元自动解码器：一种基于元学习的降阶模型用于求解参数化偏微分方程
Meta-Auto-Decoder: A Meta-Learning Based Reduced Order Model for Solving Parametric Partial Differential Equations
AI总结：
提出基于元学习的非线性降阶模型MAD，以无网格无监督方式学习试验流形，实现参数化PDE快速求解，兼顾精度与收敛速度。
摘要：
科学和工程中的许多重要问题需要求解所谓的参数化偏微分方程（PDE），即具有不同物理参数、边界条件、计算域形状等的PDE。典型的降阶建模技术通过将参数化PDE投影到离线阶段构造的线性试验流形上来加速求解。这些方法通常需要预定义的网格以及一系列预先计算的解快照，并且由于线性ansatz的局限性，可能难以在效率和精度之间取得平衡。利用神经网络的非线性表示，我们提出了元自动解码器（MAD）来构造非线性试验流形，其最佳可能性能理论上由解码器宽度衡量。基于元学习概念，试验流形可以在预训练阶段以无网格和无监督的方式学习。通过在该试验流形上进行搜索，并可选地同时微调试验流形，实现了对新（可能异构）PDE参数的快速适应。大量数值实验表明，MAD方法在保持精度的同时，比其他基于深度学习的方法具有更快的收敛速度。
英文摘要：
Many important problems in science and engineering require solving the so-called parametric partial differential equations (PDEs), i.e., PDEs with different physical parameters, boundary conditions, shapes of computational domains, etc. Typical reduced order modeling techniques accelarate solution of the parametric PDEs by projecting them onto a linear trial manifold constructed in the offline stage. These methods often need a predefined mesh as well as a series of precomputed solution snapshots, andmay struggle to balance between efficiency and accuracy due to the limitation of the linear ansatz. Utilizing the nonlinear representation of neural networks, we propose Meta-Auto-Decoder (MAD) to construct a nonlinear trial manifold, whose best possible performance is measured theoretically by the decoder width. Based on the meta-learning concept, the trial manifold can be learned in a mesh-free and unsupervised way during the pre-training stage. Fast adaptation to new (possibly heterogeneous) PDE parameters is enabled by searching on this trial manifold, and optionally fine-tuning the trial manifold at the same time. Extensive numerical experiments show that the MAD method exhibits faster convergence speed without losing accuracy than other deep learning-based methods.
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