PICL：用于偏微分方程的物理信息对比学习
PICL: Physics Informed Contrastive Learning for Partial Differential Equations
AI总结：
本研究提出基于广义对比损失的物理信息对比预训练框架PICL，通过控制方程系数和物理信息驱动的系统演化提升神经算子在多方程上的泛化能力，显著提高了Fourier Neural Operator在多类偏微分方程任务中的精度。
摘要：
神经算子近期作为偏微分方程（PDE）代理模型日益流行。学习解泛函而非函数已被证明是计算复杂PDE快速、精确解的强大方法。尽管已有大量工作评估神经算子在多种代理建模任务上的性能，但这些工作通常每次仅评估单个方程。在本研究中，我们开发了一种利用广义对比损失的全新对比预训练框架，该框架提升了神经算子在多个控制方程上的泛化能力。控制方程系数被用于衡量系统间的真实相似度。物理信息驱动的系统演化与潜空间模型输出的组合被锚定到输入数据中，并用于我们的距离函数。我们发现，物理信息对比预训练提升了Fourier Neural Operator在1D和2D热方程、Burgers方程及线性平流方程的固定未来与自回归展开任务中的精度。
英文摘要：
Neural operators have recently grown in popularity as Partial Differential Equation (PDE) 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 accuracy for the Fourier Neural Operator in fixed-future and autoregressive rollout tasks for the 1D and 2D Heat, Burgers', and linear advection equations.
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