论文标题
可调的复杂性基准，用于评估耦合的普通微分方程的物理信息神经网络
Tunable Complexity Benchmarks for Evaluating Physics-Informed Neural Networks on Coupled Ordinary Differential Equations
论文作者
论文摘要
在这项工作中，我们评估了物理知识神经网络（PINN）解决日益复杂的普通微分方程（ODE）的能力。我们专注于一对基准：离散的偏微分方程和谐波振荡器，每个方程都有一个可调参数来控制其复杂性。即使改变了网络体系结构并应用了一种说明“困难”培训区域的最先进的培训方法，我们表明，Pinns最终无法为这些基准的正确解决方案提供正确的解决方案，因为它们的复杂性和时间域的大小和时间域的大小 - 增加了。我们确定了可能是这种情况的几个原因，包括网络容量不足，ODE的条件差和高局部曲率，这是由Pinn损失的Laplacian衡量的。
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.
