PINNACLE: PINN Adaptive ColLocation and Experimental points selection

本文介绍了物理信息神经网络（Physics-Informed Neural Networks，PINNs），它将偏微分方程（PDEs）作为软约束条件，使用复合损失函数进行训练，该函数包含多种训练点类型：在训练期间选择的不同类型的求解点，用于强制执行每个PDE和初始/边界条件，以及通常需要通过实验或模拟获得的实验点。使用这种损失函数训练PINNs具有挑战性，因为它通常需要选择大量不同类型的点，每个点都具有不同的训练动态。与过去专注于选择求解点或实验点的工作不同，本文介绍了PINN自适应求解点和实验点选择（PINNACLE）算法，它是第一个联合优化所有训练点类型选择的算法，同时在训练过程中自动调整求解点类型的比例。PINNACLE利用训练点类型之间的相互作用信息，这在以前没有考虑过，基于对PINN训练动态的神经切向核（Neural Tangent Kernel，NTK）分析。我们在理论上表明，PINNACLE使用的标准与PINN泛化误差有关，并在实证上证明，PINNACLE能够优于现有的前向、反向和转移学习问题的点选择方法。

PINNACLE aims to solve the challenge of selecting multiple types of training points with different training dynamics, including collocation points and experimental points, to train Physics-Informed Neural Networks (PINNs). The goal is to optimize the selection of all training point types while adjusting the proportion of collocation point types as training progresses.

The key idea of the paper is to introduce PINNACLE, the first algorithm that jointly optimizes the selection of all training point types while automatically adjusting the proportion of collocation point types. The algorithm uses information on the interaction among training point types based on an analysis of PINN training dynamics via the Neural Tangent Kernel (NTK). The criterion used by PINNACLE is related to the PINN generalization error.

The paper demonstrates that PINNACLE outperforms existing point selection methods for forward, inverse, and transfer learning problems. The algorithm is evaluated on various datasets, including the heat equation, Poisson equation, and Navier-Stokes equation. The authors also provide an open-source implementation of the algorithm. The paper suggests that the proposed method can be extended to other types of PDEs and can be used to improve the generalization of other physics-informed machine learning models.

Related work includes previous studies on PINNs and their applications in solving PDEs. Some relevant papers include 'Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations' and 'Deep learning for universal linear embeddings of nonlinear dynamics'.