Many real-world systems modeled using differential equations involve unknown or uncertain parameters. Standard approaches to address parameter estimation inverse problems in this setting typically focus on estimating constants; yet some unobservable system parameters may vary with time without known evolution models. In this work, we propose a novel approximation method inspired by the Fourier series to estimate time-varying parameters in deterministic dynamical systems modeled with ordinary differential equations. Using ensemble Kalman filtering in conjunction with Fourier series-based approximation models, we detail two possible implementation schemes for sequentially updating the time-varying parameter estimates given noisy observations of the system states. We demonstrate the capabilities of the proposed approach in estimating periodic parameters, both when the period is known and unknown, as well as non-periodic time-varying parameters of different forms with several computed examples using a forced harmonic oscillator. Results emphasize the importance of the frequencies and number of approximation model terms on the time-varying parameter estimates and corresponding dynamical system predictions.

翻译：许多通过微分方程建模的实际系统涉及未知或不确定的参数。解决此类参数估计逆问题的标准方法通常集中于估计常数参数；然而某些不可观测的系统参数可能随时间变化且缺乏已知的演化模型。本文提出一种受傅里叶级数启发的新型逼近方法，用于估计常微分方程建模的确定性动力系统中的时变参数。通过结合集合卡尔曼滤波与傅里叶级数逼近模型，我们详细阐述了两种基于系统状态含噪观测值对时变参数估计进行顺序更新的实现方案。利用受迫谐振子的多个计算实例，我们展示了该方法在估计周期参数（无论周期已知或未知）以及不同形式的非周期时变参数方面的能力。研究结果强调了逼近模型的频率项数与项数对时变参数估计及其对应动力系统预测的重要影响。