Investigating Guiding Information for Adaptive Collocation Point Sampling in PINNs

TL;DR

It is shown how a number of important metrics can have an impact in improving the quality of the results obtained when using a fixed number of residual evaluations, and is illustrated through the use of two benchmark test problems.

Abstract

Physics-informed neural networks (PINNs) provide a means of obtaining approximate solutions of partial differential equations and systems through the minimisation of an objective function which includes the evaluation of a residual function at a set of collocation points within the domain. The quality of a PINNs solution depends upon numerous parameters, including the number and distribution of these collocation points. In this paper we consider a number of strategies for selecting these points and investigate their impact on the overall accuracy of the method. In particular, we suggest that no single approach is likely to be "optimal" but we show how a number of important metrics can have an impact in improving the quality of the results obtained when using a fixed number of residual evaluations. We illustrate these approaches through the use of two benchmark test problems: Burgers' equation and the Allen-Cahn equation.

Figures (8)

• Figure 1: $Y(\mathcal{X})=$ PDE Residual shown at every point in $\mathcal{X}$ (left); and the corresponding points selected for $\mathcal{T}$ (right).
• Figure 2: Burgers' Equation contour plot of example solution with $\nu = \frac{1}{100\pi}$
• Figure 3: Error against number of resamples solving Burgers' Equation with described default parameters $v=\frac{1}{100\pi}; N=2000$
• Figure 4: Error against number of collocation points for different resampling methods solving Burgers' Equation with described default parameters $v=\frac{1}{100\pi};$ 100 resamples
• Figure 5: Error versus number of resamples. Initial conditions 2 and 3 are combinations of randomly weighted sin curves of different frequency; with 2 being made up of three low frequency curves and 3 being a higher frequency combination of four curves. Initial conditions 4 and 5 are $sin(2\pi)$ and $1.5sin(\pi)$ respectively.