让神经PDE代理模型直接学习可校准的不确定性，提升数据少时的可靠性。
Direct Learning of Calibration-Aware Uncertainty for Neural PDE Surrogates
- 训练时用交叉正则化同时优化预测与不确定性参数。
- 在不同观测比例和数据量下，预测不确定性更准确且集中在误差大区域。
- 适合需要可靠置信度的科学计算与少样本场景。
神经PDE代理模型常用于数据有限或部分可观测的场景，下游决策依赖于校准后的不确定性而非仅低误差。现有方法通过集成、固定随机噪声（如丢弃）或事后校准获取不确定性。交叉正则化不确定性在训练中通过梯度路径经保留的正则化子集学习不确定性参数。预测器在训练子集上优化拟合，而低维不确定性控制在正则化子集上优化以减少训练-测试偏差，实现无需逐场景调参的自适应不确定性。该框架可在输出头、隐藏特征或特定组件（如谱模式）中学习连续噪声水平。我们在傅里叶神经算子中实现该方法，并在APEBench上对观测比例和训练集大小进行多组测试。结果表明，所学预测分布在保留子集上校准更好，且不确定性场在一阶空间诊断中集中于高误差区域。
原文摘要 · Abstract (English)
Neural PDE surrogates are often deployed in data-limited or partially observed regimes where downstream decisions depend on calibrated uncertainty in addition to low prediction error. Existing approaches obtain uncertainty through ensemble replication, fixed stochastic noise such as dropout, or post hoc calibration. Cross-regularized uncertainty learns uncertainty parameters during training using gradients routed through a held-out regularization split. The predictor is optimized on the training split for fit, while low-dimensional uncertainty controls are optimized on the regularization split to reduce train-test mismatch, yielding regime-adaptive uncertainty without per-regime noise tuning. The framework can learn continuous noise levels at the output head, within hidden features, or within operator-specific components such as spectral modes. We instantiate the approach in Fourier Neural Operators and evaluate on APEBench sweeps over observed fraction and training-set size. Across these sweeps, the learned predictive distributions are better calibrated on held-out splits and the resulting uncertainty fields concentrate in high-error regions in one-step spatial diagnostics.
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