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Machine learning algorithms have been successfully used to approximatenonlinear maps under weak assumptions on the structure and properties of themaps. We present deep neural networks using dense and convolutional layers tosolve an inverse problem, where we seek to estimate parameters of aFitzHugh-Nagumo model, which consists of a nonlinear system of ordinarydifferential equations (ODEs). We employ the neural networks to approximatereconstruction maps for model parameter estimation from observational data,where the data comes from the solution of the ODE and takes the form of a timeseries representing dynamically spiking membrane potential of a biologicalneuron. We target this dynamical model because of the computational challengesit poses in an inference setting, namely, having a highly nonlinear andnonconvex data misfit term and permitting only weakly informative priors onparameters. These challenges cause traditional optimization to fail andalternative algorithms to exhibit large computational costs. We quantify theprediction errors of model parameters obtained from the neural networks andinvestigate the effects of network architectures with and without the presenceof noise in observational data. We generalize our framework for neuralnetwork-based reconstruction maps to simultaneously estimate ODE parameters andparameters of autocorrelated observational noise. Our results demonstrate thatdeep neural networks have the potential to estimate parameters in dynamicalmodels and stochastic processes, and they are capable of predicting parametersaccurately for the FitzHugh-Nagumo model.

Parameter Estimation with Dense and Convolutional Neural Networks Applied to the FitzHugh-Nagumo ODE