将无限宽神经算子与函数空间上的高斯过程关联，提升对物理方程求解器的不确定性建模能力。
Infinite Neural Operators: Gaussian processes on functions
- 证明任意深度神经算子在特定条件下收敛为函数值高斯过程
- 推导出FNO等算子的协方差函数，实现后验分布计算
- 为物理信息机器学习提供新归纳偏置，适合科学计算领域研究者
多种无限宽神经网络架构（如全连接网络、卷积网络和Transformer）会诱导输出空间上的高斯过程先验。这一关系不仅准确刻画了先验预测分布，还使高斯过程工具可用于提升深度神经网络的不确定性量化。本文将该联系扩展至神经算子（NOs），一类用于学习函数空间间映射的模型。具体地，我们给出任意深度神经算子在高斯卷积核下收敛到函数值高斯过程的条件。基于此结果，我们推导了两种神经算子参数化形式下的协方差函数，包括流行的傅里叶神经算子（FNO）。据此，我们在回归场景（包括偏微分方程求解算子）中计算了这些高斯过程的后验分布。这项工作是揭示当前FNO架构归纳偏置的重要一步，并为基于核的方法中的算子学习开辟了引入新归纳偏置的新路径。
原文摘要 · Abstract (English)
A variety of infinitely wide neural architectures (e.g., dense NNs, CNNs, and transformers) induce Gaussian process (GP) priors over their outputs. These relationships provide both an accurate characterization of the prior predictive distribution and enable the use of GP machinery to improve the uncertainty quantification of deep neural networks. In this work, we extend this connection to neural operators (NOs), a class of models designed to learn mappings between function spaces. Specifically, we show conditions for when arbitrary-depth NOs with Gaussian-distributed convolution kernels converge to function-valued GPs. Based on this result, we show how to compute the covariance functions of these NO-GPs for two NO parametrizations, including the popular Fourier neural operator (FNO). With this, we compute the posteriors of these GPs in regression scenarios, including PDE solution operators. This work is an important step towards uncovering the inductive biases of current FNO architectures and opens a path to incorporate novel inductive biases for use in kernel-based operator learning methods.
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