Physics Informed Neural Networks (PINNs) have gained immense popularity as an alternate method for numerically solving PDEs. Despite their empirical success we are still building an understanding of the convergence properties of training on such constraints with gradient descent. It is known that, in the absence of an explicit inductive bias, Neural Networks can struggle to learn or approximate even simple and well known functions in a sample efficient manner. Thus the numerical approximation induced from few collocation points may not generalize over the entire domain. Meanwhile, a symbolic form can exhibit good generalization, with interpretability as a useful byproduct. However, symbolic approximations can struggle to simultaneously be concise and accurate. Therefore in this work we explore a NeuroSymbolic approach to approximate the solution for PDEs. We observe that our approach work for several simple cases. We illustrate the efficacy of our approach on Navier Stokes: Kovasznay flow where there are multiple physical quantities of interest governed with non-linear coupled PDE system. Domain splitting is now becoming a popular trick to help PINNs approximate complex functions. We observe that a NeuroSymbolic approach can help such complex functions as well. We demonstrate Domain-splitting assisted NeuroSymbolic approach on a temporally varying two-dimensional Burger's equation. Finally we consider the scenario where PINNs have to be solved for parameterized PDEs, for changing Initial-Boundary Conditions and changes in the coefficient of the PDEs. Hypernetworks have shown to hold promise to overcome these challenges. We show that one can design Hyper-NeuroSymbolic Networks which can combine the benefits of speed and increased accuracy. We observe that that the NeuroSymbolic approximations are consistently 1-2 order of magnitude better than just the neural or symbolic approximations.
翻译:物理智能神经网络(PINNs)作为数字解析 PDEs 的替代方法,获得了巨大的受欢迎程度。 尽管它们取得了经验性的成功, 我们仍在建立对关于这些限制的训练的趋同特性的理解。 众所周知, 在缺乏明确的感官偏差的情况下, 神经网络可以以抽样有效的方式努力学习或近似简单和众所周知的功能。 因此, 少数合用点引出的数值近差可能无法覆盖整个域。 同时, 一个象征性的形式可以显示良好的概括化, 并且可以作为有用的副产品加以解释。 然而, 象征性近似可以同时努力做到简洁和准确。 因此, 在此工作中, 我们探索一种NeuroSymbic 方法来接近 PDEs 的解决方案。 我们观察我们对于几个简单案例的处理办法。 我们用Navier Stokes: Kovasnay 流来说明我们的方法的功效, 与非线性连接的 PDESDI 系统有多种实际的调控量。 而 Dome dialdial dial 方法现在可以帮助 PINN 接近复杂功能。 我们观察了 NuralSilalSilal 的内的变变的变的内变的内变的变。