This paper presents a mathematics-informed approach to neural operator design, building upon the theoretical framework established in our prior work. By integrating rigorous mathematical analysis with practical design strategies, we aim to enhance the stability, convergence, generalization, and computational efficiency of neural operators. We revisit key theoretical insights, including stability in high dimensions, exponential convergence, and universality of neural operators. Based on these insights, we provide detailed design recommendations, each supported by mathematical proofs and citations. Our contributions offer a systematic methodology for developing next-gen neural operators with improved performance and reliability.

翻译：本文提出了一种基于数学知识的神经算子设计方法，该方法建立在我们先前工作中确立的理论框架之上。通过将严格的数学分析与实际设计策略相结合，我们旨在提升神经算子的稳定性、收敛性、泛化能力及计算效率。我们重新审视了关键的理论见解，包括高维稳定性、指数收敛性以及神经算子的普适性。基于这些见解，我们提供了详细的设计建议，每项建议均辅以数学证明和文献引用。我们的贡献为开发具有更高性能和可靠性的下一代神经算子提供了一套系统的方法论。