Title: ELM-FBPINN: efficient finite-basis physics-informed neural networks Show Chinese title

Title: ELM-FBPINN：高效有限基物理信息神经网络

Abstract: Physics Informed Neural Networks (PINNs) offer several advantages when compared to traditional numerical methods for solving PDEs, such as being a mesh-free approach and being easily extendable to solving inverse problems. One promising approach for allowing PINNs to scale to multi-scale problems is to combine them with domain decomposition; for example, finite basis physics-informed neural networks (FBPINNs) replace the global PINN network with many localised networks which are summed together to approximate the solution. In this work, we significantly accelerate the training of FBPINNs by linearising their underlying optimisation problem. We achieve this by employing extreme learning machines (ELMs) as their subdomain networks and showing that this turns the FBPINN optimisation problem into one of solving a linear system or least-squares problem. We test our workflow in a preliminary fashion by using it to solve an illustrative 1D problem. Show Chinese abstract

Abstract: 物理信息神经网络（PINNs）在与传统数值方法求解偏微分方程（PDEs）相比时具有多种优势，例如无需网格的处理方式，并且易于扩展以求解逆问题。 让PINNs能够扩展到多尺度问题的一种有前景的方法是将它们与区域分解相结合；例如，有限基物理信息神经网络（FBPINNs）用许多局部网络代替全局PINN网络，这些网络相加在一起以近似求解。 在本工作中，我们通过线性化其底层优化问题，显著加速了FBPINNs的训练。 我们通过使用极端学习机（ELMs）作为子域网络，并证明这将FBPINN的优化问题转化为求解线性系统或最小二乘问题。 我们通过使用该工作流来求解一个示例性的1D问题，初步测试了我们的方法。