Probabilistic solution of chaotic dynamical system inverse problems using Bayesian Artificial Neural Networks

This paper demonstrates the application of Bayesian Artificial Neural Networks to Ordinary Differential Equation (ODE) inverse problems. We consider the case of estimating an unknown chaotic dynamical system transition model from state observation data. Inverse problems for chaotic systems are numerically challenging as small perturbations in model parameters can cause very large changes in estimated forward trajectories. Bayesian Artificial Neural Networks can be used to simultaneously fit a model and estimate model parameter uncertainty. Knowledge of model parameter uncertainty can then be incorporated into the probabilistic estimates of the inferred system's forward time evolution. The method is demonstrated numerically by analysing the chaotic Sprott B system. Observations of the system are used to estimate a posterior predictive distribution over the weights of a parametric polynomial kernel Artificial Neural Network. It is shown that the proposed method is able to perform accurate time predictions. Further, the proposed method is able to correctly account for model uncertainties and provide useful prediction uncertainty bounds.

Model inference for Ordinary Differential Equations by parametric polynomial kernel regression

Model inference for dynamical systems aims to estimate the future behaviour of a system from observations. Purely model-free statistical methods, such as Artificial Neural Networks, tend to perform poorly for such tasks. They are therefore not well suited to many questions from applications, for example in Bayesian filtering and reliability estimation. This work introduces a parametric polynomial kernel method that can be used for inferring the future behaviour of Ordinary Differential Equation models, including chaotic dynamical systems, from observations. Using numerical integration techniques, parametric representations of Ordinary Differential Equations can be learnt using Backpropagation and Stochastic Gradient Descent. The polynomial technique presented here is based on a nonparametric method, kernel ridge regression. However, the time complexity of nonparametric kernel ridge regression scales cubically with the number of training data points. Our parametric polynomial method avoids this manifestation of the curse of dimensionality, which becomes particularly relevant when working with large time series data sets. Two numerical demonstrations are presented. First, a simple regression test case is used to illustrate the method and to compare the performance with standard Artificial Neural Network techniques. Second, a more substantial test case is the inference of a chaotic spatio-temporal dynamical system, the Lorenz--Emanuel system, from observations. Our method was able to successfully track the future behaviour of the system over time periods much larger than the training data sampling rate. Finally, some limitations of the method are presented, as well as proposed directions for future work to mitigate these limitations.