Weak form Estimation of Nonlinear Dynamics (WENDy) for Nonlinear-in-Parameters ODEs

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 demon-

strated 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 approx-

imates 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 accom-

modate 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.