# Respecting causality is all you need for training physics-informed neural networks

**Authors:** Sifan Wang, Shyam Sankaran, Paris Perdikaris  
**arXiv:** [2203.07404](https://arxiv.org/abs/2203.07404)  
**Date:** 2022-03-14  
**Summary page:** https://www.summarizepaper.com/en/paper/2203.07404v1/

## Key points

- Physics-informed neural networks (PINNs) are used for simulating complex physical systems
- PINN models have difficulty accurately simulating dynamical systems that exhibit multi-scale, chaotic or turbulent behavior
- Researchers proposed a re-formulation of PINN loss functions that explicitly account for physical causality during model training
- This modification introduced significant accuracy improvements and provided a practical quantitative mechanism for assessing the convergence of a PINN model
- The modified PINNs demonstrated state-of-the-art numerical results across several benchmarks where existing formulations fail, including the chaotic Lorenz system, the Kuramoto-Sivashinsky equation in the chaotic regime, and the Navier-Stokes equations in the turbulent regime
- Respecting causality is fundamental to accurately simulating physical systems and overcoming challenges that currently limit the success of PINNs to relatively simple problems
- Classical numerical methods typically discretize PDEs sequentially over time using algorithms that ensure full resolution at each time step before approximating subsequent steps
- Respecting causality can significantly improve the accuracy and applicability of PINNs to problems with industrial complexity.
- These findings could pave the way for further advancements in physics-informed machine learning techniques.

## Summary

Physics-informed neural networks (PINNs) have gained popularity in recent years as a promising tool for simulating complex physical systems. To address the issue of PINN models not accurately simulating dynamical systems that exhibit multi-scale, chaotic or turbulent behavior, a team of researchers proposed a simple re-formulation of PINN loss functions that explicitly account for physical causality during model training. This modification introduced significant accuracy improvements and provided a practical quantitative mechanism for assessing the convergence of a PINN model. The researchers demonstrated state-of-the-art numerical results across several benchmarks where existing PINN formulations fail, including the chaotic Lorenz system, the Kuramoto-Sivashinsky equation in the chaotic regime, and the Navier-Stokes equations in the turbulent regime. The team argued that respecting causality is fundamental to accurately simulating physical systems and overcoming challenges that currently limit the success of PINNs to relatively simple problems. Physical systems possess an inherent causal structure that dictates how local changes in initial/boundary data are reflected in corresponding states at later times. Classical numerical methods typically discretize PDEs sequentially over time using algorithms that ensure full resolution at each time step before approximating subsequent steps. Overall, this work highlights how respecting causality can significantly improve the accuracy and applicability of PINNs to problems with industrial complexity. By introducing new opportunities for their use across various fields, these findings could pave the way for further advancements in physics-informed machine learning techniques.