In various engineering and applied science applications, repetitive numerical
simulations of partial differential equations (PDEs) for varying input
parameters are often required (e.g., aircraft shape optimization over many
design parameters) and solvers are required to perform rapid execution. In this
study, we suggest a path that potentially opens up a possibility for
physics-informed neural networks (PINNs), emerging deep-learning-based solvers,
to be considered as one such solver. Although PINNs have pioneered a proper
integration of deep-learning and scientific computing, they require repetitive
time-consuming training of neural networks, which is not suitable for
many-query scenarios. To address this issue, we propose a lightweight low-rank
PINNs containing only hundreds of model parameters and an associated
hypernetwork-based meta-learning algorithm, which allows efficient
approximation of solutions of PDEs for varying ranges of PDE input parameters.
Moreover, we show that the proposed method is effective in overcoming a
challenging issue, known as "failure modes" of PINNs.

通过提出一种轻量级低秩 PINN 和相关的超网络元学习算法，本研究有效地解决了在各种工程和应用科学应用中需要对多个输入参数进行重复数值模拟的问题，并展示了该方法在克服 PINN 的 “失败模式” 方面的有效性。