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：本文提出一种用于非线性常微分方程组模型参数估计的弱形式估计方法（WENDy）。核心数学思想是将模型的强形式表示高效转换为弱形式，进而通过求解回归问题实现参数推断。统计基础基于误差变量框架，需采用迭代重加权最小二乘算法。通过使用由不同支撑尺度的$C^{\infty}$ 光滑函数构造的正交测试函数，进一步提升了方法性能。我们证明WENDy是微分方程参数推断中高度稳健且高效的方法。无需依赖任何数值微分方程求解器，WENDy即可计算精确估计值，并能耐受生物学相关量级的高测量噪声。对于数据量适中的低维系统，WENDy在速度和精度上均能与基于正向求解器的传统非线性最小二乘法相媲美。对于高维系统和刚性系统，WENDy通常在速度上快数个数量级且精度更高。我们通过逻辑增长模型、洛特卡-沃尔泰拉模型、FitzHugh-Nagumo模型、Hindmarsh-Rose模型及蛋白质转导基准模型等常见种群与神经科学模型展示了该方法及其性能。重现示例的软件与代码见https://github.com/MathBioCU/WENDy。