This paper empirically studies commonly observed training difficulties of Physics-Informed Neural Networks (PINNs) on dynamical systems. Our results indicate that fixed points which are inherent to these systems play a key role in the optimization of the in PINNs embedded physics loss function. We observe that the loss landscape exhibits local optima that are shaped by the presence of fixed points. We find that these local optima contribute to the complexity of the physics loss optimization which can explain common training difficulties and resulting nonphysical predictions. Under certain settings, e.g., initial conditions close to fixed points or long simulations times, we show that those optima can even become better than that of the desired solution.

翻译：本文通过实证研究，探讨了物理信息神经网络（PINNs）在动力系统建模中常见的训练难题。研究结果表明，动力系统固有的不动点在PINNs嵌入式物理损失函数的优化过程中发挥着关键作用。我们观察到，损失景观中存在受不动点影响的局部最优解。这些局部最优解增加了物理损失优化的复杂性，从而解释了常见的训练困难及由此产生的非物理预测结果。在特定条件下（如初始条件接近不动点或模拟时间较长），我们发现这些局部最优解甚至可能优于目标解对应的最优解。