Design of the monodomain model by artificial neural networks

Sébastien Court; Karl Kunisch

doi:10.3934/dcds.2022137

Article Contents

2022, Volume 42, Issue 12: 6031-6061. Doi: 10.3934/dcds.2022137

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Design of the monodomain model by artificial neural networks

Sébastien Court^1,2, , and
Karl Kunisch^3,4,

1.
Department of Mathematics, University of Innsbruck, Technikerstrasse 13, A-6020 Innsbruck, Austria
2.
Digital Science Center, University of Innsbruck, Innrain 15, A-6020 Innsbruck, Austria
3.
Institute for Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, A-8010 Graz, Austria
4.
Radon Institute, Austrian Academy of Sciences, Altenbergstrasse 69, A-4040 Linz, Austria

^*Corresponding author: Sébastien Court
^*Corresponding author: Sébastien Court

Received: July 2021

Revised: August 2022

Early access: October 2022

Published: December 2022

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Abstract

We propose an optimal control approach in order to identify the nonlinearity in the monodomain model, from given data. This data-driven approach gives an answer to the problem of selecting the model when studying phenomena related to cardiac electrophysiology. Instead of determining coefficients of a prescribed model (like the FitzHugh-Nagumo model for instance) from empirical observations, we design the model itself, in the form of an artificial neural network. The relevance of this approach relies on the approximation capacities of neural networks. We formulate this inverse problem as an optimal control problem, and provide mathematical analysis and derivation of optimality conditions. One of the difficulties comes from the lack of smoothness of activation functions which are classically used for training neural networks. Numerical simulations demonstrate the feasibility of the strategy proposed in this work.

Keywords:

Optimal control problem,
model approximation,
data-driven approach,
artificial neural networks,
monodomain model,
semilinear partial differential equations,
$ L^p $-maximal regularity,
optimality conditions,
non-smooth optimization.

Mathematics Subject Classification: Primary: 49K20, 35A01, 41A46, 68T07; Secondary: 35D30, 35K40, 35K45, 35K58, 35B30, 35M99, 41A99, 49J45, 49N15.

Citation:

Full Text(HTML)

Figure 1. With $ (v_0,w_0) = (v_{{\rm{ FH}}},w_{{\rm{ FH}}})(0) = (2,0) $: The solution $ v_{{\rm{ FH}}} $ of the FitzHugh-Nagumo model (47) in blue, and in red the solution $ v $ of (46) with the neural network as $ \Phi = (\phi_v,\phi_w) $. (For interpretation of the references to color in this figure legend, the reader is invited to refer to the digital version of this article.)

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Figure 2. With $ (v_0,w_0) = (v_{{\rm{ AP}}},w_{{\rm{ AP}}})(0) = (0.75,0.75) $: The solution $ (v_{{\rm{ AP}}},w_{{\rm{ AP}}}) $ of the Aliev-Panfilov model (49) in dashed blue and green respectively, in red and magenta respectively, the solution $ (v,w) $ of (46) with the trained neural networks as $ \Phi = (\phi_v,\phi_w) $. Left: with Architecture 1. Right: with Architecture 2

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Figure 3. At time $ t = 9.00 $, the computed solution $ (v_{{\rm{ AP}}},w_{{\rm{ AP}}}) $ of the Aliev-Panfilov PDE model corresponding to (51) represented in dashed blue and green respectively, compared with the computed solution $ (v,w) $ of (50) with the neural network as $ \Phi = (\phi_v,\phi_w) $, represented in red and magenta, respectively. Here the neural network was trained with Architecture 1

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Figure 4. Same as Figure 3, but with the neural network trained with Architecture 2 as $ \Phi = (\phi_v,\phi_w) $

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Table 1. Description of the two types of architectures

Architecture act. function nb of layers width of hidden layers

1 GCU 7 2

2 Tanh 5 8

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