MetaNO: How to Transfer Your Knowledge on Learning Hidden Physics.
MetaNO: How to Transfer Your Knowledge on Learning Hidden Physics.
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MetaNO：如何转移您学习隐藏物理的知识。
DOI：
10.1016/j.cma.2023.116280
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发表时间：
2023
影响因子：
7.2
通讯作者：
Yu,Yue
中科院分区：
文献类型：
--
作者：
Zhang,Lu;You,Huaiqian;Gao,Tian;Yu,Mo;Lee,Chung-Hao;Yu,Yue
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。最近，偏微分方程（PDE）求解深度学习方法（例如神经算子）开始对直接从观测数据学习和预测复杂物理系统的响应产生重要影响。以材料建模问题为例，神经算子方法学习从加载场到相应材料响应场的代理映射，这可以看作是学习隐藏偏微分方程的解算子。每个材料样本的微观结构和机械参数对应于该隐藏偏微分方程中的（可能是异质的）参数场。由于实验测量技术的限制，每个材料样本的数据采集通常具有挑战性，而且成本也可能很高。这一事实要求利用现有知识并将其转移到新的和未见过的材料样本中，这对应于对具有不同参数场的隐偏微分方程的解算子进行采样有效学习。在这里，我们提出了一种新颖的神经算子元学习方法，可以看作是在具有不同参数场的控制（未知）偏微分方程之间转移解算子的知识。我们的方法是一种可证明通用的解决算子，适用于多个偏微分方程求解任务，其关键的理论观察是，可以在神经算子模型的第一层中捕获底层参数字段，这与现有元学习方法中典型的最后层迁移形成鲜明对比。作为应用，我们展示了我们提出的方法在基于偏微分方程的数据集和现实世界材料建模问题上的有效性，说明我们的方法可以处理复杂和非线性的物理响应学习任务，同时大大提高看不见的任务中的采样效率。
Gradient-based meta-learning methods have primarily been applied to classical machine learning tasks such as image classification. Recently, partial differential equation (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. Taking the material modeling problems for example, the neural operator approach learns a surrogate mapping from the loading field to the corresponding material response field, which can be seen as learning the solution operator of a hidden PDE. The microstructure and mechanical parameters of each material specimen correspond to the (possibly heterogeneous) parameter field in this hidden PDE. Due to the limitation on experimental measurement techniques, the data acquisition for each material specimen is commonly challenging and may also be costly. This fact calls for the utilization and transfer of existing knowledge to new and unseen material specimens, which corresponds to sampling efficient learning of the solution operator of a hidden PDE with a different parameter field. Herein, we propose a novel meta-learning approach for neural operators that 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.