线性化将神经算子转化为函数值高斯过程
Linearization Turns Neural Operators into Function-Valued Gaussian Processes
AI总结：
提出LUNO框架，通过模型线性化将贝叶斯深度学习方法应用于神经算子，实现高效、分辨率无关的函数值高斯过程不确定性量化。
摘要：
神经算子推广了神经网络，用于从数据中学习函数空间之间的映射。它们通常被用于学习参数化偏微分方程（PDE）的解算子，或时间依赖性PDE的传播算子。然而，为了使其在高风险模拟场景中发挥作用，必须可靠地量化其固有的预测误差。我们引入了LUNO，一种用于训练后的神经算子近似贝叶斯不确定性量化的新框架。我们的方法利用模型线性化，将（高斯）权重空间的不确定性前向传播到神经算子的预测中。我们证明这可以解释为函数式编程中柯里化概念的概率版本，从而产生函数值（高斯）随机过程信念。我们的框架提供了一种实用且理论上可靠的方式，将现有的贝叶斯深度学习方法（如线性化拉普拉斯近似）应用于神经算子。与底层的神经算子一样，我们的方法在设计上是分辨率无关的。该方法增加了极少的预测开销，可以在不重新训练网络的情况下事后应用，并且能够扩展到大型模型和数据集。我们在傅里叶神经算子的案例研究中评估了这些方面。
英文摘要：
Neural operators generalize neural networks to learn mappings between function spaces from data. They are commonly used to learn solution operators of parametric partial differential equations (PDEs) or propagators of time-dependent PDEs. However, to make them useful in high-stakes simulation scenarios, their inherent predictive error must be quantified reliably. We introduce LUNO, a novel framework for approximate Bayesian uncertainty quantification in trained neural operators. Our approach leverages model linearization to push (Gaussian) weight-space uncertainty forward to the neural operator's predictions. We show that this can be interpreted as a probabilistic version of the concept of currying from functional programming, yielding a function-valued (Gaussian) random process belief. Our framework provides a practical yet theoretically sound way to apply existing Bayesian deep learning methods such as the linearized Laplace approximation to neural operators. Just as the underlying neural operator, our approach is resolution-agnostic by design. The method adds minimal prediction overhead, can be applied post-hoc without retraining the network, and scales to large models and datasets. We evaluate these aspects in a case study on Fourier neural operators.
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