单域模型的人工神经网络设计
Design of the monodomain model by artificial neural networks
摘要：
我们提出了一个最优控制方法，以确定在单域模型的非线性，从给定的数据。这种数据驱动的方法给出了一个答案，选择模型的问题时，研究与心脏电生理学的现象。我们不是从经验观察中确定指定模型（例如FitzHugh-Nagumo模型）的系数，而是以人工神经网络的形式设计模型本身。这种方法的相关性依赖于神经网络的逼近能力。我们制定了这个逆问题作为一个最优控制问题，并提供数学分析和推导的最优性条件。其中一个困难来自缺乏光滑的激活函数，这是经典的用于训练神经网络。数值模拟证明了本文所提出的策略的可行性。
英文摘要：
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.
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