新模型可精准预测复杂区域的时空变化，突破传统方法只能处理相同区域的限制。
A general reduced-order neural operator for spatio-temporal predictive learning on complex spatial domains
- 用基函数分解时空数据，将不同区域映射转为相同区域处理
- 在6个基准测试中精度和训练效率均优于现有方法
- 适合物理模拟、工程与生物医学中的复杂系统预测
复杂空间域上的时空过程预测学习（PL-STP）在众多科学与工程领域具有关键作用，其核心在于构建无限维函数空间间的算子。本文聚焦于不等域映射问题，将其分为增域与减域两类。近年来深度学习揭示了神经算子（NOs）从观测数据中直接学习算子的巨大潜力，但现有方法要求输入与输出空间为同一域，难以保证不等域映射下的预测精度与稳定性。为此，本文提出一种通用降阶神经算子——黎曼流形上的降阶神经算子（RO-NORM），由不等域编码器/解码器与同域近似器两部分构成。受经典模态分解变量分离思想启发，不等域编码器/解码器利用预计算基函数，将时空函数重写为时空基函数与对应权重函数乘积之和，从而将原不等域映射转化为同域映射。随后，同域近似器NORM用于建模转换后的映射。实验在六个基准案例上验证，涵盖参数化偏微分方程、工程与生物医学应用，对比四种基线算法（DeepONet、POD-DeepONet、PCA-Net、vanilla NORM），结果表明RO-NORM在预测精度与训练效率上均具显著优势。
原文摘要 · Abstract (English)
Predictive learning for spatio-temporal processes (PL-STP) on complex spatial domains plays a critical role in various scientific and engineering fields, with its essence being the construction of operators between infinite-dimensional function spaces. This paper focuses on the unequal-domain mappings in PL-STP and categorising them into increase-domain and decrease-domain mapping. Recent advances in deep learning have revealed the great potential of neural operators (NOs) to learn operators directly from observational data. However, existing NOs require input space and output space to be the same domain, which pose challenges in ensuring predictive accuracy and stability for unequal-domain mappings. To this end, this study presents a general reduced-order neural operator named Reduced-Order Neural Operator on Riemannian Manifolds (RO-NORM), which consists of two parts: the unequal-domain encoder/decoder and the same-domain approximator. Motivated by the variable separation in classical modal decomposition, the unequal-domain encoder/decoder uses the pre-computed bases to reformulate the spatio-temporal function as a sum of products between spatial (or temporal) bases and corresponding temporally (or spatially) distributed weight functions, thus the original unequal-domain mapping can be converted into a same-domain mapping. Consequently, the same-domain approximator NORM is applied to model the transformed mapping. The performance of our proposed method has been evaluated on six benchmark cases, including parametric PDEs, engineering and biomedical applications, and compared with four baseline algorithms: DeepONet, POD-DeepONet, PCA-Net, and vanilla NORM. The experimental results demonstrate the superiority of RO-NORM in prediction accuracy and training efficiency for PL-STP.
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