用扩散模型提升神经算子的不确定性预测能力
Generative Neural Operators through Diffusion Last Layer
- 在算子输出层引入条件扩散模型，通过低秩系数建模目标函数
- 在随机微分方程任务中实现高保真分布生成，优于像素与潜空间基线
- 提升长时滚动预测稳定性，并给出可信的预测误差估计
神经算子能学习函数空间间的离散不变映射，但传统确定性模型无法捕捉预测不确定性。本文提出扩散最后层（DLL），作为神经算子主干的模块化概率输出头。DLL通过受Karhunen-Loève展开启发的输入相关低秩展开表示目标场，并在对应系数空间上学习条件扩散模型。该设计实现了高效分布建模，同时保留算子学习的结构优势。在带随机激励的随机微分方程基准测试中，DLL实现强分布保真度，性能媲美像素空间与传统潜空间扩散基线。在确定性长时滚动任务中，DLL提升了主干模型的滚动稳定性，并在自回归误差累积下提供有效的预测不确定性估计。结果表明，在学习系数空间中采用扩散建模是实现具备不确定性感知能力的神经算子的可行路径。
原文摘要 · Abstract (English)
Neural operators provide a powerful framework for learning discretization invariant mappings between function spaces, but standard deterministic models do not capture predictive uncertainty. We introduce diffusion last layer (DLL), a modular probabilistic output head for neural operator backbones. DLL represents target fields through an input dependent low rank expansion inspired by the Karhunen-Loéve expansion and learns a conditional diffusion model over the corresponding coefficient space. This design enables efficient distributional modeling while preserving the structural advantages of operator learning. On stochastic PDE benchmarks with random forcing, DLL achieves strong distributional fidelity and performs competitively with pixel space and conventional latent diffusion baselines. In deterministic long horizon rollout tasks, DLL improves rollout stability over the underlying backbone and provides useful estimates of predictive uncertainty under compounding autoregressive errors. These results suggest that diffusion modeling in learned coefficient spaces offers a practical route to uncertainty aware neural operators.
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