Although Physics-Informed Neural Networks (PINNs) have been successfully applied in a wide variety of science and engineering fields, they can fail to accurately predict the underlying solution in slightly challenging convection-diffusion-reaction problems. In this paper, we investigate the reason of this failure from a domain distribution perspective, and identify that learning multi-scale fields simultaneously makes the network unable to advance its training and easily get stuck in poor local minima. We show that the widespread experience of sampling more collocation points in high-loss layer regions hardly help optimize and may even worsen the results. These findings motivate the development of a novel curriculum learning method that encourages neural networks to prioritize learning on easier non-layer regions while downplaying learning on harder layer regions. The proposed method helps PINNs automatically adjust the learning emphasis and thereby facilitate the optimization procedure. Numerical results on typical benchmark equations show that the proposed curriculum learning approach mitigates the failure modes of PINNs and can produce accurate results for very sharp boundary and interior layers. Our work reveals that for equations whose solutions have large scale differences, paying less attention to high-loss regions can be an effective strategy for learning them accurately.

翻译：尽管物理信息神经网络（PINNs）已在众多科学与工程领域得到成功应用，但在处理具有轻微挑战性的对流-扩散-反应问题时，其可能无法准确预测潜在解。本文从域分布视角出发，探究了这一失效的原因，并发现同时学习多尺度场会导致网络无法推进训练进程，且易陷入较差的局部极小值。我们证明，在高损失层区域采样更多配置点这一普遍做法，不仅难以优化，甚至可能使结果恶化。这些发现催生了一种新型课程学习方法——该方法鼓励神经网络优先学习较简单的非层区域，同时淡化对较难的层区域的学习。所提方法有助于PINNs自动调整学习重点，从而优化训练流程。典型基准方程的数值结果表明，该课程学习方法缓解了PINNs的失效模式，并能在极陡峭的边界层与内部层问题上产生精确结果。我们的研究揭示，对于解存在显著尺度差异的方程，减少对高损失区域的关注可成为准确学习该类方程的有效策略。