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。近年来，诸如神经算子等求解偏微分方程的深度学习方法，在直接通过观测数据学习和预测复杂物理系统响应方面开始发挥重要作用。由于此类场景中数据采集通常具有挑战性且成本高昂，因此将现有知识迁移至新的、未见物理系统的需求显得尤为迫切。本文提出了一种适用于神经算子的新型元学习方法，可视为在具有可变参数场的未知偏微分方程之间迁移解算子的知识。该方法是一个可证明为通用解算器的多元偏微分方程求解方案，其关键理论发现是：与现有元学习方法中典型的最后一层迁移不同，底层参数场可以被神经算子模型的第一层捕获。作为应用，我们在基于偏微分方程的数据集和真实材料建模问题上验证了所提方法的有效性，表明该方法能够处理复杂非线性物理响应学习任务，同时显著提升未见任务中的采样效率。