A novel comparison is presented of the effect of optimiser choice on the accuracy of physics-informed neural networks (PINNs). To give insight into why some optimisers are better, a new approach is proposed that tracks the training trajectory curvature and can be evaluated on the fly at a low computational cost. The linear advection equation is studied for several advective velocities, and we show that the optimiser choice substantially impacts PINNs model performance and accuracy. Furthermore, using the curvature measure, we found a negative correlation between the convergence error and the curvature in the optimiser local reference frame. It is concluded that, in this case, larger local curvature values result in better solutions. Consequently, optimisation of PINNs is made more difficult as minima are in highly curved regions.

翻译：本文针对优化器选择对物理学指导神经网络（PINNs）精度的影响提出了新颖的比较研究。为揭示某些优化器表现更优的内在机理，我们提出了一种可实时追踪训练轨迹曲率且计算成本低廉的新方法。通过研究不同对流速度下的线性平流方程，发现优化器选择显著影响PINNs模型的性能与精度。进一步利用曲率度量发现，在优化器局部参考系中，收敛误差与曲率呈负相关。结论表明：在本研究案例中，更大的局部曲率值对应更优的求解效果。由此揭示，由于极小值点位于高曲率区域，PINNs的优化过程变得更加困难。