We introduce the Weak-form Estimation of Nonlinear Dynamics (WENDy) method for estimating model parameters for non-linear systems of ODEs. 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 $C^{\infty}$ bump functions of varying support sizes. We demonstrate that WENDy is a highly robust and efficient method for parameter inference in differential equations. 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. We illustrate the method and its performance in some common population and neuroscience models, 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).

翻译：我们提出了一种用于估计非线性常微分方程组模型参数的弱形式非线性动力学（WENDy）方法。其核心数学思想是将模型的强形式表示高效转换为弱形式，然后通过求解回归问题进行参数推断。核心统计思想基于误差变量框架，这要求使用迭代重加权最小二乘算法。通过使用由一组具有不同支撑大小的$C^{\infty}$ bump函数构造的正交测试函数，可获得进一步改进。我们证明WENDy是一种高度稳健且高效的微分方程参数推断方法。无需依赖任何数值微分方程求解器，WENDy即可计算出精确估计，并对大（生物学相关）水平的测量噪声具有鲁棒性。对于数据量适中的低维系统，WENDy在速度和精度上与传统的基于正向求解器的非线性最小二乘方法具有竞争力。对于高维系统和刚性系统，WENDy通常比基于正向求解器的方法更快（通常快几个数量级）且更准确。我们在一些常见的群体和神经科学模型中展示了该方法及其性能，包括逻辑增长、Lotka-Volterra、FitzHugh-Nagumo、Hindmarsh-Rose以及一个蛋白质转导基准模型。用于重现示例的软件和代码可在(https://github.com/MathBioCU/WENDy)获取。