基于SWAN的共形神经算子时空预测不确定性量化

Conformal Neural Operators with SWAN for Spatio-Temporal Forecasting Uncertainty Quantification

摘要: 时空预测中的不确定性量化长期面临着边缘有效性与条件适应性之间的权衡：共形预测提供了分布自由的有限样本覆盖保证，但标准方法在网格化场上产生空间恒定宽度的预测带，难以反映预报难度在不同地理区域间的显著差异。为此，本文提出谱–小波自适应非一致性(SWAN)分数，一种为球面时空场设计的新型非一致性函数。SWAN通过球谐变换将预报残差分解为$J$ 个频带，对平方频带分量进行空间高斯平滑以计算局部能量，并通过可学习权重自适应聚合，从而根据局部谱可预测性调节预测带宽度。将SWAN分数与球面傅里叶神经算子(SFNO)集成及分裂共形校准相结合，构成SWAN-CP方法；针对气候时间序列的不可交换性，引入加权共形校正以保证有限样本边缘有效性，并从理论上证明了$\beta$ -混合条件下的近似有效性和条件覆盖逼近性质。在ERA5再分析数据(2010~2024年，128 × 256网格，6小时间隔)上的实验表明，与持续预报、气候态、普通共形预测、分层共形预测、共形分位数回归及时间加权共形预测等六种基线方法相比，SWAN-CP在$J=4$ 配置下实现了所有基于模型方法中最窄的预测带，$J=8$ 配置下获得了最优Winkler区间评分(0.828)，覆盖率维持86.3%；跨纬度条件覆盖率变异从8.5个百分点降至1.5个百分点，条件覆盖均匀性提升5.7倍。消融实验确认谱分解是性能提升的主要驱动力，可学习频带权重贡献了额外4.4%的效率增益。上述结果表明，通过频域分解实现的空间自适应预测带在所比较的六种基线方法中展现了最优的带宽效率和综合评分性能，为场结构和函数型数据的共形推断提供了一种新的技术路径。

Abstract: Accurate uncertainty quantification in spatio-temporal forecasting is fundamentally challenged by the tension between marginal validity and conditional adaptivity. Conformal prediction provides distribution-free finite-sample coverage guarantees, yet standard split-conformal methods applied to gridded spatio-temporal fields produce prediction bands of constant spatial width that ignore the substantial geographic heterogeneity in forecast difficulty. This paper introduces the Spectral-Wavelet Adaptive Nonconformity (SWAN) score, a principled nonconformity function designed specifically for spatio-temporal fields on the sphere. Drawing on the insight that predictability is inherently scale-dependent, SWAN decomposes forecast residuals via spherical harmonic transforms into $J$ power-law spaced frequency bands, computes locally smoothed per-band energy through Gaussian spatial smoothing, and aggregates these band-wise energy fields through learnable weights optimized for bandwidth efficiency. The SWAN-CP method integrates the SWAN score with an ensemble of Spherical Fourier Neural Operators (SFNOs) within a split conformal calibration framework, augmented by a weighted conformal correction to address the temporal non-exchangeability of climate time series. Theoretical analysis establishes approximate marginal validity under $\beta$ -mixing conditions and conditional coverage approximation guarantees within the class of band-structured nonconformity scores. Comprehensive evaluation on ERA5 reanalysis data (2010~2024, 128 × 256 grid, 6-hourly resolution) benchmarks SWAN-CP against six baselines: persistence, climatology, vanilla conformal prediction, Mondrian conformal prediction, Conformalized Quantile Regression (CQR), and temporal-weighted conformal prediction. SWAN-CP with $J=4$ frequency bands achieves the narrowest average prediction bands among all model-based methods, while the $J=8$ configuration attains the best overall Winkler interval score (0.828) with 86.3% empirical coverage. Cross-latitude conditional coverage variation is reduced from 8.5 percentage points to 1.5 percentage points, a 5.7-fold improvement in uniformity. Ablation experiments confirm that spectral decomposition is the primary driver of performance gains, with learnable band weights contributing an additional 4.4% efficiency improvement. The computational overhead remains modest and acceptable for operational deployment. These results demonstrate that frequency-domain decomposition achieves the best bandwidth efficiency and overall scoring performance among the six baselines compared, advancing conformal inference for field-structured and functional data.