Learning Interactions in Reaction Diffusion Equations by Neural Networks.

Area of Science:

• Computational Biology
• Mathematical Biology
• Systems Biology

Background:

• Partial differential equations (PDEs) are crucial for modeling complex biological phenomena.
• Deriving accurate PDEs and estimating parameters from biological data remains a significant challenge.
• Modeling species interactions often requires accounting for saturation effects and other complex regimes.

Purpose of the Study:

• To develop a novel neural network-based framework, Frac-PDE-Net, for discovering underlying partial differential equation (PDE) systems from observed data.
• To enhance existing PDE discovery methods by incorporating capabilities to learn fractional and integration terms.
• To address the identifiability challenge in learning complex PDE terms from data.

Main Methods:

• Frac-PDE-Net adapts the PDE-Net 2.0 architecture with specialized layers for learning fractional and integration terms.
• Identifiability issues are managed by incorporating biologically realistic assumptions.
• An L2-norm based term selection criterion and sparse regression are employed for parsimonious model identification.

Main Results:

• The Frac-PDE-Net framework successfully recovers the main terms of underlying PDEs with accurate coefficients.
• The method demonstrates capability for effective long-term prediction of biological system dynamics.
• The approach was validated on a biological PDE model for pollen tube growth.

Conclusions:

• Frac-PDE-Net offers a powerful approach for discovering complex, data-driven partial differential equations in biology.
• The framework effectively handles fractional and integration terms, overcoming key identifiability challenges.
• This method facilitates more accurate modeling and prediction of biological processes, such as pollen tube growth.