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.

翻译：在各种工程与应用科学领域中，常需针对变化的输入参数对偏微分方程进行重复数值模拟（例如，针对多个设计参数的飞行器形状优化），且求解器需具备快速执行能力。本研究提出了一条潜在路径，使得新兴的基于深度学习的物理信息神经网络（PINN）有望成为此类求解器之一。尽管PINN开创了深度学习与科学计算的深度融合，但其需要重复进行耗时的神经网络训练，这并不适用于多查询场景。为解决此问题，我们提出了一种轻量级低秩PINN，仅含数百个模型参数，并配套提出了基于超网络的元学习算法，能够高效逼近不同偏微分方程输入参数范围内的解。此外，我们证明该方法能有效克服PINN中被称为"失效模式"的挑战性难题。