Cubature Kalman Filter Based Training of Hybrid Differential Equation Recurrent Neural Network Physiological Dynamic Models

TL;DR

This paper introduces a novel method combining the Cubature Kalman Filter with neural networks to model and estimate missing components in physiological dynamic systems, demonstrated on blood circulation models.

Contribution

It presents a Bayesian filtering approach to train neural networks within hybrid differential equation models, improving estimation accuracy over traditional backpropagation methods.

Findings

Neural networks can effectively approximate missing ODEs in physiological models.

The proposed Bayesian filtering method outperforms backpropagation in state and parameter estimation.

Joint estimation captures the impact of unmeasured state variables accurately.

Abstract

Modeling biological dynamical systems is challenging due to the interdependence of different system components, some of which are not fully understood. To fill existing gaps in our ability to mechanistically model physiological systems, we propose to combine neural networks with physics-based models. Specifically, we demonstrate how we can approximate missing ordinary differential equations (ODEs) coupled with known ODEs using Bayesian filtering techniques to train the model parameters and simultaneously estimate dynamic state variables. As a study case we leverage a well-understood model for blood circulation in the human retina and replace one of its core ODEs with a neural network approximation, representing the case where we have incomplete knowledge of the physiological state dynamics. Results demonstrate that state dynamics corresponding to the missing ODEs can be approximated well using a neural network trained using a recursive Bayesian filtering approach in a fashion coupled with the known state dynamic differential equations. This demonstrates that dynamics and impact of missing state variables can be captured through joint state estimation and model parameter estimation within a recursive Bayesian state estimation (RBSE) framework. Results also indicate that this RBSE approach to training the NN parameters yields better outcomes (measurement/state estimation accuracy) than training the neural network with backpropagation through time in the same setting.