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).

翻译：我们提出了弱形式非线性动力学估计方法（Weak-form Estimation of Nonlinear Dynamics，简称WENDy），用于估计非线性常微分方程组模型参数。无需依赖任何数值微分方程求解器，WENDy即可计算出精确的估计值，并对大尺度（生物学相关）测量噪声具有鲁棒性。对于数据量适中的低维系统，WENDy在速度和精度方面可与基于传统正向求解器的非线性最小二乘法相媲美。对于高维系统和刚性系统，WENDy通常比基于正向求解器的方法更快（通常快几个数量级）且更精确。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。核心统计思想基于变量含误差框架，这需要采用迭代重加权最小二乘算法。通过使用由不同支撑大小C无穷碰撞函数生成的正交测试函数，可进一步改进性能。我们通过将WENDy应用于种群生物学、神经科学和生物化学中的若干常见模型（包括逻辑增长、Lotka-Volterra、FitzHugh-Nagumo、Hindmarsh-Rose模型以及蛋白质转导基准模型）的参数估计，展示了其高鲁棒性和计算效率。复现示例的软件和代码可在（https://github.com/MathBioCU/WENDy）获取。