Adapting Physics-Informed Neural Networks to Improve ODE Optimization in Mosquito Population Dynamics

TL;DR

A PINN framework with several improvements for forward and inverse problems for ODE systems with a case study application in modelling the dynamics of mosquito populations, and preliminary results indicate that physics-informed machine learning holds significant potential for advancing the study of ecological systems.

Abstract

Physics informed neural networks have been gaining popularity due to their unique ability to incorporate physics laws into data-driven models, ensuring that the predictions are not only consistent with empirical data but also align with domain-specific knowledge in the form of physics equations. The integration of physics principles enables the method to require less data while maintaining the robustness of deep learning in modelling complex dynamical systems. However, current PINN frameworks are not sufficiently mature for real-world ODE systems, especially those with extreme multi-scale behavior such as mosquito population dynamical modelling. In this research, we propose a PINN framework with several improvements for forward and inverse problems for ODE systems with a case study application in modelling the dynamics of mosquito populations. The framework tackles the gradient imbalance and stiff problems posed by mosquito ordinary differential equations. The method offers a simple but effective way to resolve the time causality issue in PINNs by gradually expanding the training time domain until it covers entire domain of interest. As part of a robust evaluation, we conduct experiments using simulated data to evaluate the effectiveness of the approach. Preliminary results indicate that physics-informed machine learning holds significant potential for advancing the study of ecological systems.

Figures (11)

• Figure 1: Lorenz ODE system for the forward problem, with $U$ approximation of the system state. The blue line u_true represents the target for each of the 5 models used in the experiment. Graphs illustrate the performance across the x,y and z dimensions.
• Figure 2: Loss analysis of OdePINN framework with Lorenz system, $t \in [0, 2.0]$. (a) Loss Terms across final models selecting through Early Stopping; the bars are presented in a logscale pointing downwards, with the lower the value the better the performance. (b) ODE errors of the $\text{OdePINN}_\text{grad}$ model at different time $t$ during the training, motivating the need for Causal Training; (c) The $x$-value Approximation Solution $U$ of the model $\text{OdePINN}_\text{grad+causal}$ at different steps during the training.
• Figure 3: Lorenz ODE system, forward problem, $U$ approximation of the system state from the first 5 models, using the first five models, excluding the Domain Decomposition model.
• Figure 4: Lorenz ODE system, forward problem, $U$ approximation of the system state, using the domain composition model.
• Figure 5: Trade-off between the number of subdomains and accuracy.