  We introduce the Weak-form Estimation of Nonlinear Dynamics (WENDy) method
for estimating model parameters for non-linear systems of ODEs. Without relying
on any numerical differential equation solvers, WENDy computes accurate
estimates and is robust to large (biologically relevant) levels of measurement
noise. For low dimensional systems with modest amounts of data, WENDy is
competitive with conventional forward solver-based nonlinear least squares
methods in terms of speed and accuracy. For both higher dimensional systems and
stiff systems, WENDy is typically both faster (often by orders of magnitude)
and more accurate than forward solver-based approaches.
  The core mathematical idea involves an efficient conversion of the strong
form representation of a model to its weak form, and then solving a regression
problem to perform parameter inference. The core statistical idea rests on the
Errors-In-Variables framework, which necessitates the use of the iteratively
reweighted least squares algorithm. Further improvements are obtained by using
orthonormal test functions, created from a set of C-infinity bump functions of
varying support sizes.
  We demonstrate the high robustness and computational efficiency by applying
WENDy to estimate parameters in some common models from population biology,
neuroscience, and biochemistry, including logistic growth, Lotka-Volterra,
FitzHugh-Nagumo, Hindmarsh-Rose, and a Protein Transduction Benchmark model.
Software and code for reproducing the examples is available at
(https://github.com/MathBioCU/WENDy).
