# Reduced-PINN: An Integration-Based Physics-Informed Neural Networks for Stiff ODEs

**Authors:** Pouyan Nasiri, Roozbeh Dargazany  
**arXiv:** [2208.12045](https://arxiv.org/abs/2208.12045)  
**Date:** 2022-08-23  
**Summary page:** https://www.summarizepaper.com/en/paper/2208.12045v1/

## Key points

- Physics-informed neural networks (PINNs) can solve both forward and inverse problems
- PINNs face challenges with stiff equations and simulating stiff chemical kinetics
- The authors propose a Reduced-PINN architecture to address this issue
- Reduced-PINN uses a reduced-order integration method for solving stiff chemical kinetics
- It can be applied to various reaction-diffusion systems with stiff dynamics
- Initial value problems (IVPs) are transformed into equivalent integral forms and solved using physics-informed neural networks
- The approach allows for adaptive increments, addressing the limitations of traditional PINNs in solving stiff equations
- Multiple second order ODEs are simulated to demonstrate the capabilities of Reduced-PINN
- Results show accurate solutions for a stiff scalar ODE and validation against a system of linear ODEs
- Reduced-PINN offers an integration-based approach that overcomes challenges faced by traditional PINNs in simulating stiff chemical kinetics
- It provides an efficient and accurate solution for reaction-diffusion systems with stiff dynamics at lower computational cost.

## Summary

Physics-informed neural networks (PINNs) have gained attention for their ability to solve both forward and inverse problems. However, they face challenges when dealing with stiff equations and simulating stiff chemical kinetics. Stiffness is a primary cause of the failure of PINNs in these scenarios. To address this issue, the authors propose a Reduced-PINN architecture which utilizes a reduced-order integration method to enable the solution of stiff chemical kinetics. It can be applied to various reaction-diffusion systems involving stiff dynamics. The authors transform initial value problems (IVPs) into equivalent integral forms and solve the resulting integral equations using physics-informed neural networks. In the derived integral-based optimization process, there is only one term without explicitly incorporating loss terms associated with ordinary differential equations (ODEs) and initial conditions (ICs). This approach allows for adaptive increments which addresses the limitations faced by traditional PINNs in solving stiff equations. To demonstrate the capabilities of Reduced-PINN, multiple second order ODEs are simulated; results show that it accurately captures solutions for a stiff scalar ODE and validates well against a system of linear ODEs. The proposed Reduced-PINN offers an integration-based approach that overcomes the challenges faced by traditional PINNs in simulating stiff chemical kinetics. It provides an efficient and accurate solution for reaction-diffusion systems with stiff dynamics at a lower computational cost.