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.

翻译：科学与工程中的许多重要问题需要求解所谓的参数偏微分方程（PDEs），即具有不同物理参数、边界条件、计算域形状等的PDEs。典型的降阶建模技术通过将PDEs投影到离线阶段构建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计算的解快照，并且由于线性假设的局限性，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能力，我们提出元自动解码器（Meta-Auto-Decoder, MAD）来构建非线性试验流形，其最佳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，可以实现对新（可能异构的）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，具有比其他基于深度学习的方法更快的收敛速度。