@ARTICLE{Borzyszkowski_Bartłomiej_Physics-guided_2021,
 author={Borzyszkowski, Bartłomiej and Damaszke, Karol and Romankiewicz, Jakub and Świniarski, Marcin and Moszyński, Marek},
 volume={69},
 number={6},
 journal={Bulletin of the Polish Academy of Sciences Technical Sciences},
 pages={e139391},
 howpublished={online},
 year={2021},
 abstract={Numerous examples of physically unjustified neural networks, despite satisfactory performance, generate contradictions with logic and lead to many inaccuracies in the final applications. One of the methods to justify the typical black-box model already at the training stage involves extending its cost function by a relationship directly inspired by the physical formula. This publication explains the concept of Physics-guided neural networks (PGNN), makes an overview of already proposed solutions in the field and describes possibilities of implementing physics-based loss functions for spatial analysis. Our approach shows that the model predictions are not only optimal but also scientifically consistent with domain specific equations. Furthermore, we present two applications of PGNNs and illustrate their advantages in theory by solving Poisson’s and Burger’s partial differential equations. The proposed formulas describe various real-world processes and have numerous applications in the area of applied mathematics. Eventually, the usage of scientific knowledge contained in the tailored cost functions shows that our methods guarantee physics-consistent results as well as better generalizability of the model compared to classical, artificial neural networks.},
 type={Article},
 title={Physics-guided neural networks (PGNNs) to solve differential equations for spatial analysis},
 URL={http://www.journals.pan.pl/Content/121316/PDF/Z_17_02322_Bpast.No.69(6)_OK.pdf},
 doi={10.24425/bpasts.2021.139391},
 keywords={physics-guided neural networks, spatial analysis, differential equations, machine learning},
}