WENDy for Nonlinear-in-Parameters ODEs

TL;DR

This paper extends the WENDy framework to nonlinear-in-parameters ODEs, introducing WENDy-MLE, which improves accuracy, convergence, and speed, and handles multiplicative noise, demonstrated through extensive numerical benchmarks.

Contribution

The paper introduces WENDy-MLE, a novel extension of WENDy for nonlinear-in-parameters ODEs, with analytic likelihood expressions and improved performance.

Findings

WENDy-MLE achieves higher accuracy and larger convergence domain.

The method is often faster than existing weak form and least squares methods.

Numerical benchmarks show superior performance on various ODE systems.

Abstract

The Weak-form Estimation of Non-linear Dynamics (WENDy) framework is a recently developed approach for parameter estimation and inference of systems of ordinary differential equations (ODEs). Prior work demonstrated WENDy to be robust, computationally efficient, and accurate, but only works for ODEs which are linear-in-parameters. In this work, we derive a novel extension to accommodate systems of a more general class of ODEs that are nonlinear-in-parameters. Our new WENDy-MLE algorithm approximates a maximum likelihood estimator via local non-convex optimization methods. This is made possible by the availability of analytic expressions for the likelihood function and its first and second order derivatives. WENDy-MLE has better accuracy, a substantially larger domain of convergence, and is often faster than other weak form methods and the conventional output error least squares method. Moreover, we extend the framework to accommodate data corrupted by multiplicative log-normal noise. The WENDy.jl algorithm is efficiently implemented in Julia. In order to demonstrate the practical benefits of our approach, we present extensive numerical results comparing our method, other weak form methods, and output error least squares on a suite of benchmark systems of ODEs in terms of accuracy, precision, bias, and coverage.