WENDy for Nonlinear-in-Parameters ODEs
This work addresses the problem of nonlinear-in-parameters ODEs for researchers and practitioners in the field of dynamical systems, providing an incremental yet significant improvement over existing methods.
The authors tackled the problem of parameter estimation and inference for nonlinear-in-parameters ODEs, achieving better accuracy and a larger domain of convergence with their WENDy-MLE algorithm. The approach outperforms other weak form methods and the conventional output error least squares method in terms of accuracy, precision, bias, and coverage.
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.