  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.
