Neural operators are widely used as surrogate solution maps for partial differential equations (PDEs), but full-size models can be costly to store, deploy, and evaluate in many-query scientific workflows. This work introduces Operator Boosting, a stagewise residual-learning framework for constructing compact neural-operator surrogates directly, rather than training a large model and compressing it afterward. Starting from the empirical mean predictor in normalized output coordinates, the method trains a sequence of tiny same-family neural operators on residual fields and incorporates each correction through validation-selected shrinkage. We instantiate the framework with Fourier neural operators (FNOs), DeepONets, and convolutional neural operators (CNOs), and compare boosted tiny stacks against full-size monolithic baselines across one-, two-, and three-dimensional PDE benchmarks from PDEBench, APEBench, and The Well. Across 30 dataset-architecture pairs, 21 show positive mean accuracy gains and 17 have positive confidence intervals, while all boosted stacks reduce trainable parameter count by approximately 72-95%. Best-model comparisons show empirical Pareto improvements on 7 of 10 completed PDE benchmarks, including two-dimensional Navier-Stokes, shallow-water dynamics, Darcy flow, one-dimensional transport and reaction systems, and three-dimensional compressible Navier-Stokes. These results show that Operator Boosting often improves the empirical accuracy-parameter Pareto frontier of neural PDE surrogates, while also exposing PDE- and architecture-dependent regimes where residual boosting fails to offset compression.

翻译：神经算子广泛用作偏微分方程的替代解映射，但在多查询科学工作流中，全尺寸模型的存储、部署和评估成本较高。本文提出算子提升（Operator Boosting）——一种逐阶段残差学习框架，可直接构建紧凑的神经算子替代模型，而非先训练大型模型再进行压缩。该方法从归一化输出坐标中的经验均值预测器出发，在残差场上训练一系列同族小型神经算子，并通过验证选择的收缩机制整合每次修正。我们以傅里叶神经算子（FNO）、DeepONet和卷积神经算子（CNO）为例实例化该框架，并在来自PDEBench、APEBench和The Well的一维、二维和三维偏微分方程基准上，将提升后的小型堆栈与全尺寸单一基线模型进行比较。在30个数据集-架构组合中，21个显示出正向平均精度提升，17个置信区间为正值，同时所有提升堆栈的可训练参数量减少约72-95%。最佳模型对比显示，在10个已完成的偏微分方程基准中有7个出现实证帕累托改进，包括二维纳维-斯托克斯方程、浅水动力学、达西流、一维输运与反应系统，以及三维可压缩纳维-斯托克斯方程。结果表明，算子提升常能改善神经偏微分方程替代模型在精度-参数上的经验帕累托前沿，同时揭示了残差提升无法抵消压缩损失的特定偏微分方程和架构依赖情形。