Neural Networks for Bayesian Inverse Problems Governed by a Nonlinear ODE

We investigate the use of neural networks (NNs) for the estimation of hidden model parameters and uncertainty quantification from noisy observational data for inverse parameter estimation problems. We formulate the parameter estimation as a Bayesian inverse problem. We consider a parametrized system of nonlinear ordinary differential equations (ODEs), which is the FitzHugh--Nagumo model. The considered problem exhibits significant mathematical and computational challenges for classical parameter estimation methods, including strong nonlinearities, nonconvexity, and sharp gradients. We explore how NNs overcome these challenges by approximating reconstruction maps for parameter estimation from observational data. The considered data are time series of the spiking membrane potential of a biological neuron. We infer parameters controlling the dynamics of the model, noise parameters of autocorrelated additive noise, and noise modeled via stochastic differential equations, as well as the covariance matrix of the posterior distribution to expose parameter uncertainties--all with just one forward evaluation of an appropriate NN. We report results for different NN architectures and study the influence of noise on prediction accuracy. We also report timing results for training NNs on dedicated hardware. Our results demonstrate that NNs are a versatile tool to estimate parameters of the dynamical system, stochastic processes, as well as uncertainties, as they propagate through the governing ODE.