轻量级扩散神经算子实现高效不确定性量化
Light-Weight Diffusion Multiplier and Uncertainty Quantification for Fourier Neural Operators
- 用扩散机制替代FNO的稠密乘子，每通道仅一个可学习时间参数
- 参数量大幅减少，预测性能不降，且支持空间相关不确定性输出
- 适合需要可靠置信度估计的科学计算与工程仿真场景
算子学习是求解偏微分方程的强大范式，傅里叶神经算子（FNO）是其主流基础。但FNO存在过度参数化问题，且缺乏原生不确定性量化能力——这对科学与工程应用的可靠性至关重要。现有方法依赖事后不确定性估算，忽略几何归纳偏置。本文提出DINOZAUR：一种基于扩散的神经算子参数化方法，具备不确定性量化能力。受热核结构启发，DINOZAUR将FNO中的稠密张量乘子替换为维度无关的扩散乘子，每个通道仅有一个可学习的时间参数，显著降低参数量和内存占用，同时保持预测性能。通过对这些时间参数定义先验，将DINOZAUR建模为贝叶斯神经算子，实现空间相关输出与校准的不确定性估计。在多个偏微分方程基准测试中，该方法表现优于或相当，并提供高效的不确定性量化。
原文摘要 · Abstract (English)
Operator learning is a powerful paradigm for solving partial differential equations, with Fourier Neural Operators serving as a widely adopted foundation. However, FNOs face significant scalability challenges due to overparameterization and offer no native uncertainty quantification -- a key requirement for reliable scientific and engineering applications. Instead, neural operators rely on post hoc UQ methods that ignore geometric inductive biases. In this work, we introduce DINOZAUR: a diffusion-based neural operator parametrization with uncertainty quantification. Inspired by the structure of the heat kernel, DINOZAUR replaces the dense tensor multiplier in FNOs with a dimensionality-independent diffusion multiplier that has a single learnable time parameter per channel, drastically reducing parameter count and memory footprint without compromising predictive performance. By defining priors over those time parameters, we cast DINOZAUR as a Bayesian neural operator to yield spatially correlated outputs and calibrated uncertainty estimates. Our method achieves competitive or superior performance across several PDE benchmarks while providing efficient uncertainty quantification.
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