用泛函测量替代点采样，提升DeepONet在复杂空间的精度与泛化能力
Fixed and Adaptive Topological DeepONets: Functional Measurements on Hausdorff Locally Convex Spaces
- 以拓扑空间的连续线性泛函作为输入函数的测量方式
- 在非赋范空间上实现5.5%-5.6%的近似不变误差，且适应性测量使均值误差低于1.2%
- 适合需要可解释、网格无关建模的科学计算场景
Deep Operator Networks (DeepONets) 通常通过固定离散化上的点值编码输入函数。基于Ismailov提出的拓扑DeepONet框架，本文将点采样替换为来自Hausdorff局部凸空间$({V},\{p_α\}_{α\in A})$连续对偶空间的连续线性泛函，该空间由分离点的半范数族生成而非单一范数，并构建了固定与自适应的泛函测量系统。测量结果结合Lee和Shin的系数空间两步法，配合仅训练时使用的解码器与正则化，稳定自适应坐标。推导出分离测量、输出基与神经逼近误差的离散误差分解，并引入Barron率改进。在反导算子（非赋范局部凸空间）、异质达西流（控制算子）、固定时间与时间演化纳维-斯托克斯涡度算子上评估。在异质达西问题中，泛函模型在未见网格上保持5.5%-5.6%的近似不变误差；在控制问题中，自适应测量使均值误差降至1.2%以下。固定时间纳维-斯托克斯问题中，自适应拓扑DeepONet为最准确的DeepONet模型，使用128个泛函坐标时均相对$L^2$误差为1.685% ± 0.017%。对比同等规模的傅里叶神经算子（FNO）虽误差更低（0.832% ± 0.172%），但需完整64x64输入场，训练时间翻倍，峰值显存高10.7倍。该框架在连续对偶空间$V'$中提供紧凑、可解释且网格可移植的坐标，适用于非赋范输入空间。
原文摘要 · Abstract (English)
Deep Operator Networks (DeepONets; arXiv:1910.03193) typically encode an input function through point values on a fixed discretization. Building on the Topological DeepONet framework of Ismailov (arXiv:2603.11972), we replace point samples by continuous linear functionals drawn from the continuous dual of a Hausdorff locally convex space $({V},\{p_α\}_{α\in A})$, whose topology is generated by a point-separating family of seminorms rather than a single norm, and develop fixed and adaptive functional measurement systems. Measurements are combined with the coefficient-space Two-Step procedure of Lee and Shin (arXiv:2309.01020), while a training-only decoder and regularization stabilize the adaptive coordinates. We derive a discrete error decomposition separating measurement, output-basis, and neural-approximation errors, together with a Barron-rate refinement. The framework is evaluated on the antiderivative operator, a non-normable locally convex input space, heterogeneous Darcy flow, a controlled operator, and fixed-time and time-evolving Navier-Stokes vorticity operators. In the heterogeneous Darcy problem, the functional models retain nearly resolution-independent errors of 5.5-5.6% on unseen grids, while in the controlled problem adaptive measurements reduce the mean error below 1.2%. For the fixed-time Navier-Stokes problem, the Adaptive Topological DeepONet is the most accurate DeepONet-based model, attaining a mean relative $L^2$ error of 1.685% +/- 0.017% using 128 functional coordinates. A comparably sized Fourier neural operator (FNO; arXiv:2010.08895) achieves the lower error 0.832% +/- 0.172%, but requires the full 64x64 input field, twice the training time, and 10.7x greater peak GPU memory. The formulation provides compact, interpretable, and discretization-portable coordinates in the continuous dual $V'$, including for non-normable input spaces.
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