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, three 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 can achieve comparable performance in input generalization and achieves superior performance in both spatial super-resolution and discretization robustness, while also offering novel insights into temporal super-resolution scenarios and ROM-based approaches for learning on time-dependent data.

翻译：时变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（如实时预测、最优控制和不确定性量化）中，其计算仍然非常密集。降阶建模通过构建低维代理模型来应对这些挑战，但依赖于固定的离散化方案，这限制了在评估过程中跨不同网格的灵活性。算子学习方法（如神经算子）提供了一种替代方案，它通过参数化无限维函数空间之间的映射，实现了对不同分辨率数据的适应。虽然降阶建模提供了严格的数值误差估计，但神经算子学习主要关注离散化收敛性和不变性，而未量化无限维算子与离散化算子之间的误差。本文提出了降阶神经算子建模框架，该框架融合了降阶建模与算子学习的概念。我们建立了类似于降阶建模中的离散化误差界，并深入分析了RONOM的离散化收敛性和离散化鲁棒性。此外，本文通过三个数值算例，将RONOM与现有神经算子在求解偏微分方程方面进行了比较。结果表明，使用标准向量到向量神经网络的RONOM在输入泛化方面可以达到相当的性能，并在空间超分辨率和离散化鲁棒性方面均表现出更优的性能，同时为时间超分辨率场景以及基于降阶建模的时变数据学习方法提供了新的见解。