Neural Symbolic Regression of Complex Network Dynamics

TL;DR

Physically Inspired Neural Dynamics Symbolic Regression (PI-NDSR), a method based on neural networks and genetic programming to automatically learn the symbolic expression of dynamics, which outperforms the existing method in terms of both recovery probability and error.

Abstract

Complex networks describe important structures in nature and society, composed of nodes and the edges that connect them. The evolution of these networks is typically described by dynamics, which are labor-intensive and require expert knowledge to derive. However, because the complex network involves noisy observations from multiple trajectories of nodes, existing symbolic regression methods are either not applicable or ineffective on its dynamics. In this paper, we propose Physically Inspired Neural Dynamics Symbolic Regression (PI-NDSR), a method based on neural networks and genetic programming to automatically learn the symbolic expression of dynamics. Our method consists of two key components: a Physically Inspired Neural Dynamics (PIND) to augment and denoise trajectories through observed trajectory interpolation; and a coordinated genetic search algorithm to derive symbolic expressions. This algorithm leverages references of node dynamics and edge dynamics from neural dynamics to avoid overfitted expressions in symbolic space. We evaluate our method on synthetic datasets generated by various dynamics and real datasets on disease spreading. The results demonstrate that PI-NDSR outperforms the existing method in terms of both recovery probability and error.

Figures (5)

• Figure 1: Our method is designed for symbolic regression of complex network dynamics. The proposed method interpolates and denoises complex network observations with neural dynamics to avoid inaccurate estimation of time derivatives and applies a coordinated genetic search algorithm to derive the symbolic expressions of complex network dynamics.
• Figure 2: Visualizing the predicted number of newly reported cases in two regions using symbolic expressions from TP-SINDy and PI-NDSR.
• Figure 3: Evaluation of robustness. The shaded areas correspond to 95% confidence interval. (a) and (b) show the recovery probability and MSE when adding noise to the observations. (c) and (d) show the recovery probability and MSE when increasing the time interval between observations.
• Figure 4: Visualization of neural dynamics estimation for LV dynamics in the BA graph. For node dynamics, we show the values at intervals of 0.1 between 0 and 1. For edge dynamics, we use a heatmap to show the values in (b) and (c).
• Figure 5: Visualization of interpolated and denoised observations and the estimated time derivative. (a) The interpolated observations are very close to the ground truth when noise exists. (b) The estimated time derivative is inaccurate with noisy observation. (c) The interpolated observations are close to the ground truth with a large time interval (0.1). (d) The estimated time derivative is inaccurate when the sample time interval is large.