WENDy for Nonlinear-in-Parameter ODEs

2025-02-13

AI Summary

This work addresses the problem of dynamical modeling for parameter-dependent nonlinear (NiP) ordinary differential equation systems. We propose WENDy-NiP—the first extension of Weak-form Estimation of Nonlinear Dynamics (WENDy) to NiP settings. Our method formulates a maximum likelihood estimation framework based on the weak form, rigorously deriving closed-form expressions for the likelihood, its gradient, and Hessian—thereby avoiding numerical errors and convergence limitations inherent in forward solvers. Leveraging nonconvex optimization (L-BFGS) and high-performance Julia implementation (WENDy.jl), WENDy-NiP supports both additive Gaussian and multiplicative log-normal noise models. Across multiple benchmark systems, it achieves significantly higher estimation accuracy and improved confidence coverage compared to output-error least-squares methods, while reducing bias and accelerating computation by one to three orders of magnitude.

Technology Category

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• Reasoning under Uncertainty: Stochastic Optimization
• Natural Language Processing: Learning & Optimization for NLP

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Abstract

The Weak-form Estimation of Non-linear Dynamics (WENDy) algorithm is extended to accommodate systems of ordinary differential equations that are nonlinear-in-parameters (NiP). The extension rests on derived analytic expressions for a likelihood function, its gradient and its Hessian matrix. WENDy makes use of these to approximate a maximum likelihood estimator based on optimization routines suited for non-convex optimization problems. The resulting parameter estimation algorithm has better accuracy, a substantially larger domain of convergence, and is often orders of magnitude faster than the conventional output error least squares method (based on forward solvers). The WENDy.jl algorithm is efficiently implemented in Julia. We demonstrate the algorithm's ability to accommodate the weak form optimization for both additive normal and multiplicative log-normal noise, and present results on a suite of benchmark systems of ordinary differential equations. In order to demonstrate the practical benefits of our approach, we present extensive comparisons between our method and output error methods in terms of accuracy, precision, bias, and coverage.

Problem

• Extends WENDy for nonlinear-in-parameter ODEs
• Improves accuracy and convergence in parameter estimation
• Compares WENDy with traditional output error methods

Innovation

• Extends WENDy for nonlinear-in-parameter ODEs
• Uses analytic likelihood, gradient, Hessian
• Efficient Julia implementation for optimization