Understanding the impact of numerical solvers on inference for differential equation models

Area of Science:

• Computational Mathematics
• Mathematical Modeling
• Scientific Computing

Background:

• Ordinary differential equation (ODE) models are widely used in biological and physical sciences.
• Numerical methods are essential for solving ODEs, but approximations can introduce errors.
• Accuracy for forward simulations does not guarantee accuracy for inverse problems (parameter inference).

Purpose of the Study:

• To investigate how numerical solver inaccuracies affect parameter inference in ODE models.
• To identify conditions where these inaccuracies are most severe.
• To provide guidance on setting solver tolerances for reliable inference.

Main Methods:

• Analysis of fixed and adaptive step ODE solvers.
• Demonstration of likelihood surface distortion due to numerical approximation.
• Reanalysis of a COVID-19 ODE change-point model.
• Fitting a hydrological rainfall-run-off model to data.

Main Results:

• Insufficient solver accuracy can create jagged likelihood surfaces, trapping inference algorithms in local optima.
• Inference biases are most pronounced in low-noise, rapid nonlinear dynamical systems.
• Step size significantly impacts simulation and inference outcomes in ODE models.
• Solver tolerances must be carefully tuned to avoid distorted likelihoods.

Conclusions:

• Numerical approximations in ODE solvers pose a significant challenge for parameter inference.
• Adaptive step size solver tolerances require cautious setting during inference.
• Inspection of likelihood surfaces for numerical artifacts is recommended.
• Ensuring solver accuracy is critical for reliable ODE model parameter estimation.