Gradient-based meta-learning methods have primarily been applied to classical machine learning tasks such as image classification. Recently, PDE-solving deep learning methods, such as neural operators, are starting to make an important impact on learning and predicting the response of a complex physical system directly from observational data. Since the data acquisition in this context is commonly challenging and costly, the call of utilization and transfer of existing knowledge to new and unseen physical systems is even more acute. Herein, we propose a novel meta-learning approach for neural operators, which can be seen as transferring the knowledge of solution operators between governing (unknown) PDEs with varying parameter fields. Our approach is a provably universal solution operator for multiple PDE solving tasks, with a key theoretical observation that underlying parameter fields can be captured in the first layer of neural operator models, in contrast to typical final-layer transfer in existing meta-learning methods. As applications, we demonstrate the efficacy of our proposed approach on PDE-based datasets and a real-world material modeling problem, illustrating that our method can handle complex and nonlinear physical response learning tasks while greatly improving the sampling efficiency in unseen tasks.

翻译：基于梯度的元学习方法主要应用于图像分类等经典机器学习任务。近期，求解偏微分方程的深度学习方法（如神经算子）开始对直接从观测数据学习并预测复杂物理系统响应产生重要影响。由于该场景中数据采集通常具有挑战性且成本高昂，将现有知识迁移至新的未见物理系统的需求愈发迫切。本文提出一种面向神经算子的新型元学习方法，该方法可视为在参数场变化但控制方程未知的偏微分方程间迁移解算子知识。我们的方法是一个可证明通用的多PDE求解任务统一解算子，其关键理论发现是：底层参数场可被神经算子模型的第一层捕获——这与现有元学习方法中典型的末层迁移形成对比。在应用层面，我们通过在基于偏微分方程的数据集和真实材料建模问题上验证所提方法的有效性，表明该方法能处理复杂非线性物理响应学习任务，同时显著提升未知任务的采样效率。