arXiv subjects

Andrea Arnold

Publications and source records attributed to Andrea Arnold.

14 recordsLinked to original sources

arXiv2026

Dynamics-Encoded Deep Learning for Robust System Identification and Parameter Estimation

Incorporating a priori physics knowledge into machine learning leads to more robust and interpretable algorithms. In this work, we combine deep learning techniques and classic numerical methods for differential equations to address two challenging missing physics problems in dynamical systems theory: dynamics discovery and parameter estimation. The presented methods encode available information relating to the system dynamics into deep learning architectures, incorporating different assumptions on the known inputs and desired outputs in each case. Results demonstrate the effectiveness of the proposed approaches in making data-driven model predictions given corrupt system observations on a suite of test problems exhibiting oscillatory and chaotic dynamics. When comparing the performance of various numerical schemes, such as the Runge-Kutta and linear multistep families of methods, we observe promising results in predicting the system dynamics and estimating physical parameters, given appropriate choices of spatial and temporal discretization schemes and numerical method orders.

cs.LG↗

arXiv2025

A Systematic Particle Filter for Estimating Time-Varying Parameters in Advection-Diffusion Equations with Source Terms

Many real-world systems modeled using partial differential equations (PDEs) involve unknown parameters that must be estimated from limited, noisy system observations. While typically assumed to be constants, some of these unobserved parameters may vary with time. This work proposes a two-phase, offline-online numerical procedure for systematically estimating and quantifying uncertainty in time-varying parameters (TVPs) in time-dependent PDEs, specifically focusing on advection-diffusion models with TVPs involved in the source terms. Numerical results on a set of one-dimensional test problems demonstrate the effectiveness of the proposed estimation procedure in tracking unknown TVPs of different forms, while simultaneously estimating variance parameters affecting the TVP evolution model, from partial, noisy observations of the solution at discrete spatial locations and times.

stat.ME↗

arXiv2025

Optimal Linear Baseline Models for Scientific Machine Learning

Across scientific domains, a fundamental challenge is to characterize and compute the mappings from underlying physical processes to observed signals and measurements. While nonlinear neural networks have achieved considerable success, they remain theoretically opaque, which hinders adoption in contexts where interpretability is paramount. In contrast, linear neural networks serve as a simple yet effective foundation for gaining insight into these complex relationships. In this work, we develop a unified theoretical framework for analyzing linear encoder-decoder architectures through the lens of Bayes risk minimization for solving data-driven scientific machine learning problems. We derive closed-form, rank-constrained linear and affine linear optimal mappings for forward modeling and inverse recovery tasks. Our results generalize existing formulations by accommodating rank-deficiencies in data, forward operators, and measurement processes. We validate our theoretical results by conducting numerical experiments on datasets from simple biomedical imaging, financial factor analysis, and simulations involving nonlinear fluid dynamics via the shallow water equations. This work provides a robust baseline for understanding and benchmarking learned neural network models for scientific machine learning problems.

cs.LG↗

arXiv2025

Data-driven modeling and prediction of microglial cell dynamics in the ischemic penumbra

Neuroinflammation immediately follows the onset of ischemic stroke. During this process, microglial cells are activated in and recruited to the tissue surrounding the irreversibly injured infarct core, referred to as the penumbra. Microglial cells can be activated into two distinct phenotypes; however, the dynamics between the detrimental M1 phenotype and beneficial M2 phenotype are not fully understood. Using phenotype-specific cell count data obtained from experimental studies on middle cerebral artery occlusion-induced stroke in mice, we employ sparsity-promoting system identification techniques combined with Bayesian statistical methods for uncertainty quantification to generate continuous and discrete-time predictive models of the M1 and M2 microglial cell dynamics. The resulting data-driven models include constant and linear terms but do not include nonlinear interactions between the cells. Results emphasize an initial M2 dominance followed by a takeover of M1 cells, capture potential long-term dynamics of microglial cells, and suggest a persistent inflammatory response.

q-bio.CB↗

arXiv2024

A data-informed mathematical model of microglial cell dynamics during ischemic stroke in the middle cerebral artery

Neuroinflammation immediately follows the onset of ischemic stroke in the middle cerebral artery. During this process, microglial cells are activated in and recruited to the penumbra. Microglial cells can be activated into two different phenotypes: M1, which can worsen brain injury; or M2, which can aid in long-term recovery. In this study, we contribute a summary of experimental data on microglial cell counts in the penumbra following ischemic stroke induced by middle cerebral artery occlusion (MCAO) in mice and compile available data sets into a single set suitable for time series analysis. Further, we formulate a mathematical model of microglial cells in the penumbra during ischemic stroke due to MCAO. Through use of global sensitivity analysis and Markov Chain Monte Carlo (MCMC)-based parameter estimation, we analyze the effects of the model parameters on the number of M1 and M2 cells in the penumbra and fit identifiable parameters to the compiled experimental data set. We utilize results from MCMC parameter estimation to ascertain uncertainty bounds and forward predictions for the number of M1 and M2 microglial cells over time. Results demonstrate the significance of parameters related to M1 and M2 activation on the number of M1 and M2 microglial cells. Simulations further suggest that potential outliers in the observed data may be omitted and forecast predictions suggest a lingering inflammatory response.

q-bio.CB↗

arXiv2023

Fourier Series-Based Approximation of Time-Varying Parameters in Ordinary Differential Equations

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.

math.DS↗

arXiv2022

When Artificial Parameter Evolution Gets Real: Particle Filtering for Time-Varying Parameter Estimation in Deterministic Dynamical Systems

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).

stat.ME↗

arXiv2021

Identification of Tissue Optical Properties During Thermal Laser-Tissue Interactions: An Ensemble Kalman Filter-Based Approach

In this paper, we propose a computational framework to estimate the physical properties that govern the thermal response of laser-irradiated tissue. We focus in particular on two quantities, the absorption and scattering coefficients, which describe the optical absorption of light in the tissue and whose knowledge is vital to correctly plan medical laser treatments. To perform the estimation, we utilize an implementation of the Ensemble Kalman Filter (EnKF), a type of Bayesian filtering algorithm for data assimilation. Unlike prior approaches, in this work we estimate the tissue optical properties based on observations of the tissue thermal response to laser irradiation. This method has the potential for straightforward implementation in a clinical setup, as it would only require a simple thermal sensor, e.g., a miniaturized infrared camera. Because the optical properties of tissue can undergo shifts during laser exposure, we employ a variant of EnKF capable of tracking time-varying parameters. Through simulated experimental studies, we demonstrate the ability of the proposed technique to identify the tissue optical properties and track their dynamic changes during laser exposure, while simultaneously tracking changes in the tissue temperature at locations beneath the surface. We further demonstrate the framework's capability in estimating additional unknown tissue properties (i.e., the volumetric heat capacity and thermal conductivity) along with the optical properties of interest.

physics.med-ph↗

arXiv2021

Analyzing the Effects of Observation Function Selection in Ensemble Kalman Filtering for Epidemic Models

The Ensemble Kalman Filter (EnKF) is a popular sequential data assimilation method that has been increasingly used for parameter estimation and forecast prediction in epidemiological studies. The observation function plays a critical role in the EnKF framework, connecting the unknown system variables with the observed data. Key differences in observed data and modeling assumptions have led to the use of different observation functions in the epidemic modeling literature. In this work, we present a novel computational analysis demonstrating the effects of observation function selection when using the EnKF for state and parameter estimation in this setting. In examining the use of four epidemiologically-inspired observation functions of different forms in connection with the classic Susceptible-Infectious-Recovered (SIR) model, we show how incorrect observation modeling assumptions (i.e., fitting incidence data with a prevalence model, or neglecting under-reporting) can lead to inaccurate filtering estimates and forecast predictions. Results demonstrate the importance of choosing an observation function that well interprets the available data on the corresponding EnKF estimates in several filtering scenarios, including state estimation with known parameters, and combined state and parameter estimation with both constant and time-varying parameters. Numerical experiments further illustrate how modifying the observation noise covariance matrix in the filter can help to account for uncertainty in the observation function in certain cases.

stat.ME↗

arXiv2020

Modeling Microglia Activation and Inflammation-Based Neuroprotectant Strategies During Ischemic Stroke

Neural inflammation immediately follows the onset of ischemic stroke. During this process, microglial cells can be activated into two different phenotypes: the M1 phenotype, which can worsen brain injury by producing pro-inflammatory cytokines; or the M2 phenotype, which can aid in long term recovery by producing anti-inflammatory cytokines. In this study, we formulate a nonlinear system of differential equations to model the activation of microglia post-ischemic stroke, which includes bidirectional switching between the microglia phenotypes, as well as the interactions between these cells and the cytokines that they produce. Further, we explore neuroprotectant-based modeling strategies to suppress the activation of the detrimental M1 phenotype, while promoting activation of the beneficial M2 phenotype. Through use of global sensitivity techniques, we analyze the effects of the model parameters on the ratio of M1 to M2 microglia and the total number of activated microglial cells in the system over time. Results demonstrate the significance of bidirectional microglia phenotype switching on the ratio of M1 to M2 microglia, in both the absence and presence of neuroprotectant terms. Simulations further suggest that early inhibition of M1 activation and support of M2 activation leads to a decreased minimum ratio of M1 to M2 microglia and allows for a larger number of M2 than M1 cells for a longer time period.

q-bio.CB↗

arXiv2019

Estimating Time-Varying Applied Current in the Hodgkin-Huxley Model

The classic Hodgkin-Huxley model is widely used for understanding the electrophysiological dynamics of a single neuron. While applying a constant current to the system results in a single voltage spike, it is possible to produce more interesting dynamics by applying time-varying currents, which may not be experimentally measurable. The aim of this work is to estimate time-varying applied currents of different deterministic forms given noisy voltage data. In particular, we utilize an augmented ensemble Kalman filter with parameter tracking to estimate four different deterministic applied currents, analyzing how the model dynamics change in each case. We test the efficiency of the parameter tracking algorithm in this setting by exploring the effects of changing the standard deviation of the parameter drift and the frequency of data available on the resulting time-varying applied current estimates and related uncertainty.

q-bio.QM↗

arXiv2018

An approach to periodic, time-varying parameter estimation using nonlinear filtering

Many systems arising in biological applications are subject to periodic forcing. In these systems the forcing parameter is not only time-varying but also known to have a periodic structure. We present an approach to estimating periodic, time-varying parameters that imposes periodic structure by treating the time-varying parameter as a piecewise function with unknown coefficients. This method allows the resulting parameter estimate more flexibility in shape than prescribing a specific functional form (e.g., sinusoidal) to model its behavior, while still maintaining periodicity. We employ nonlinear filtering, more specifically, a version of the augmented ensemble Kalman filter (EnKF), to estimate the unknown coefficients comprising the piecewise approximation of the periodic, time-varying parameter. This allows for straightforward comparison of the proposed method with an EnKF-based parameter tracking algorithm, where periodicity is not guaranteed. We demonstrate the effectiveness of the proposed approach on two biological examples: a synthetic example with data generated from the nonlinear FitzHugh-Nagumo system, modeling the excitability of a nerve cell, to estimate the external voltage parameter, and a case study using reported measles incidence data from three locations during the pre-vaccine era to estimate the seasonal transmission parameter. The formulation of the proposed approach also allows for simultaneous estimation of initial conditions and other static system parameters, such as the reporting probability of measles cases, which is vital for predicting under-reported incidence data.

q-bio.QM↗

arXiv2017

Practical Identifiability and Uncertainty Quantification of a Pulsatile Cardiovascular Model

Mathematical models are essential tools to study how the cardiovascular system maintains homeostasis. The utility of such models is limited by the accuracy of their predictions, which can be determined by uncertainty quantification (UQ). A challenge associated with the use of UQ is that many published methods assume that the underlying model is identifiable (e.g. that a one-to-one mapping exists from the parameter space to the model output). In this study we present a novel methodology that is used here to calibrate a lumped-parameter model to left ventricular pressure and volume time series data sets. Key steps include using (1) literature and available data to determine nominal parameter values; (2) sensitivity analysis and subset selection to determine a set of identifiable parameters; (3) optimization to find a point estimate for identifiable parameters; and (4) frequentist and Bayesian UQ calculations to assess the predictive capability of the model. Our results show that it is possible to determine 5 identifiable model parameters that can be estimated to our experimental data from three rats, and that computed UQ intervals capture the measurement and model error.

q-bio.QM↗

arXiv2014

Vectorized and Parallel Particle Filter SMC Parameter Estimation for Stiff ODEs

Particle filter (PF) sequential Monte Carlo (SMC) methods are very attractive for the estimation of parameters of time dependent systems where the data is either not all available at once, or the range of time constants is wide enough to create problems in the numerical time propagation of the states. The need to evolve a large number of particles makes PF-based methods computationally challenging, the main bottlenecks being the time propagation of each particle and the large number of particles. While parallelization is typically advocated to speed up the computing time, vectorization of the algorithm on a single processor may result in even larger speedups for certain problems. In this paper we present a formulation of the PF-SMC class of algorithms proposed in Arnold et al. (2013), which is particularly amenable to a parallel or vectorized computing environment, and we illustrate the performance with a few computed examples in MATLAB.

stat.CO↗