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Scientific machine learning in ecological systems: A study on the predator-prey dynamics

Ranabir Devgupta, Raj Abhijit Dandekar, Rajat Dandekar, Sreedath Panat

TL;DR

This work demonstrates that both Neural ODEs and UDEs can be effectively utilized for prediction and forecasting of the Lotka-Volterra system, and introduces the forecasting breakdown point: the time at which forecasting fails for both Neural ODEs and UDEs.

Abstract

In this study, we apply two pillars of Scientific Machine Learning: Neural Ordinary Differential Equations (Neural ODEs) and Universal Differential Equations (UDEs) to the Lotka Volterra Predator Prey Model, a fundamental ecological model describing the dynamic interactions between predator and prey populations. The Lotka-Volterra model is critical for understanding ecological dynamics, population control, and species interactions, as it is represented by a system of differential equations. In this work, we aim to uncover the underlying differential equations without prior knowledge of the system, relying solely on training data and neural networks. Using robust modeling in the Julia programming language, we demonstrate that both Neural ODEs and UDEs can be effectively utilized for prediction and forecasting of the Lotka-Volterra system. More importantly, we introduce the forecasting breakdown point: the time at which forecasting fails for both Neural ODEs and UDEs. We observe how UDEs outperform Neural ODEs by effectively recovering the underlying dynamics and achieving accurate forecasting with significantly less training data. Additionally, we introduce Gaussian noise of varying magnitudes (from mild to high) to simulate real-world data perturbations and show that UDEs exhibit superior robustness, effectively recovering the underlying dynamics even in the presence of noisy data, while Neural ODEs struggle with high levels of noise. Through extensive hyperparameter optimization, we offer insights into neural network architectures, activation functions, and optimizers that yield the best results. This study opens the door to applying Scientific Machine Learning frameworks for forecasting tasks across a wide range of ecological and scientific domains.

Scientific machine learning in ecological systems: A study on the predator-prey dynamics

TL;DR

This work demonstrates that both Neural ODEs and UDEs can be effectively utilized for prediction and forecasting of the Lotka-Volterra system, and introduces the forecasting breakdown point: the time at which forecasting fails for both Neural ODEs and UDEs.

Abstract

In this study, we apply two pillars of Scientific Machine Learning: Neural Ordinary Differential Equations (Neural ODEs) and Universal Differential Equations (UDEs) to the Lotka Volterra Predator Prey Model, a fundamental ecological model describing the dynamic interactions between predator and prey populations. The Lotka-Volterra model is critical for understanding ecological dynamics, population control, and species interactions, as it is represented by a system of differential equations. In this work, we aim to uncover the underlying differential equations without prior knowledge of the system, relying solely on training data and neural networks. Using robust modeling in the Julia programming language, we demonstrate that both Neural ODEs and UDEs can be effectively utilized for prediction and forecasting of the Lotka-Volterra system. More importantly, we introduce the forecasting breakdown point: the time at which forecasting fails for both Neural ODEs and UDEs. We observe how UDEs outperform Neural ODEs by effectively recovering the underlying dynamics and achieving accurate forecasting with significantly less training data. Additionally, we introduce Gaussian noise of varying magnitudes (from mild to high) to simulate real-world data perturbations and show that UDEs exhibit superior robustness, effectively recovering the underlying dynamics even in the presence of noisy data, while Neural ODEs struggle with high levels of noise. Through extensive hyperparameter optimization, we offer insights into neural network architectures, activation functions, and optimizers that yield the best results. This study opens the door to applying Scientific Machine Learning frameworks for forecasting tasks across a wide range of ecological and scientific domains.

Paper Structure

This paper contains 36 sections, 6 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Methodology
  3. Data Generation
  4. Neural Ordinary Differential Equation (Neural ODE)
  5. Universal Differential Equation (UDE)
  6. Model Training and Optimization
  7. Neural Network Architecture and Performance
  8. Neural ODE Architecture
  9. UDE Architecture and the Advantage of a Shallow Network
  10. Model Optimization
  11. Neural ODE Optimization
  12. UDE Optimization
  13. Hyperparameters
  14. Neural ODE Hyperparameters
  15. UDE Hyperparameters
  16. ...and 21 more sections

Figures (7)

  • Figure 1: Generated Lotka-Volterra Synthetic Data
  • Figure 2: Loss variation during training for both ODE and UDE models.
  • Figure 3: Comparison of Loss Variation with Hidden Units, Activation Functions, and Step Size between Neural ODE and UDE. The first row shows loss variation with hidden units, the second row with activation functions, and the third row with step size.
  • Figure 4: Neural ODE Forecast Breakdown: Forecasts based on varying training data percentages.
  • Figure 5: UDE Forecast Breakdown: (a) 35% training data with 65% forecasting, (b) Partial breakdown at 31% training data, (c) Complete breakdown at 30% training data.
  • ...and 2 more figures