arXiv:2603.11052cs.LGcs.AI2026-03被引 3

给神经算子的不确定性量化加了结构感知,更准更省时。

Structure-Aware Epistemic Uncertainty Quantification for Neural Operator PDE Surrogates

  • 只在输入映射层加随机扰动,保留核心推理模块确定性
  • 在多个复杂方程上验证,不确定性分布更贴合真实残差
  • 适合需要可靠风险评估的科学计算场景

神经算子(NOs)能快速、分辨率不变地预测偏微分方程(PDE)解场,但受限于数据有限、优化不完善和分布偏移,其预测存在显著认知不确定性。为实现科学计算中的实用部署,不确定性量化(UQ)需兼顾计算效率与空间精度——即不确定性带应与影响下游风险的关键局部残差结构对齐。本文提出一种结构感知的认知不确定性量化方法,利用现代神经算子普遍存在的分层结构(升维-传播-还原)。不同于对全网络进行无结构权重扰动(如简单丢弃),本方法将蒙特卡洛采样限制在模块对齐的子空间中:仅在升维模块注入随机性,而将学习到的求解动态(传播与还原)视为确定性。我们设计了两种轻量级升维层扰动:通道级乘性特征丢弃与方差匹配的高斯特征扰动,并通过标准校准构建不确定性带。在多个挑战性PDE基准测试中(包括系数不连续的达西流及几何偏移的3D汽车气动仿真代理模型),结果表明该结构感知设计在保持运行效率的同时,实现了更可靠的覆盖率、更紧的不确定性带以及更强的残差-不确定性对齐效果,优于常见基线方法。

原文摘要 · Abstract (English)

Neural operators (NOs) provide fast, resolution-invariant surrogates for mapping input fields to PDE solution fields, but their predictions can exhibit significant epistemic uncertainty due to finite data, imperfect optimization, and distribution shift. For practical deployment in scientific computing, uncertainty quantification (UQ) must be both computationally efficient and spatially faithful, i.e., uncertainty bands should align with the localized residual structures that matter for downstream risk management. We propose a structure-aware epistemic UQ scheme that exploits the modular anatomy common to modern NOs (lifting-propagation-recovering). Instead of applying unstructured weight perturbations (e.g., naive dropout) across the entire network, we restrict Monte Carlo sampling to a module-aligned subspace by injecting stochasticity only into the lifting module, and treat the learned solver dynamics (propagation and recovery) as deterministic. We instantiate this principle with two lightweight lifting-level perturbations, including channel-wise multiplicative feature dropout and a Gaussian feature perturbation with matched variance, followed by standard calibration to construct uncertainty bands. Experiments on challenging PDE benchmarks (including discontinuous-coefficient Darcy flow and geometry-shifted 3D car CFD surrogates) demonstrate that the proposed structure-aware design yields more reliable coverage, tighter bands, and improved residual-uncertainty alignment compared with common baselines, while remaining practical in runtime.

神经算子不确定性量化偏微分方程结构感知

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