Unified Spatiotemporal Physics-Informed Learning (USPIL): A Framework for Modeling Complex Predator-Prey Dynamics

Abstract

Ecological systems exhibit complex multi-scale dynamics that challenge traditional modeling. New methods must capture temporal oscillations and emergent spatiotemporal patterns while adhering to conservation principles. We present the Unified Spatiotemporal Physics-Informed Learning (USPIL) framework, a deep learning architecture integrating physics-informed neural networks (PINNs) and conservation laws to model predator-prey dynamics across dimensional scales. The framework provides a unified solution for both ordinary (ODE) and partial (PDE) differential equation systems, describing temporal cycles and reaction-diffusion patterns within a single neural network architecture. Our methodology uses automatic differentiation to enforce physics constraints and adaptive loss weighting to balance data fidelity with physical consistency. Applied to the Lotka-Volterra system, USPIL achieves 98.9% correlation for 1D temporal dynamics (loss: 0.0219, MAE: 0.0184) and captures complex spiral waves in 2D systems (loss: 4.7656, pattern correlation: 0.94). Validation confirms conservation law adherence within 0.5% and shows a 10-50x computational speedup for inference compared to numerical solvers. USPIL also enables mechanistic understanding through interpretable physics constraints, facilitating parameter discovery and sensitivity analysis not possible with purely data-driven methods. Its ability to transition between dimensional formulations opens new avenues for multi-scale ecological modeling. These capabilities make USPIL a transformative tool for ecological forecasting, conservation planning, and understanding ecosystem resilience, establishing physics-informed deep learning as a powerful and scientifically rigorous paradigm.

Extracted equations

• du/dt = alpha*u - beta*u*v
• dv/dt = delta*u*v - gamma*v
• du/dt = alpha*u - beta*u*v + D_u*nabla^2(u)
• dv/dt = delta*u*v - gamma*v + D_v*nabla^2(v)

Paper claims vs. our run

• 98.9% correlation for 1D temporal dynamics with loss 0.0219 and MAE 0.0184
  not-testable
  no fidelity score recorded
• captures complex spiral waves in 2D systems with loss 4.7656 and pattern correlation 0.94
  not-testable
  no fidelity score recorded
• conservation law adherence within 0.5%
  not-testable
  no fidelity score recorded
• 10-50x computational speedup for inference compared to numerical solvers
  not-testable
  no fidelity score recorded

Parameters