快速训练且带不确定性量化的随机神经算子，解决科学计算中反复求解难题。
Randomized neural operator for parametric PDEs with fast training and conformal uncertainty quantification
- 用随机特征+主成分降维，将训练转为线性回归，大幅提速
- 在多个方程上保持精度，训练时间缩短1~3个数量级
- 支持在线更新和置信区间，适合需多次求解的科研场景
重复求解参数化偏微分方程（PDE）对不确定性量化、设计优化和反问题至关重要，但传统神经算子需昂贵的非凸训练。本文提出PCA--RaNN，一种结合主成分分析（PCA）降维与固定随机特征的随机隐空间神经算子，采用闭式最小二乘读出层。它将隐空间算子学习转化为固定特征线性回归，使多个基准测试中的训练时间减少1至3个数量级，同时保持良好精度。引入能量匹配缩放规则与轻量级两参数BFGS优化以校正特征尺度偏差。通过集成平均降低预测方差。在Burgers、Darcy、Navier-Stokes及逆热方程基准上，PCA--RaNN相较现有算子学习方法展现出更优的速度-精度权衡。其集成支持分割共形预测区间，线性读出结构可借助递归最小二乘实现无需重训练的快速在线适应。该方法为多查询科学工作流提供高效且带不确定性的代理模型。
原文摘要 · Abstract (English)
Repeatedly solving parametric PDEs is essential for uncertainty quantification, design optimization and inverse problems, but conventional neural operators require expensive non-convex training. We introduce PCA--RaNN, a randomized latent neural operator that combines PCA-based dimensionality reduction with fixed random features and a closed-form least-squares readout. It recasts latent operator learning as fixed-feature linear regression, reducing training time by one to three orders of magnitude across benchmarks while maintaining competitive accuracy. We introduce an energy-matched scaling rule and a lightweight two-parameter BFGS refinement to correct suboptimal feature scales. Ensemble averaging reduces predictive variance. On Burgers, Darcy, Navier--Stokes and backward heat equation benchmarks, PCA--RaNN provides a favorable speed--accuracy trade-off against operator-learning baselines. The ensemble supports split-conformal prediction intervals, and the linear readout enables rapid online adaptation via recursive least squares without retraining hidden features. This provides an efficient, uncertainty-aware surrogate for many-query scientific workflows.
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