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

arXiv: https://arxiv.org/abs/2509.13425 · HTML: https://mlsanity.com/paper/2509.13425/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 52 references that are in the map.
- First posted on arXiv 16 Sep 2025.

## Builds on (most cited first, 28 of 28)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [Tackling Climate Change with Machine Learning](https://mlsanity.com/paper/1906.05433/) (2019)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [Seeing biodiversity: perspectives in machine learning for wildlife conservation](https://mlsanity.com/paper/2110.12951/) (2021)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEs](https://mlsanity.com/paper/2007.01138/) (2020)
- [All-in-one simulation-based inference](https://mlsanity.com/paper/2404.09636/) (2024)
- [Deep Kronecker neural networks: A general framework for neural networks with adaptive activation functions](https://mlsanity.com/paper/2105.09513/) (2021)
- [Nonseparable Symplectic Neural Networks](https://mlsanity.com/paper/2010.12636/) (2020)
- [Physics guided neural networks for modelling of non-linear dynamics](https://mlsanity.com/paper/2205.06858/) (2022)
- [Stochastic Physics-Informed Neural Ordinary Differential Equations](https://mlsanity.com/paper/2109.01621/) (2021)
- [Forecasting Hamiltonian dynamics without canonical coordinates](https://mlsanity.com/paper/2010.15201/) (2020)
- [Multilinear Operator Networks](https://mlsanity.com/paper/2401.17992/) (2024)
- [Machine Learning and Deep Learning -- A review for Ecologists](https://mlsanity.com/paper/2204.05023/) (2022)
- [Learning Governing Equations of Unobserved States in Dynamical Systems](https://mlsanity.com/paper/2404.18572/) (2024)
- [Bayesian polynomial neural networks and polynomial neural ordinary differential equations](https://mlsanity.com/paper/2308.10892/) (2023)
- [Augmenting Neural Differential Equations to Model Unknown Dynamical Systems with Incomplete State Information](https://mlsanity.com/paper/2008.08226/) (2020)
- [Machine learning in parameter estimation of nonlinear systems](https://mlsanity.com/paper/2308.12393/) (2023)
- [Physics-informed nonlinear vector autoregressive models for the prediction of dynamical systems](https://mlsanity.com/paper/2407.18057/) (2024)
- [Scientific machine learning in ecological systems: A study on the predator-prey dynamics](https://mlsanity.com/paper/2411.06858/) (2024)
- [Learning Monotone Dynamics by Neural Networks](https://mlsanity.com/paper/2006.06417/) (2020)
- [A parametric framework for kernel-based dynamic mode decomposition using deep learning](https://mlsanity.com/paper/2409.16817/) (2024)
- [Modelling Mosquito Population Dynamics using PINN-derived Empirical Parameters](https://mlsanity.com/paper/2412.07514/) (2024)
- [Predator Prey Scavenger Model using Holling's Functional Response of Type III and Physics-Informed Deep Neural Networks](https://mlsanity.com/paper/2412.18344/) (2024)
- [A Unified Framework for Simultaneous Parameter and Function Discovery in Differential Equations](https://mlsanity.com/paper/2505.16996/) (2025)

## Built on by (most cited first, 1 of 1)

- [Learning Coupled System Dynamics under Incomplete Physical Constraints and Missing Data](https://mlsanity.com/paper/2512.23761/) (2025)

## Abstract (arXiv)

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.