发表机构
Penn State University; School of Electrical Engineering and Computer Science, Penn State University（宾夕法尼亚州立大学; 宾夕法尼亚州立大学电气工程与计算机科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出灵敏度约束神经算子（SC-NO），通过采样灵敏度监督改进神经PDE代理，在正演预测、反演重建等任务中提升精度与效率，实现了更优的精度-成本权衡。
AI 中文摘要
神经算子可为偏微分方程（PDE）求解器提供快速代理模型，但其在高维空间输入及反演或重复推理场景下可靠性会下降。仅状态训练仅约束解的数值，却不约束学习到的输入-输出响应。本文研究灵敏度约束神经算子（SC-NO），其通过添加采样自求解器的雅可比监督来增强标准训练。训练过程中匹配来自可微求解器或离散伴随的选定灵敏度，使得响应信息可在小批量间分摊，无需在每次更新时施加完整雅可比矩阵。我们在平流-扩散、RANS-Spalart-Allmaras基准测试、输入维度缩放测试、长时自回归滚动以及东北浅海海啸源反演案例中评估SC-NO。灵敏度监督提升了正演预测，并在分布式场的基于梯度反演重建中带来更大收益。缩放实验显示，其在高维网格输入下实现了更优的精度-成本权衡； ablation实验表明，状态值与雅可比信息提供互补监督。在海啸案例中，SC-FNO从稀疏早期监测站观测中重建网格状海底变形，并在近实时概念验证工作流中预测后续波传播。这些结果支持采样灵敏度监督作为一种实用方法，用于在同时考虑正演精度、反演稳定性、鲁棒性及计算成本时改进神经PDE代理模型。
英文摘要
Neural operators provide fast surrogates for partial differential equation (PDE) solvers, but their reliability can degrade for high-dimensional spatial inputs and inverse or repeated inference. State-only training constrains solution values but not the learned input--output response. We study sensitivity-constrained neural operators (SC-NOs), which augment standard training with sampled solver-derived Jacobian supervision. Selected sensitivities from differentiable solvers or discrete adjoints are matched during training, allowing response information to be amortized across minibatches without imposing the full Jacobian at every update. We evaluate SC-NO on advection--diffusion and RANS--Spalart--Allmaras benchmarks, input-dimensionality scaling tests, long-horizon autoregressive rollout, and a shallow-water Tohoku tsunami source-inversion case. Sensitivity supervision improves forward prediction and yields larger gains in gradient-based inverse reconstruction of distributed fields. Scaling experiments show an improved accuracy--cost tradeoff for high-dimensional gridded inputs, while ablations indicate that state values and Jacobian information provide complementary supervision. In the tsunami case, SC-FNO reconstructs gridded seafloor deformation from sparse early gauge observations and forecasts subsequent wave propagation in a near-real-time proof-of-concept workflow. These results support sampled sensitivity supervision as a practical way to improve neural PDE surrogates when forward accuracy, inverse stability, robustness, and computational cost must be considered together.
