Neural operators have recently grown in popularity as Partial Differential Equation (PDEs) surrogate models. Learning solution functionals, rather than functions, has proven to be a powerful approach to calculate fast, accurate solutions to complex PDEs. While much work has been done evaluating neural operator performance on a wide variety of surrogate modeling tasks, these works normally evaluate performance on a single equation at a time. In this work, we develop a novel contrastive pretraining framework utilizing Generalized Contrastive Loss that improves neural operator generalization across multiple governing equations simultaneously. Governing equation coefficients are used to measure ground-truth similarity between systems. A combination of physics-informed system evolution and latent-space model output are anchored to input data and used in our distance function. We find that physics-informed contrastive pretraining improves both accuracy and generalization for the Fourier Neural Operator in fixed-future task, with comparable performance on the autoregressive rollout, and superresolution tasks for the 1D Heat, Burgers', and linear advection equations.

翻译：近年来，神经算子作为偏微分方程（PDE）的替代模型日益受到关注。学习解泛函而非函数本身已被证明是快速精确求解复杂PDE的有效方法。尽管已有大量研究评估了神经算子在各类替代建模任务中的性能，但这些工作通常针对单一方程进行性能评估。本文提出了一种利用广义对比损失的新型对比预训练框架，可同时提升神经算子在多个控制方程间的泛化能力。我们使用控制方程系数来度量系统间的真实相似性，并将物理信息驱动的系统演化与潜在空间模型输出相结合，以输入数据为锚点构建距离函数。实验表明，物理信息对比预训练能够显著提升傅里叶神经算子在固定未来预测任务中的准确性与泛化能力，同时在自回归推演和一维热方程、伯格斯方程及线性平流方程的超分辨率任务中均展现出可比的性能表现。