标题： RONOM：降阶神经算子建模 显示英文标题

标题： RONOM: Reduced-Order Neural Operator Modeling

摘要： 时间依赖的偏微分方程在基于物理的建模中无处不在，但在许多查询场景中仍然计算密集，例如实时预测、最优控制和不确定性量化。 降阶建模（ROM）通过构建低维代理模型来解决这些挑战，但依赖于固定的离散化，这限制了在评估过程中不同网格之间的灵活性。 算子学习方法，如神经算子，通过参数化无限维函数空间之间的映射提供了一种替代方案，使模型能够适应不同分辨率的数据。 尽管ROM提供了严格的数值误差估计，神经算子学习主要关注离散化收敛性和不变性，而没有量化无限维算子与离散化算子之间的误差。 本文介绍了降阶神经算子建模（RONOM）框架，该框架结合了ROM和算子学习的概念。 我们建立了类似于ROM中的离散化误差界限，并深入了解RONOM的离散化收敛性和离散化鲁棒性。 此外，给出了两个数值例子，将RONOM与现有的神经算子进行比较，以求解偏微分方程。 结果表明，RONOM使用标准的向量到向量神经网络在输入泛化方面表现出可比的性能，并在空间超分辨率和离散化鲁棒性方面表现出更优的性能，同时为时间超分辨率场景提供了新的见解。 显示英文摘要

摘要： Time-dependent partial differential equations are ubiquitous in physics-based modeling, but they remain computationally intensive in many-query scenarios, such as real-time forecasting, optimal control, and uncertainty quantification. Reduced-order modeling (ROM) addresses these challenges by constructing a low-dimensional surrogate model but relies on a fixed discretization, which limits flexibility across varying meshes during evaluation. Operator learning approaches, such as neural operators, offer an alternative by parameterizing mappings between infinite-dimensional function spaces, enabling adaptation to data across different resolutions. Whereas ROM provides rigorous numerical error estimates, neural operator learning largely focuses on discretization convergence and invariance without quantifying the error between the infinite-dimensional and the discretized operators. This work introduces the reduced-order neural operator modeling (RONOM) framework, which bridges concepts from ROM and operator learning. We establish a discretization error bound analogous to those in ROM, and get insights into RONOM's discretization convergence and discretization robustness. Moreover, two numerical examples are presented that compare RONOM to existing neural operators for solving partial differential equations. The results demonstrate that RONOM using standard vector-to-vector neural networks achieves comparable performance in input generalization and superior performance in both spatial super-resolution and discretization robustness, while also offering novel insights into temporal super-resolution scenarios.