Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Machine Learning for Complex Systems Dynamics: Detecting Bifurcations in Dynamical Systems with Deep Neural Networks

Swadesh Pal, Roderick Melnik

TL;DR

A novel machine learning approach based on deep neural networks called equilibrium-informed neural networks (EINNs) to identify critical thresholds associated with catastrophic regime shifts is proposed, showing that EINNs can recover the parameter regions associated with impending transitions.

Abstract

Critical transitions are the abrupt shifts between qualitatively different states of a system, and they are crucial to understanding tipping points in complex dynamical systems across ecology, climate science, and biology. Detecting these shifts typically involves extensive forward simulations or bifurcation analyses, which are often computationally intensive and limited by parameter sampling. In this study, we propose a novel machine learning approach based on deep neural networks (DNNs) called equilibrium-informed neural networks (EINNs) to identify critical thresholds associated with catastrophic regime shifts. Rather than fixing parameters and searching for solutions, the EINN method reverses this process by using candidate equilibrium states as inputs and training a DNN to infer the corresponding system parameters that satisfy the equilibrium condition. By analyzing the learned parameter landscape and observing abrupt changes in the feasibility or continuity of equilibrium mappings, critical thresholds can be effectively detected. We demonstrate this capability on nonlinear systems exhibiting saddle-node bifurcations and multi-stability, showing that EINNs can recover the parameter regions associated with impending transitions. This method provides a flexible alternative to traditional techniques, offering new insights into the early detection and structure of critical shifts in high-dimensional and nonlinear systems.

Machine Learning for Complex Systems Dynamics: Detecting Bifurcations in Dynamical Systems with Deep Neural Networks

TL;DR

A novel machine learning approach based on deep neural networks called equilibrium-informed neural networks (EINNs) to identify critical thresholds associated with catastrophic regime shifts is proposed, showing that EINNs can recover the parameter regions associated with impending transitions.

Abstract

Critical transitions are the abrupt shifts between qualitatively different states of a system, and they are crucial to understanding tipping points in complex dynamical systems across ecology, climate science, and biology. Detecting these shifts typically involves extensive forward simulations or bifurcation analyses, which are often computationally intensive and limited by parameter sampling. In this study, we propose a novel machine learning approach based on deep neural networks (DNNs) called equilibrium-informed neural networks (EINNs) to identify critical thresholds associated with catastrophic regime shifts. Rather than fixing parameters and searching for solutions, the EINN method reverses this process by using candidate equilibrium states as inputs and training a DNN to infer the corresponding system parameters that satisfy the equilibrium condition. By analyzing the learned parameter landscape and observing abrupt changes in the feasibility or continuity of equilibrium mappings, critical thresholds can be effectively detected. We demonstrate this capability on nonlinear systems exhibiting saddle-node bifurcations and multi-stability, showing that EINNs can recover the parameter regions associated with impending transitions. This method provides a flexible alternative to traditional techniques, offering new insights into the early detection and structure of critical shifts in high-dimensional and nonlinear systems.
Paper Structure (7 sections, 17 equations, 5 figures)

This paper contains 7 sections, 17 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. EINNs Approach and its Applications
  4. Single-equation dynamical models
  5. Models based on two coupled equations
  6. Extended systems of equations
  7. Conclusions

Figures (5)

  • Figure 1: (Color online) Schematic diagram of the DNN-based inversion approach. The input and output layers are denoted by $I$ and $O$, respectively, while $H_{1}$, $H_{2}$, and $H_{3}$ represent the hidden layers of the DNN. CTCS refers to the critical thresholds for catastrophic shifts.
  • Figure 2: (Color online) (a) Plots of the functions (\ref{['F1']}) for two different values of $r$: $r_{1} = 1.7869$ and $r_{2} = 2.6049$. (b) Bifurcation diagram of the system (\ref{['SSME']}) with respect to the parameter $r$.
  • Figure 3: (Color online) Prediction of potential catastrophic shifts in the system (\ref{['SSME']}) using the EINNs approach. (a) Bifurcation diagram as predicted by EINNs. (b) Reoriented version of (a) with axes interchanged, overlaid with the linear stability behaviour of the equilibrium points to highlight stability transitions.
  • Figure 4: (Color online) (a) Bifurcation diagram and critical transition thresholds ($\beta_{1}$ and $\beta_{2}$) of system (\ref{['SSME1']}) derived using the EINNs method, and (b) plots of the curves from equation (\ref{['NE2']}) corresponding to two selected critical threshold values.
  • Figure 5: (Color online) Comparison of equilibrium point estimates for system (\ref{['ABCI1']}) obtained via (a) traditional and (b) EINNs approaches.