In this work, we assess the ability of physics-informed neural networks (PINNs) to solve increasingly-complex coupled ordinary differential equations (ODEs). We focus on a pair of benchmarks: discretized partial differential equations and harmonic oscillators, each of which has a tunable parameter that controls its complexity. Even by varying network architecture and applying a state-of-the-art training method that accounts for "difficult" training regions, we show that PINNs eventually fail to produce correct solutions to these benchmarks as their complexity -- the number of equations and the size of time domain -- increases. We identify several reasons why this may be the case, including insufficient network capacity, poor conditioning of the ODEs, and high local curvature, as measured by the Laplacian of the PINN loss.
翻译:在这项工作中,我们评估了物理学知情神经网络(PINNs)解决日益复杂、相互交织的普通差异方程式(ODEs)的能力。我们侧重于一对基准:分离的局部偏差方程式和调和振荡器,每个均有一个可以控制其复杂性的可捕量参数。即使网络结构各异,并且采用了计算“困难”培训区域的最先进的培训方法,我们也表明,PINNs最终未能对这些基准的复杂性 -- -- 方程式的数量和时间范围 -- -- 提出正确的解决方案。我们找出了造成这种情况的几个原因,包括网络能力不足、模型的缺陷以及按PINN损失的拉普拉克测量的当地曲线高。