**Natural Numerical Network - forward-backward diffusion on directed graphs for supervised image classification**

The main objective of the presentation is to introduce a novel method for PDE-based supervised deep learning data classification, in our case, the classification of Natura 2000 protected habitats using Sentinel-2 satellite optical data. The Natural Numerical Network [1] is based on the numerical solution of nonlinear partial differential equations of forward-backward diffusion type on a semi-complete directed graph [2]. Due to the model nonlinearity, for time discretization, we use a semi-implicit scheme, and for discretization of the graph diffusion, we use the finite volume method principles, i.e. the mass balance of diffused quantity in graph vertices. In the Natural Numerical Network, the forward diffusion causes the movement of points in a feature space toward each other and thus clustering of desired vertices. The opposite effect, keeping the points away from each other, is caused by a slight backward diffusion and gives a repulsion of vertices belonging to different clusters. The Natural Numerical Network contains a few parameters that are optimized in the learning phase of the method. After learning parameters and optimizing the topology of the network graph, the classification of Natura 2000 habitats is performed. A relevancy map for each habitat is created as a tool for validating the classification, and then it is used for automatic identification of Natura 2000 habitat areas in satellite images as well as for finding new protected habitat appearances by using satellite imagery.

This is a joint work with K. Mikula, M. Kollár, M. Ambroz, J. Šibík and M. Šibíková.

[1] Karol Mikula, Michal Kollár, Aneta A. Ožvat, Martin Ambroz, Lucia Čahojová, Ivan Jarolímek, Jozef Šibík, Mária Šibíková, Natural numerical networks for Natura 2000 habitats classification by satellite images, Applied Mathematical Modelling, Volume 116 (2023) pp. 209-235

[2] Karol Mikula, Michal Kollár, Aneta A. Ožvat, Lucia Čahojová, Mária Šibíková, Natural numerical networks on directed graphs in satellite image classification, L.Calatroni et al.(Eds.): Scale Space and Variational Methods in Computer Vision. SSVM 2023. Lecture Notes in Computer Science, Vol 14009, pp. 339–351, Springer, Cham, 2023