发表机构
School of Mechanical Engineering, Purdue University; Department of Mathematics, Purdue University（普渡大学机械工程学院; 普渡大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
FAST-DeepONet 解决小样本下高维 PDE 输入导致 DeepONet 统计不稳定的问题，通过结合固定谱路径与带方向惩罚的正交残差投影，在降低误差的同时减少可训练参数，适用于坐标查询架构。
AI 中文摘要
当从数千个强相关传感器观测到偏微分方程（PDE）输入，但仅能获取少量算子样本时，深度算子网络（DeepONet）会出现统计不稳定问题。本文提出 FAST-DeepONet，其分支表示结合了固定谱路径与正交残差的正则化投影，其中方向惩罚作用于有效残差映射，且该映射的每一行均经过归一化处理。在纳维-斯托克斯（Navier--Stokes）流任务中，普通 DeepONet 的分支坐标从 129 增长至 8193 时，其平均相对 L₂ 误差从 0.0394 降至 0.1556，而 FAST-DeepONet 的误差始终维持在 0.04 附近，这意味着可对传感器网格进行细化而不会产生统计性能损失。在纳维-斯托克斯流、达西（Darcy）流及符号终端波场预测的独立测试集上，FAST-DeepONet 可将平均相对 L₂ 误差降低 4.7% 至 37.0%，同时可训练参数仅为原有模型的 1/3 至 1/7。通过分离出共享同一基的纯谱分支与残差分支可知：固定谱路径实现了在纳维-斯托克斯流和达西流任务上的性能提升，而终端波预测任务则需结合残差路径及其方向惩罚项。FAST-DeepONet 针对坐标查询架构设计，仅基于解值进行训练。
英文摘要
Deep operator networks can become statistically unstable when partial differential equation inputs are observed at thousands of strongly correlated sensors but only a small number of operator samples is available. We introduce FAST-DeepONet, a branch representation combining a fixed spectral path with a regularized projection of the orthogonal residual, in which the directional penalty acts on the effective residual map after each of its rows is normalized. On Navier--Stokes flow a plain DeepONet degrades from $0.0394$ to $0.1556$ mean relative $L_2$ error as the branch grows from $129$ to $8193$ coordinates, while FAST-DeepONet stays near $0.04$, so the sensor grid can be refined without a statistical penalty. Across independent test sets for Navier--Stokes flow, Darcy flow, and signed terminal wavefield prediction it lowers mean relative $L_2$ error by $4.7\%$ to $37.0\%$ with three to seven times fewer trainable parameters. A spectral-only branch sharing the same basis separates the two paths: the fixed spectral path carries the improvement on Navier--Stokes and Darcy, while terminal wave prediction requires the residual path together with its directional penalty. FAST-DeepONet targets coordinate-query architectures and trains on solution values alone.
