arXiv subjects

Animikh Biswas

Publications and source records attributed to Animikh Biswas.

At least 19 recordsLinked to original sources

arXiv2026

Data-driven techniques for translational neuroscience and personalized neuro-health

Neurodegenexrative diseases such as Alzheimer's disease and Parkinson's disease are diagnosed most reliably only after substantial, often irreversible, neuronal loss has already occurred, creating an urgent need for quantitative tools that can detect subtle, early, and individual-specific brain changes from neuroimaging data. This review surveys a broad and rapidly evolving toolkit of data-driven techniques for translational neuroscience and personalized neuro-health, organized around four complementary methodological pillars. Throughout, we emphasize how these methodologically diverse approaches converge on a common translational goal: personalized, mechanistically grounded, and clinically actionable models of individual brain health, and we close by discussing the principal open statistical, computational, and clinical challenges that remain.

q-bio.NC↗

arXiv2026

A Quantum Optimization Framework for Data-Assimilation-Augmented Parameter Estimation

Parameter estimation is a fundamental challenge in the calibration of ordinary differential equation (ODE) models, where repeated numerical integration can lead to high computational cost. In this work, we investigate whether quantum algorithms can be leveraged to assist parameter estimation in nonlinear dynamical systems. We develop a hybrid classical-quantum framework that reformulates a data-assimilation-augmented parameter estimation problem as a combinatorial optimization task. Model dynamics and data assimilation are enforced entirely on the classical side, while the resulting parameter estimation cost functional is discretized and approximated by a quadratic unconstrained binary optimization (QUBO) surrogate. This surrogate is mapped to an Ising Hamiltonian, and quantum optimizers are used to search for low-energy configurations corresponding to candidate parameter estimates. We apply the framework to SIS and SIR epidemic models, the chaotic Lorenz-63 system, and a high-dimensional two-layer Lorenz-96 system. In this setting, the method is used to recover classical system parameters from partial state observations across steady-state, chaotic, and high-dimensional multiscale dynamical systems. Numerical experiments with synthetic data show that the proposed approach accurately recovers parameters while requiring data-assimilation solves only on a prescribed coarse grid. The framework avoids quantum state tomography, illustrating a viable pathway for integrating quantum optimization into data-driven parameter estimation for nonlinear dynamical systems.

quant-ph↗

arXiv2026

Data Assimilation for Chemical Reaction Networks and Population Models via a Tunable Observer

We consider the problem of state reconstruction for a nonlinear dynamical system from observations of a linear function of the state. We present a design method for a tunable observer and provide a general theorem which under certain conditions guarantees exponential convergence of the observer regardless of initial error. Additional results are provided that apply this theorem to chemical reaction network models. Moreover, these results are illustrated via examples of mass action form of chemical reaction networks where a subset of the species concentrations are observed. Numerical results are provided to show the efficacy of our proposed observer. Numerical results are also shown for the case of noisy observations and our observer is compared favorably with the particle filter when the observation noise is small.

math.DS↗

arXiv2026

A Data-Assimilation-Augmented Optimization Framework for Parameter Estimation in Dynamical Systems

Parameter estimation in nonlinear dynamical systems from observational data is a fundamental inverse problem with applications in many disciplines. In practice, this is further complicated by the fact that observations are often noisy, sparse, and available only for a subset of the state variables. Furthermore, the initial condition (IC) may be unknown or inaccurate, causing further complications for chaotic systems with sensitive dependence on initial conditions. In this work, we develop a data-assimilation-augmented optimization framework for parameter estimation in ordinary differential equations using partial state observations. The method introduces a nudged system driven by the available observed component and estimates the unknown parameters by minimizing a cost functional, defined as a time-delayed mismatch between the observations and the corresponding observed component of the nudged solution over the admissible parameter space. Since the nudged system can be arbitrarily initialized, this approach eliminates the dependence on accurate IC. Using the Lorenz-63 system as a test case, we establish theoretical results showing synchronization of the nudged solution under parameter agreement, stability under parameter mismatch, and well-posedness of the data-to-parameter inverse map under suitable nondegeneracy conditions. Structural & practical identifiability, and Sobol sensitivity analyses are incorporated to assess which parameters can be reliably estimated from the observations. Numerical experiments in both chaotic and non-chaotic regimes show that this framework accurately recovers parameters from noisy partial observations. Comparisons with an on-the-fly parameter learning method and with Bayesian MCMC estimation demonstrate that the proposed method remains accurate under partial observations and higher noise levels while requiring substantially lower computational cost.

math.DS↗

arXiv2026

Reformulation and Interpretation of the Regularity Criterion for 3D NSE Based on Finitely Many Observations

We revisit and sharpen a recent observable regularity criterion for the three-dimensional Navier-Stokes equations on the periodic cube by requiring only finitely many measurements of the flow on a given time interval. Two data models are treated: (i) modal observations (a finite set of low Fourier modes), and (ii) nodal observations, i.e. values of the velocity field sampled at finitely many points on a uniform grid. The key upgrade is a piecewise linear interpolant built on a fixed five-tetrahedra subdivision of each grid cube, which removes the mollification step used previously and yields an explicit control of the derivatives of the interpolation operator purely in terms of the measured data. The criterion is also shown to be both necessary and sufficient for regularity.

math.AP↗

arXiv2024

A unified framework for the analysis of accuracy and stability of a class of approximate Gaussian filters for the Navier-Stokes Equations

Bayesian state estimation of a dynamical system utilising a stream of noisy measurements is important in many geophysical and engineering applications. We establish rigorous results on (time-asymptotic) accuracy and stability of these algorithms with general covariance and observation operators. The accuracy and stability results for EnKF and EnSRKF for dissipative PDEs are, to the best of our knowledge, completely new in this general setting. It turns out that a hitherto unexploited cancellation property involving the ensemble covariance and observation operators and the concept of covariance localization in conjunction with covariance inflation play a pivotal role in the accuracy and stability for EnKF and EnSRKF. Our approach also elucidates the links, via determining functionals, between the approximate-Bayesian and control-theoretic approaches to data assimilation. We consider the `model' dynamics governed by the two-dimensional incompressible Navier-Stokes equations and observations given by noisy measurements of averaged volume elements or spectral/modal observations of the velocity field. In this setup, several continuous-time data assimilation techniques, namely the so-called 3DVar, EnKF and EnSRKF reduce to a stochastically forced Navier-Stokes equations. For the first time, we derive conditions for accuracy and stability of EnKF and EnSRKF. The derived bounds are given for the limit supremum of the expected value of the $L^2$ norm and of the $\mathbb{H}^1$ Sobolev norm of the difference between the approximating solution and the actual solution as the time tends to infinity. Moreover, our analysis reveals an interplay between the resolution of the observations associated with the observation operator underlying the data assimilation algorithms and covariance inflation and localization which are employed in practice for improved filter performance.

math.AP↗

arXiv2022

A Novel Regularity Criterion For The three-dimensional Navier-Stokes Equations Based On Finitely many observations

In this paper we present two results: (1) A data assimilation algorithm for the 3D Navier-Stokes equation (3D NSE) using nodal data, and, as a consequence (2) a novel regularity criterion for the 3D NSE based on finitely many observations of the velocity. The data assimilation algorithm we employ utilizes nudging, a method based on a Newtonian relaxation scheme motivated by feedback-control. The observations, which may be either modal, nodal or volume elements, are drawn from a weak solution of the 3D NSE and are collected almost everywhere in time over a finite grid and our results, including the regularity criterion, hold for data of any of the aforementioned forms. The regularity criterion we propose follows from our data assimilation algorithm and is hence intimately connected to the notion of determining functionals (modes, nodes and volume elements). To the best of our knowledge, all existing regularity criteria require knowing the solution of the 3D NSE almost everywhere in space. Our regularity criterion is fundamentally different from any preexisting regularity criterion as it is based on finitely many observations (modes, nodes and volume elements). We further prove that the regularity criterion we propose is both a necessary and sufficient condition for regularity. Thus our result can be viewed as a natural generalization of the notion of determining modes, nodes and volume elements as well as the asymptotic tracking property of the nudging algorithm for the 2D NSE to the 3D setting.

math.AP↗

arXiv2022

Determining the viscosity of the Navier-Stokes equations from observations of finitely many modes

In this work, we ask and answer the question: when is the viscosity of a fluid uniquely determined from spatially sparse measurements of its velocity field? We pose the question mathematically as an optimization problem using the determining map (the mapping of data to an approximation made via a nudging algorithm) to define a loss functional, the minimization of which solves the inverse problem of identifying the true viscosity given the measurement data. We give explicit a priori conditions for the well-posedness of this inverse problem. In addition, we show that smallness of the loss functional implies proximity to the true viscosity. We then present an algorithm for solving the inverse problem and prove its convergence.

math.AP↗

arXiv2022

Convergence of a mobile data assimilation scheme for the 2D Navier-Stokes equations

We introduce a localized version of the nudging data assimilation algorithm for the periodic 2D Navier-Stokes equations in which observations are confined (i.e., localized) to a window that moves across the entire domain along a predetermined path at a given speed. We prove that, if the movement is fast enough, then the algorithm perfectly synchronizes with a reference solution. The analysis suggests an informed scheme in which the subdomain moves according to a region where the error is dominant is optimal. Numerical simulations are presented that compare the efficacy of movement that follows a regular pattern, one guided by the dominant error, and one that is random.

math.AP↗

arXiv2021

Joint modeling of geometric features of longitudinal process and discrete survival time measured on nested timescales: an application to fecundity studies

In biomedical studies, longitudinal processes are collected till time-to-event, sometimes on nested timescales (example, days within months). Most of the literature in joint modeling of longitudinal and time-to-event data has focused on modeling the mean or dispersion of the longitudinal process with the hazard for time-to-event. However, based on the motivating studies, it may be of interest to investigate how the cycle-level {\it geometric features} (such as the curvature, location and height of a peak), of a cyclical longitudinal process is associated with the time-to-event being studied. We propose a shared parameter joint model for a cyclical longitudinal process and a discrete survival time, measured on nested timescales, where the cycle-varying geometric feature is modeled through a linear mixed effects model and a proportional hazards model for the discrete survival time. The proposed approach allows for prediction of survival probabilities for future subjects based on their available longitudinal measurements. Our proposed model and approach is illustrated through simulation and analysis of Stress and Time-to-Pregnancy, a component of Oxford Conception Study. A joint modeling approach was used to assess whether the cycle-specific geometric features of the lutenizing hormone measurements, such as its peak or its curvature, are associated with time-to-pregnancy (TTP).

stat.ME↗

arXiv2021

Mesh-Free Interpolant Observables for Continuous Data Assimilation

This paper considers a nudging-based scheme for data assimilation for the two-dimensional (2D) Navier-Stokes equations (NSE) with periodic boundary conditions and studies the synchronization of the signal produced by this algorithm with the true signal, to which the observations correspond, in all higher-order Sobolev topologies. This work complements previous results in the literature where conditions were identified under which synchronization is guaranteed either with respect to only the $H^1$--topology, in the case of general observables, or to the analytic Gevrey topology, in the case of spectral observables. To accommodate the property of synchronization in the stronger topologies, the framework of general interpolant observable operators, originally introduced by Azouani, Olson, and Titi, is expanded to a far richer class of operators. A significant effort is dedicated to the development of this more expanded framework, specifically, their basic approximation properties, the identification of subclasses of such operators relevant to obtaining synchronization, as well as the detailed relation between the structure of these operators and the system regarding the syncrhonization property. One of the main features of this framework is its "mesh-free" aspect, which allows the observational data itself to dictate the subdivision of the domain. Lastly, estimates for the radius of the absorbing ball of the 2D NSE in all higher-order Sobolev norms are obtained, thus properly generalizing previously known bounds; such estimates are required for establishing the synchronization property of the algorithm in the higher-order topologies.

math.AP↗

arXiv2021

Determining Map, Data Assimilation and an Observable Regularity Criterion for the Three-Dimensional Boussinesq System

In this paper, we provide conditions, \emph{based solely on the observed velocity data}, for the global well-posedness, regularity and convergence of the Azouni-Olson-Titi data assimilation algorithm (AOT algorithm) for a Leray-Hopf weak solutions of the three dimensional Boussinesq system. This condition also guarantees the construction of the {\it determining map}. The aforementioned conditions on the (finite-dimensional) velocity observations, which in this case comprise either of a finite-dimensional \emph{modal} projection or finitely many \emph{volume element observations}, are automatically satisfied for solutions that are globally regular and are uniformly bounded in the $H^1$-norm. However, neither regularity nor uniqueness is {\it a priori} assumed on the solutions. To the best of our knowledge, this is the first such rigorous analysis of the AOT data assimilation algorithm for the three-dimensional Boussinesq system. As a corollary, we obtain that the condition that we imposed is in fact {\it a new observable regularity criterion on the weak global attractor.} The proof of this fact proceeds through the construction of the determining map.

math.AP↗

arXiv2020

Data assimilation for the Navier-Stokes equations using local observables

We develop, analyze, and test an approximate, global data assimilation/synchronization algorithm based on purely local observations for the two-dimensional Navier-Stokes equations on the torus. We prove that, for any error threshold, if the reference flow is analytic with sufficiently large analyticity radius, then it can be recovered within that threshold. Numerical computations are included to demonstrate the effectiveness of this approach, as well as variants with data on moving subdomains. In particular, we demonstrate numerically that machine precision synchronization is achieved for mobile data collected from a small fraction of the domain.

math.AP↗

arXiv2020

Error Estimates for Deep Learning Methods in Fluid Dynamics

In this study, we provide error estimates and stability analysis of deep learning techniques for certain partial differential equations including the incompressible Navier-Stokes equations. In particular, we obtain explicit error estimates (in suitable norms) for the solution computed by optimizing a loss function in a Deep Neural Network (DNN) approximation of the solution, with a fixed complexity.

math.AP↗

arXiv2020

Continuous Data Assimilation for the Three Dimensional Navier-Stokes Equations

In this paper, we provide conditions, \emph{based solely on the observed data}, for the global well-posedness, regularity and convergence of the Azouni-Olson-Titi data assimilation algorithm (AOT algorithm) for a Leray-Hopf weak solutions of the three dimensional Navier-Stokes equations (3D NSE). The aforementioned conditions on the observations, which in this case comprise either of \emph{modal} or \emph{volume element observations}, are automatically satisfied for solutions that are globally regular and are uniformly bounded in the $H^1$-norm. However, neither regularity nor uniqueness is necessary for the efficacy of the AOT algorithm. To the best of our knowledge, this is the first such rigorous analysis of the AOT data assimilation algorithm for the 3D NSE.

math.AP↗

arXiv2019

Persistence time of solutions of the three-dimensional Navier-Stokes equations in Sobolev-Gevrey classes

In this paper, we study existence times of strong solutions of the three-dimensional Navier-Stokes equations in time-varying analytic Gevrey classes based on Sobolev spaces $H^s, s> \frac{1}{2}$. This complements the seminal work of Foias and Temam (1989) on $H^1$ based Gevrey classes, thus enabling us to improve estimates of the analyticity radius of solutions for certain classes of initial data. The main thrust of the paper consists in showing that the existence times in the much stronger Gevrey norms (i.e. the norms defining the analytic Gevrey classes which quantify the radius of real-analyticity of solutions) match the best known persistence times in Sobolev classes. Additionally, as in the case of persistence times in the corresponding Sobolev classes, our existence times in Gevrey norms are optimal for $\frac{1}{2} < s < \frac{5}{2}$.

math-ph↗

arXiv2018

Downscaling data assimilation algorithm with applications to statistical solutions of the Navier-Stokes equations

Based on a previously introduced downscaling data assimilation algorithm, which employs a nudging term to synchronize the coarse mesh spatial scales, we construct a determining map for recovering the full trajectories from their corresponding coarse mesh spatial trajectories, and investigate its properties. This map is then used to develop a downscaling data assimilation scheme for statistical solutions of the two-dimensional Navier-Stokes equations, where the coarse mesh spatial statistics of the system is obtained from discrete spatial measurements. As a corollary, we deduce that statistical solutions for the Navier-Stokes equations are determined by their coarse mesh spatial distributions. Notably, we present our results in the context of the Navier-Stokes equations; however, the tools are general enough to be implemented for other dissipative evolution equations.

math.AP↗

arXiv2017

Space and time analyticity for inviscid equations of fluid dynamics

We show that solutions to a large class of inviscid equations, in Eulerian variables, extend as holomorphic functions of time, with values in a Gevrey class (thus space-analytic), and are solutions of complexified versions of the said equations. The class of equations we consider includes those of fluid dynamics such as the Euler, surface quasi-geostrophic, Boussinesq and magnetohydrodynamic equations, as well as other equations with analytic nonlinearities. The initial data are assumed to belong to a $\mathit{Gevrey\:class}$, i.e., analytic in the space variable. Our technique follows that of the seminal work of Foias and Temam (1989), where they introduced the so-called Gevrey class technique for the Navier-Stokes equations to show that the solutions of the Navier-Stokes equations extend as holomorphic functions of time, in a complex neighborhood of $(0,T)$, with values in a Gevrey class of functions (in the space variable). We show a similar result for a wide class of inviscid models, while obtaining an $\mathit{explicit \: estimate \: of \: the \: domain \: of \: analyticity}$.

math.AP↗