Solving Parameterized PDEs with Low-rank Structured Physics-Informed Neural Networks

In various engineering and applied science applications, repetitive numerical sim- ulations 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 lightweight low-rank PINNs containing only hundreds of model parameters and an associated hypernetwork-based meta-learning algorithm, which allow efficient solution approximations for varying PDE input parameters. Moreover, we show that the proposed method is effective in overcoming a challenging issue, known as “failure modes” of PINNs.

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https://klee44.github.io/