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:我们介绍了一种名为弱形式估计非线性动力学(WENDy)的方法,用于估计非线性ODE系统的模型参数。不依赖于任何数值微分方程求解器,WENDy能够计算准确的估计值,并且对大量(具有生物学意义的)测量噪声具有鲁棒性。对于低维系统和适度数据量的情况,WENDy在速度和准确性方面与传统的向前求解器非线性最小二乘方法相当。对于高维系统和刚性系统,通常比向前求解器方法更快(通常是数量级的)和更准确。其核心数学思想涉及将模型的强形式表示有效地转换为其弱形式,然后通过解决回归问题来执行参数推断。其核心统计思想则依托于误差在变量框架,这需要使用迭代重新加权最小二乘算法。通过使用从一组支持大小不同的C无穷大隆起函数中创建的正交测试函数,可以获得更好的改进。我们通过应用WENDy来估计人口生物学,神经科学和生物化学等一些常见模型中的参数来展示其高可靠性和计算效率,包括对数增长,Lotka-Volterra,FitzHugh-Nagumo,Hindmarsh-Rose以及某个蛋白转导基准模型。代码和软件来复制这些例子可在(https://github.com/MathBioCU/WENDy)网址下获得。