🤖 AI Summary
Standard MCMC methods for Bayesian parameter estimation in ODEs suffer from high rejection rates, slow convergence, and excessive computational cost due to strong nonlinear dependencies among parameters and states. To address this, we propose a constraint-satisfying Langevin dynamics method that directly samples on the joint state-parameter manifold, embedding the ODE dynamics as hard constraints into the sampling process—thereby eliminating repeated forward numerical integration. Our approach uniquely integrates Langevin dynamics, trajectory optimization, and numerical continuation techniques within a constrained differential equation framework, enabling efficient posterior sampling while rigorously preserving the structural integrity of ODE solutions. Evaluated on a biochemical oscillator model, the method achieves over 100× improvement in sampling efficiency compared to standard MCMC. It accurately resolves Hopf bifurcations and characterizes limit-cycle regions, significantly enhancing uncertainty quantification and model selection capabilities.
📝 Abstract
Fitting models to data to obtain distributions of consistent parameter values is important for uncertainty quantification, model comparison, and prediction. Standard Markov Chain Monte Carlo (MCMC) approaches for fitting ordinary differential equations (ODEs) to time-series data involve proposing trial parameter sets, numerically integrating the ODEs forward in time, and accepting or rejecting the trial parameter sets. When the model dynamics depend nonlinearly on the parameters, as is generally the case, trial parameter sets are often rejected, and MCMC approaches become prohibitively computationally costly to converge. Here, we build on methods for numerical continuation and trajectory optimization to introduce an approach in which we use Langevin dynamics in the joint space of variables and parameters to sample models that satisfy constraints on the dynamics. We demonstrate the method by sampling Hopf bifurcations and limit cycles of a model of a biochemical oscillator in a Bayesian framework for parameter estimation, and we obtain more than a hundred fold speedup relative to a leading ensemble MCMC approach that requires numerically integrating the ODEs forward in time. We describe numerical experiments that provide insight into the speedup. The method is general and can be used in any framework for parameter estimation and model selection.