Practical Algebraic Parameter Estimation for Noisy Data via Gaussian Process Regression

2026-09-24

AI Summary

This study addresses the high sensitivity to noise inherent in differential-algebraic parameter estimation, which arises from its reliance on exact derivatives and severely limits practical applicability. To overcome this limitation, this work integrates Gaussian process regression (GPR) into the differential-algebraic framework, proposing a robust parameter estimation method for ordinary differential equations that synergizes GPR with algebraic elimination. Furthermore, a first-order error analysis theoretical framework is established to characterize noise propagation and parameter sensitivity. Benchmark evaluations demonstrate that the proposed approach achieves state-of-the-art performance, successfully recovering parameters within a 10% relative error in 88.5% of experimental runs. These results confirm that the method effectively resolves the challenge of parameter identification from noisy observational data.

Abstract

Parameter estimation for ordinary differential equation (ODE) models is a fundamental task that is often complicated by the limitations of conventional optimization-based methods. In theory, differential-algebraic approaches offer an appealing alternative: they reduce the problem to polynomial system solving and do not require user-supplied initial guesses for parameter values. In practice, however, algebraic methods have been limited by their sensitivity to measurement noise, because they require accurate derivatives of observed outputs. In this work, we integrate Gaussian Process Regression (GPR) into the differential-algebraic method and derive a first-order error analysis in terms of noise level and algebraic sensitivity. We evaluate the method across several noise levels on a benchmark of 25 dynamical systems arising in applications including mechanical engineering and systems biology. The proposed method achieves the highest aggregate performance among the methods considered, recovering all sought parameter values and initial conditions to within 10% relative error in 88.5% of runs. These results demonstrate that robust derivative estimation can make differential-algebraic parameter estimation practical for dense, noisy synthetic data while retaining key advantages of the algebraic formulation.

Problem

• parameter estimation
• ordinary differential equations
• differential-algebraic methods
• measurement noise
• Gaussian Process Regression

Innovation

• Gaussian Process Regression
• Parameter Estimation
• Differential-Algebraic Method
• Ordinary Differential Equations
• Robust Derivative Estimation