Manifold-constrained Gaussian process inference for time-varying parameters in dynamic systems

TL;DR

This paper introduces TVMAGI, a Bayesian Gaussian process-based method for efficiently estimating both static and dynamic parameters in ODE models, especially with sparse, noisy data, bypassing numerical integration.

Contribution

The paper proposes TVMAGI, a novel manifold-constrained Gaussian process approach that accurately estimates time-varying parameters in nonlinear ODEs without relying on numerical integration.

Findings

TVMAGI achieves faster computation than traditional methods.

It effectively handles missing data and unobserved components.

The method demonstrates robustness and accuracy in simulation studies.

Abstract

Identification of parameters in ordinary differential equations (ODEs) is an important and challenging task when modeling dynamic systems in biomedical research and other scientific areas, especially with the presence of time-varying parameters. This article proposes a fast and accurate method, TVMAGI (Time-Varying MAnifold-constrained Gaussian process Inference), to estimate both time-constant and time-varying parameters in the ODE using noisy and sparse observation data. TVMAGI imposes a Gaussian process model over the time series of system components as well as time-varying parameters, and restricts the derivative process to satisfy ODE conditions. Consequently, TVMAGI completely bypasses numerical integration and achieves substantial savings in computation time. By incorporating the ODE structures through manifold constraints, TVMAGI enjoys a principled statistical construct under the Bayesian paradigm, which further enables it to handle systems with missing data or unobserved components. The Gaussian process prior also alleviates the identifiability issue often associated with the time-varying parameters in ODE. Unlike existing approaches, TVMAGI can be applied to general nonlinear systems without specific structural assumptions. Three simulation examples, including an infectious disease compartmental model, are provided to illustrate the robustness and efficiency of our method compared with numerical integration and Bayesian filtering methods.