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:以渐进为基础的元学习方法主要应用于古典机器学习任务,例如图像分类。最近,像神经操作员这样的解决环境的深层次学习方法开始对直接从观测数据中学习和预测复杂物理系统的反应产生重大影响。由于这方面的数据获取通常具有挑战性和成本,因此呼吁利用现有知识并将其转让给新的和看不见的物理系统甚至更为尖锐。在这里,我们建议对神经操作员采用新的元学习方法,这可以被看作是在管理(未知的)具有不同参数的 PDE 之间转让解决方案操作员的知识。我们的方法是对于多种PDE 解决任务的一个可实现的通用解决方案操作员,其关键的理论观察是,基本参数领域可以在神经操作员模型的第一层中捕捉,而与现有元学习方法的典型最终转移相比,这更是典型的。作为应用,我们展示了我们提议的PDE 数据集和现实世界材料建模问题的方法的有效性,说明我们的方法可以处理复杂和非线性实际反应学习任务,同时大大改进了无形任务取样的效率。