We present a novel loss formulation for efficient learning of complex dynamics from governing physics, typically described by partial differential equations (PDEs), using physics-informed neural networks (PINNs). In our experiments, existing versions of PINNs are seen to learn poorly in many problems, especially for complex geometries, as it becomes increasingly difficult to establish appropriate sampling strategy at the near boundary region. Overly dense sampling can adversely impede training convergence if the local gradient behaviors are too complex to be adequately modelled by PINNs. On the other hand, if the samples are too sparse, existing PINNs tend to overfit the near boundary region, leading to incorrect solution. To prevent such issues, we propose a new Boundary Connectivity (BCXN) loss function which provides linear local structure approximation (LSA) to the gradient behaviors at the boundary for PINN. Our BCXN-loss implicitly imposes local structure during training, thus facilitating fast physics-informed learning across entire problem domains with order of magnitude sparser training samples. This LSA-PINN method shows a few orders of magnitude smaller errors than existing methods in terms of the standard L2-norm metric, while using dramatically fewer training samples and iterations. Our proposed LSA-PINN does not pose any requirement on the differentiable property of the networks, and we demonstrate its benefits and ease of implementation on both multi-layer perceptron and convolutional neural network versions as commonly used in current PINN literature.

翻译：我们提出了一种新的损失函数形式，用于通过物理信息神经网络高效学习由偏微分方程支配的复杂动力学过程。实验发现，现有版本的PINN在许多问题中学习效果较差，尤其是在复杂几何区域上，因为在边界附近建立合适的采样策略变得越来越困难。如果局部梯度行为过于复杂而无法被PINN充分建模，过度密集的采样会阻碍训练收敛；另一方面，若采样过于稀疏，现有PINN又容易在边界附近过拟合，导致错误解。为解决这些问题，我们提出了一种新的边界连接损失函数，该函数对PINN在边界处的梯度行为提供线性局部结构近似。我们的BCXN损失在训练过程中隐式施加局部结构约束，从而能够以数量级更稀疏的训练样本实现整个问题域上的快速物理信息学习。与现有方法相比，这种LSA-PINN方法在使用显著更少的训练样本和迭代次数的同时，其标准L2范数误差降低了数个数量级。我们提出的LSA-PINN不对网络的可微性提出任何要求，并在当前PINN文献中常用的多层感知机版本和卷积神经网络版本上展示了其优势与实现简便性。