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.

翻译：本研究探讨利用神经网络从含噪观测数据中估计隐藏模型参数并进行不确定性量化，以解决反参数估计问题。我们将参数估计问题表述为贝叶斯反问题框架。采用参数化的非线性常微分方程组——FitzHugh-Nagumo模型作为研究对象。该问题对经典参数估计方法构成显著的数学与计算挑战，包括强非线性、非凸性及陡峭梯度特性。我们通过神经网络逼近从观测数据到参数估计的重构映射，探究其如何克服这些挑战。所用数据为生物神经元膜电位峰电位的时间序列。我们通过单次前向计算即可推断：控制模型动力学的参数、自相关加性噪声参数、随机微分方程建模的噪声参数，以及后验分布的协方差矩阵以揭示参数不确定性。我们报告了不同神经网络架构的实验结果，研究了噪声对预测精度的影响，并给出了在专用硬件上训练神经网络的时间性能数据。研究结果表明，神经网络能够通过控制方程常微分方程传播，成为估计动力系统参数、随机过程参数及不确定性的通用工具。