A new artificial intelligence project at ELTE

Last year, a major project was launched, supported by the Hungarian National Research, Development and Innovation Office within the framework of the Thematic Excellence Program 2021 – National Research Sub programme: “Artificial intelligence, large networks, data security: mathematical foundation and applications”. This initiative allowed the Mathematical Institute at ELTE to initiate reseach on innovative artificial intelligence applications, like financial mathematics or networks, besides its more traditional subjects as computer vision or natural language processing. Here we present a work on differential equations, by group leader Ferenc Izsák and Taki Eddine Djebbar.

Neural network – based approximation of Dirichlet-to-Neumann maps

Dirichlet-to-Neumann maps can be regarded as non-trivial linear mappings between boundary spaces. Besides, they constitute the mathematical basis of the so-called Electrical Impedance Tomography.

For a given elliptic boundary value problem  div (A grad u ) = 0 on the physical or computational domain Ω, with the unknown function u and a given conductivity coefficient A, we can define the mapping

A →(u|_Γ →∂_νu)

The right hand side can be identified as the current response to some applied potential field on the surface and is called the Dirichlet-to-Neumann map. In real-life situations, this can be measured at certain points, using electrodes. Then, the real task is to compute the inverse of this map to find out the internal structure of some organ such that we face with an inverse problem. To assist this, we have constructed and trained a neural network, which can compute the Dirichlet-to-Neumann map.

For the neural network-based approximation of this mapping we took a non-trivial domain picking some boundary and outer points and the normal derivatives, as follows.

Then we defined a clever training data set based on fundamental solutions and trained a simple neural network with known  u|_Γ →∂_νu pairs.

Finally, we have smoothed the predicted Neumann data locally on the edges. The performance of the test problem is shown in the next figure, where the predicted, the smoothed and the real Neumann data is plotted vs. the number of the grid points.

For more information, contact Ferenc Izsák (ferenc.izsak@ttk.elte.hu).