Neural operators have emerged as a powerful, discretization-invariant framework for solving partial differential equations (PDEs). Although established approaches like the Deep Operator Network (DeepONet) have successfully achieved universal approximation for operators, and architectures such as Fourier Neural Operators (FNOs) have shown algebraic convergence rates, a precise theoretical connection between the continuous theory and its discrete numerical implementation remains a challenge. Specifically, the relationship between the continuous formulation and the discrete numerical stability has yet to be fully explored. In this paper, we address this gap by establishing theoretical guarantees for the discretization error and stability of neural operator approximation schemes. We prove analytical bounds that link solution regularity to input discretization, providing a formal quantification of neural operator accuracy under real-world numerical constraints. We derive these bounds to the specific cases of State Space Model-based Neural Operators (SS-NOs) and FNOs, thus providing a new discretization error theorem for these models. Additionally, through an input-to-state stability (ISS) analysis, we formally assess the impact of discretization on the stability of SS-NOs results obtained in the continuous domain. Our empirical experiments on 1D and 2D benchmarks validate our theoretical bounds and show the robustness of SS-NOs under varying resolutions.

翻译：神经算子作为求解偏微分方程（PDEs）的强大框架，具有离散化不变性。尽管深度算子网络（DeepONet）等成熟方法已成功实现算子的通用逼近，且傅里叶神经算子（FNOs）等架构展现出代数收敛速率，但其连续理论与离散数值实现之间的精确理论联系仍面临挑战。特别是连续公式与离散数值稳定性之间的关系尚未得到充分探索。本文通过建立神经算子逼近方案的离散化误差与稳定性的理论保证，填补了这一空白。我们证明了将解的正则性与输入离散化相关联的分析界，为实际数值约束下的神经算子精度提供了形式化量化。我们将这些界推导至基于状态空间模型的神经算子（SS-NOs）和FNOs的特例中，从而为这些模型提出了新的离散化误差定理。此外，通过输入-状态稳定性（ISS）分析，我们正式评估了离散化对连续域中SS-NOs结果稳定性的影响。在1D和2D基准上的实证实验验证了我们的理论界，并展示了SS-NOs在不同分辨率下的鲁棒性。