发表机构
Division of Applied Mathematics, Brown University（布朗大学应用数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出固定与自适应拓扑DeepONet，将点样本替换为Hausdorff局部凸空间对偶的连续线性泛函，在多算子任务上验证其性能，为非可赋范空间提供紧凑可移植的函数坐标。
AI 中文摘要
深度算子网络（DeepONets；arXiv:1910.03193）通常通过固定离散化上的点值编码输入函数。本文在Ismailov的拓扑DeepONet框架（arXiv:2603.11972）基础上，将点样本替换为从Hausdorff局部凸空间$({V},\{p_\alpha\}_{\alpha\in A})$的连续对偶中提取的连续线性泛函，该空间的拓扑由点分离的半范数族而非单一范数生成，并开发了固定与自适应函数测量系统。测量结果与Lee和Shin的系数空间两步法（arXiv:2309.01020）结合，同时仅训练的解码器与正则化项稳定自适应坐标。本文推导了离散误差分解，将误差分为测量、输出基与神经逼近误差，还得到了Barron速率改进项。该框架在反导数算子（非可赋范局部凸输入空间）、非均质达西流、受控算子、固定时间与随时间演化的Navier-Stokes涡度算子上进行了评估。在非均质达西问题中，函数模型在未见网格上保持近分辨率无关的5.5-5.6%误差；在受控问题中，自适应测量将平均误差降至1.2%以下。对于固定时间Navier-Stokes问题，自适应拓扑DeepONet是最准确的基于DeepONet的模型，使用128个函数坐标时平均相对$L^2$误差为1.685%±0.017%。规模相当的傅里叶神经算子（FNO；arXiv:2010.08895）实现了更低的0.832%±0.172%误差，但需要完整的64×64输入场、两倍的训练时间及10.7倍的峰值GPU内存。该公式在连续对偶空间$V'$中提供了紧凑、可解释且可离散化移植的坐标，包括针对非可赋范输入空间的情况。
英文摘要
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.
