With the increases in computational power and advances in machine learning, data-driven learning-based methods have gained significant attention in solving PDEs. Physics-informed neural networks (PINNs) have recently emerged and succeeded in various forward and inverse PDE problems thanks to their excellent properties, such as flexibility, mesh-free solutions, and unsupervised training. However, their slower convergence speed and relatively inaccurate solutions often limit their broader applicability in many science and engineering domains. This paper proposes a new kind of data-driven PDEs solver, physics-informed cell representations (PIXEL), elegantly combining classical numerical methods and learning-based approaches. We adopt a grid structure from the numerical methods to improve accuracy and convergence speed and overcome the spectral bias presented in PINNs. Moreover, the proposed method enjoys the same benefits in PINNs, e.g., using the same optimization frameworks to solve both forward and inverse PDE problems and readily enforcing PDE constraints with modern automatic differentiation techniques. We provide experimental results on various challenging PDEs that the original PINNs have struggled with and show that PIXEL achieves fast convergence speed and high accuracy. Project page: https://namgyukang.github.io/PIXEL/

翻译：随着计算能力的提升和机器学习的发展，数据驱动的学习方法在求解偏微分方程（PDEs）方面引起了广泛关注。物理信息神经网络（PINNs）因其灵活性、无网格解和无需监督训练等优异特性，近年来在各类正问题和逆问题PDE中取得了成功。然而，其收敛速度较慢且解精度相对较低，这往往限制了其在众多科学与工程领域的广泛适用性。本文提出了一种新型数据驱动PDE求解器——基于物理信息的单元表示（PIXEL），它巧妙地将经典数值方法与学习方法相结合。我们采用数值方法中的网格结构来提高精度和收敛速度，并克服PINNs中存在的谱偏差问题。此外，所提方法享有与PINNs相同的优势，例如使用相同的优化框架来求解正问题和逆问题，并能借助现代自动微分技术便捷地施加PDE约束。我们在原始PINNs难以处理的各种具有挑战性的PDE上提供了实验结果，表明PIXEL实现了快速收敛速度和高精度。项目页面：https://namgyukang.github.io/PIXEL/