论文标题
Meta-MGNET：用于求解参数化的偏微分方程的元多机网络
Meta-MgNet: Meta Multigrid Networks for Solving Parameterized Partial Differential Equations
论文作者
论文摘要
本文研究了具有深度学习（DL）的参数化部分微分方程（P-PDE）的数值解。 P-PDE在许多重要的应用领域中出现，使用传统数值方案的计算成本可以过高，尤其是当参数落入特定范围时，并且必须以高精度来解决基础PDE。最近，用DL解决PDE已成为一个新兴领域。现有作品展示了基于DL的方法在加速PDE的数值解决方案方面的巨大潜力。但是，关于P-PDE的DL方法的研究仍然有限。如果我们直接将现有的监督学习模型应用于P-PDE，则在参数更改时，必须不断进行微调或重新训练。这大大限制了这些模型在实践中的适用性和实用性。为了解决此问题，我们提出了一种基于元学习的方法，该方法可以有效地求解具有广泛参数的P-PDE，而无需重新培训。我们的主要观察结果是将求解器视为一组学习任务的p-PDE的求解器。然后，可以将求解器培训为具有不同参数的P-PDE的求解器被视为一个多任务学习问题，元学习是最有效的方法之一。这种新观点可以应用于许多现有的PDE求解器。例如，我们采用Multigrid Network（MGNET）作为基本求解器。为了实现多任务学习，我们在MGNET中介绍了一个新的HyperNetwork，称为Meta-NN，并将整个网络称为Meta-MGNET。 Meta-NN将基础P-PDE的差分运算符和右侧作为输入，并为MGNET生成适当的SmoOther，这可能会严重影响收敛速度。最后，广泛的数值实验表明，荟萃元比MG方法和MGNET更有效地求解P-PDE。
This paper studies numerical solutions for parameterized partial differential equations (P-PDEs) with deep learning (DL). P-PDEs arise in many important application areas and the computational cost using traditional numerical schemes can be exorbitant, especially when the parameters fall into a particular range and the underlying PDE is required to be solved with high accuracy. Recently, solving PDEs with DL has become an emerging field. Existing works demonstrate great potentials of the DL based approach in speeding up numerical solutions of PDEs. However, there is still limited research on the DL approach for P-PDEs. If we directly apply existing supervised learning models to P-PDEs, the models need to be constantly fine-tuned or retrained when the parameters change. This drastically limits the applicability and utility of these models in practice. To resolve this issue, we propose a meta-learning-based method that can efficiently solve P-PDEs with a wide range of parameters without retraining. Our key observation is to regard training a solver for the P-PDE with a given set of parameters as a learning task. Then, training a solver for the P-PDEs with varied parameters can be viewed as a multi-task learning problem, to which meta-learning is one of the most effective approaches. This new perspective can be applied to many existing PDE solvers. As an example, we adopt the Multigrid Network (MgNet) as the base solver. To achieve multi-task learning, we introduce a new hypernetwork, called Meta-NN, in MgNet and refer to the entire network as the Meta-MgNet. Meta-NN takes the differential operators and the right-hand-side of the underlying P-PDEs as inputs and generates appropriate smoothers for MgNet which can significantly affect the convergent speed. Finally, extensive numerical experiments demonstrate that Meta-MgNet is more efficient in solving P-PDEs than the MG methods and MgNet.
