 04853v2 Announce Type: replace Abstract: We propose HIN-LRI, a hybrid framework that augments a classical numerical solver with a neural operator trained to correct the solver's structured truncation error.
arXiv:2512. 19643v2 Announce Type: replace Abstract: Numerical simulation of time-dependent partial differential equations (PDEs) is central to scientific and engineering applications, but high-fidelity solvers are often prohibitively expensive for long-horizon or time-critical settings.
arXiv:2609. 19039v1 Announce Type: cross Abstract: We introduce the Long-Short-Range Neural Network (LSR-Net), a novel neural operator architecture designed for data-driven forward evolution modeling, and extends it to the prediction of nonlinear fluid dynamics.
arXiv:2507.07292v2 Announce Type: replace Abstract: We develop a new and general encode-approximate-reconstruct operator learning model that leverages learned neural representations of bases for inpu...
arXiv:2601. 20496v2 Announce Type: replace-cross Abstract: Generating dense physical fields from sparse measurements is a fundamental question in sampling, signal processing, and many other applications.
This chapter reviews recent advances in Scientific Machine Learning (SciML) for modeling coupled fluid flow and transport phenomena governed by the incompressible Navier-Stokes and scalar transport equations. Such systems, found in applications like turbidity currents and thermal convection, feature strong nonlinear coupling and multiscale behavior that make high-fidelity simulations computationally expensive.
arXiv:2607. 07718v1 Announce Type: cross Abstract: Neural operators have become a common approach for learning PDE solution maps and accelerating numerical simulations.
arXiv:2606. 11963v1 Announce Type: new Abstract: Neural operators provide a powerful framework for learning solution mappings of partial differential equations directly in function space.
The paper introduces Cut-DeepONet, a two‑stage training framework that explicitly models discontinuities in partial differential equations by partitioning the domain into smooth subregions and representing discontinuities as boundaries in a higher‑dimensional space. This lifting strategy aligns the operator learning task with neural network inductive biases, allowing the model to avoid directly approximating discontinuities. Experiments on benchmark PDEs demonstrate that Cut‑DeepONet outperforms state‑of‑the‑art methods, even with low‑resolution data, while using fewer trainable parameters.
arXiv:2505. 11766v4 Announce Type: replace Abstract: Neural Operators (NOs) are powerful architectures for learning mappings between function spaces.