An official website of the United States government
Estimating and quantifying uncertainty in unknown system parameters from limited data remains a challenging inverse problem in a variety of real-world applications. While many approaches focus on estimating constant parameters, a subset of these problems includes time-varying parameters with unknown evolution models that often cannot be directly observed. This work develops a systematic particle filtering approach that reframes the idea behind artificial parameter evolution to estimate time-varying parameters in nonstationary inverse problems arising from deterministic dynamical systems. Focusing on systems modeled by ordinary differential equations, we present two particle filter algorithms for time-varying parameter estimation: one that relies on a fixed value for the noise variance of a parameter random walk; another that employs online estimation of the parameter evolution noise variance along with the time-varying parameter of interest. Several computed examples demonstrate the capability of the proposed algorithms in estimating time-varying parameters with different underlying functional forms and different relationships with the system states (i.e. additive vs. multiplicative).
more »
« less
Fourier series-based approximation of time-varying parameters in ordinary differential equations
Abstract 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.
more »
« less
- Award ID(s):
- 1819203
- PAR ID:
- 10487368
- Publisher / Repository:
- IOP Publishing Ltd.
- Date Published:
- Journal Name:
- Inverse Problems
- ISSN:
- 0266-5611
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Estimating and quantifying uncertainty in system parameters remains a big challenge in applied and computational mathematics. A subset of these problems includes estimating periodic parameters that have unknown dynamics. Along with their time series, the period of these parameters may also be unknown and need to be estimated. The aim of this paper is to address the periodic parameter estimation problem, with particular focus on exploring the associated uncertainty, using Monte Carlo particle methods, such as the ensemble Kalman filter. Both parameter tracking and piecewise function approximations of periodic parameters are considered, highlighting aspects of parameter uncertainty in each approach when considering factors such as the frequency of available data and the number of piecewise segments used in the approximation. Estimation of the period of the periodic parameters and related uncertainty is also analyzed in the piecewise formulation. The pros and cons of each approach are discussed relative to a numerical example estimating the external voltage parameter in the FitzHugh-Nagumo system for modeling the spiking dynamics of neurons.more » « less
-
The estimation of static parameters in dynamical systems is well-established, but challenges remain for systems with parameters that vary over time or space. Assuming prior knowledge of the system’s dynamics, we aim to identify functions describing these varying parameters. Initially, we focus on cases where parameters are discretely switching piecewise constant functions. For this sub-class, we propose an algorithmic framework to detect discrete parameter switches and fit a piecewise constant model using optimization-based parameter estimation. Our framework is modular, allowing customization of switch detection, numerical integration and optimization steps. We demonstrate its utility through examples, including a time-varying promoter-gene expression model, a genetic toggle switch, parameter-switching manifolds, mixed fixed and switching parameters, non-uniform switching and models governed by PDEs like the heat and advection–diffusion equations. Binary segmentation is used for switch detection, and Nelder–Mead and Powell methods for optimization. In addition, we assess the framework’s robustness to measurement noise and time-series data resolution. For more complex cases, we extend the approach using dictionary-based sparse regression with trigonometric and polynomial functions to capture continuously varying parameters. We also compare our framework with two other similar methods using standard metrics under the condition of model-mismatch. This comprehensive framework offers a versatile solution for parameter estimation in varying dynamical systems.more » « less
-
Mathematical models based on systems of ordinary differential equations (ODEs) are frequently applied in various scientific fields to assess hypotheses, estimate key model parameters, and generate predictions about the system's state. To support their application, we present a comprehensive, easy‐to‐use, and flexible MATLAB toolbox,QuantDiffForecast, and associated tutorial to estimate parameters and generate short‐term forecasts with quantified uncertainty from dynamical models based on systems of ODEs. We provide software (https://github.com/gchowell/paramEstimation_forecasting_ODEmodels/) and detailed guidance on estimating parameters and forecasting time‐series trajectories that are characterized using ODEs with quantified uncertainty through a parametric bootstrapping approach. It includes functions that allow the user to infer model parameters and assess forecasting performance for different ODE models specified by the user, using different estimation methods and error structures in the data. The tutorial is intended for a diverse audience, including students training in dynamic systems, and will be broadly applicable to estimate parameters and generate forecasts from models based on ODEs. The functions included in the toolbox are illustrated using epidemic models with varying levels of complexity applied to data from the 1918 influenza pandemic in San Francisco. A tutorial video that demonstrates the functionality of the toolbox is included.more » « less
-
Abstract Time-periodic dynamical systems occur commonly both in nature and as engineered systems. Large-scale linear time-periodic dynamical systems, for example, may arise through linearization of a nonlinear system about a given periodic solution (possibly as a consequence of a baseline periodic forcing) with subsequent spatial discretization. The potential need to simulate responses to a wide variety of input profiles (viewed as perturbations off a baseline periodic forcing) creates a potent incentive for effective model reduction strategies applicable to linear time-periodic (ltp) systems. Classical approaches that take into account the underlying time-periodic system structure often utilize the Floquet transform; however, computation of the Floquet transform is typically intractable for large-order systems. In this paper, we develop the notion of apartial Floquet transformationconnected to selected invariant subspaces of a time-varying differential operator associated with theltpsystem. We modify and repurpose theDominant Pole Algorithmof Rommes to identify effective invariant subspaces useful for model reduction. We discuss the construction of associated partial Floquet transformations and time-varying reduction bases with which to produce effective reduced-orderltpmodels and illustrate the process on a simple time-periodic system.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript
- Journal Article:
- https://doi.org/10.1088/1361-6420/ad1fe5
