We develop a class of data-driven generative models that approximate the solution operator for parameter-dependent partial differential equations (PDE). We propose a novel probabilistic formulation of the operator learning problem based on recently developed generative denoising diffusion probabilistic models (DDPM) in order to learn the input-to-output mapping between problem parameters and solutions of the PDE. To achieve this goal we modify DDPM to supervised learning in which the solution operator for the PDE is represented by a class of conditional distributions. The probabilistic formulation combined with DDPM allows for an automatic quantification of confidence intervals for the learned solutions. Furthermore, the framework is directly applicable for learning from a noisy data set. We compare computational performance of the developed method with the Fourier Network Operators (FNO). Our results show that our method achieves comparable accuracy and recovers the noise magnitude when applied to data sets with outputs corrupted by additive noise.

翻译：我们开发了一类数据驱动的生成模型，用于近似参数依赖性偏微分方程（PDE）的解算子。基于近期发展的生成式去噪扩散概率模型（DDPM），我们提出了一种算子学习问题的概率化新框架，以学习问题参数与PDE解之间的输入-输出映射。为实现此目标，我们将DDPM改进为监督学习形式，其中PDE的解算子通过一类条件分布表示。这种概率化框架结合DDPM能够自动量化学习解的信赖区间。此外，该框架可直接应用于含噪数据集的训练。我们将所开发方法的计算性能与傅里叶神经算子（FNO）进行了比较。结果表明，当应用于输出数据被加性噪声污染的数据集时，本方法能达到相当的计算精度，并能有效恢复噪声量级。