# Physics-Informed Neural Network Method for Solving One-Dimensional Advection Equation Using PyTorch

arXiv: https://arxiv.org/abs/2103.09662 · HTML: https://mlsanity.com/paper/2103.09662/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 22 references that are in the map.
- First posted on arXiv 15 Mar 2021.

## Builds on (most cited first, 5 of 5)

- [Energy and Policy Considerations for Deep Learning in NLP](https://mlsanity.com/paper/1906.02243/) (2019)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [On weight initialization in deep neural networks](https://mlsanity.com/paper/1704.08863/) (2017)
- [Efficient hinging hyperplanes neural network and its application in nonlinear system identification](https://mlsanity.com/paper/1905.06518/) (2019)
- [Big Data and Learning Analytics in Higher Education: Demystifying Variety, Acquisition, Storage, NLP and Analytics](https://mlsanity.com/paper/1801.06052/) (2018)

## Built on by (most cited first, 10 of 10)

- [Respecting causality is all you need for training physics-informed neural networks](https://mlsanity.com/paper/2203.07404/) (2022)
- [Generalizing Weather Forecast to Fine-grained Temporal Scales via Physics-AI Hybrid Modeling](https://mlsanity.com/paper/2405.13796/) (2024)
- [A Review of Physics-based Machine Learning in Civil Engineering](https://mlsanity.com/paper/2110.04600/) (2021)
- [Physics-informed Neural Networks with Periodic Activation Functions for Solute Transport in Heterogeneous Porous Media](https://mlsanity.com/paper/2212.08965/) (2022)
- [Burgers' pinns with implicit euler transfer learning](https://mlsanity.com/paper/2310.15343/) (2023)
- [Predicting the spread of COVID-19 in Delhi, India using Deep Residual Recurrent Neural Networks](https://mlsanity.com/paper/2110.05477/) (2021)
- [Surface Flux Transport Modeling using Physics Informed Neural Networks](https://mlsanity.com/paper/2409.01744/) (2024)
- [Wastewater Pipe Condition Rating Model Using K- Nearest Neighbors](https://mlsanity.com/paper/2202.11049/) (2022)
- [Wastewater Pipe Rating Model Using Natural Language Processing](https://mlsanity.com/paper/2202.13871/) (2022)
- [Use of BNNM for interference wave solutions of the gBS-like equation and comparison with PINNs](https://mlsanity.com/paper/2210.12154/) (2022)

## Abstract (arXiv)

Numerical solutions to the equation for advection are determined using different finite-difference approximations and physics-informed neural networks (PINNs) under conditions that allow an analytical solution. Their accuracy is examined by comparing them to the analytical solution. We used a machine learning framework like PyTorch to implement PINNs. PINNs approach allows training neural networks while respecting the PDEs as a strong constraint in the optimization as apposed to making them part of the loss function. In standard small-scale circulation simulations, it is shown that the conventional approach incorporates a pseudo diffusive effect that is almost as large as the effect of the turbulent diffusion model; hence the numerical solution is rendered inconsistent with the PDEs. This oscillation causes inaccuracy and computational uncertainty. Of all the schemes tested, only the PINNs approximation accurately predicted the outcome. We assume that the PINNs approach can transform the physics simulation area by allowing real-time physics simulation and geometry optimization without costly and time-consuming simulations on large supercomputers.