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：我们提出一种新颖的损失函数形式，用于从物理规律（通常由偏微分方程描述）中高效学习复杂动力学，通过物理信息神经网络实现。实验中，现有版本的PINNs在许多问题中学习效果较差，尤其在复杂几何区域中，因在临近边界区域建立适当采样策略变得愈发困难。若采样过密，局部梯度行为过于复杂而无法被PINNs充分建模时，会阻碍训练收敛；反之，若采样过稀疏，现有PINNs易在临近边界区域过拟合，导致错误解。为避免此类问题，我们提出一种新的边界连接损失函数，为PINNs的边界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，从而以数量级更稀疏的训练样本实现全问题域的快速物理信息学习。在标准L2范数度量下，该方法比现有方法误差降低数个数量级，同时显著减少训练样本与迭代次数。提出的LSA-PINN方法对网络可微性无任何要求，我们分别在当前PINN文献常用的多层感知机与卷积神经网络版本上展示了其优势与易实现性。