Unknown Diffusion Coefficient Research Articles

Learning the flux and diffusion function for degenerate convection-diffusion equations using different types of observations

In recent years, there has been an increasing interest in utilizing deep learning-based techniques to predict solutions to various partial differential equations. In this study, we investigate the identification of an unknown flux function and diffusion coefficient in a one-dimensional convection-diffusion equation. The diffusion function is allowed to vanish on intervals implying that solutions generally possess low regularity, i.e., are discontinuous. Therefore, solutions must be interpreted in the sense of entropy solutions which combine a weak formulation with an additional constraint (entropy condition). We explore a methodology that utilizes symbolic neural networks (S-Nets) in combination with an entropy-consistent discrete numerical scheme (ECDNS). Different types of observation data are explored. Extensive experiments in this paper demonstrate that the proposed method is a robust tool to identify the unknown flux and diffusion function. The flux and diffusion functions are restricted to be rational functions.

Unveiling moisture transport mechanisms in cellulosic materials: Vapor vs. bound water.

Natural textiles, hair, paper, wool, or bio-based walls possess the remarkable ability to store humidity from sweat or the environment through "bound water" absorption within nanopores, constituting up to 30% of their dry mass. The knowledge of the induced water transfers is pivotal for advancing industrial processes and sustainable practices in various fields such as wood drying, paper production and use, moisture transfers in clothes or hair, humidity regulation of bio-based construction materials, etc. However, the transport and storage mechanisms of this moisture remain poorly understood, with modeling often relying on an assumption of dominant vapor transport with an unknown diffusion coefficient. Our research addresses this knowledge gap, demonstrating the pivotal role of bound water transport within interconnected fiber networks. Notably, at low porosity, bound water diffusion dominates over vapor diffusion. By isolating diffusion processes and deriving diffusion coefficients through rigorous experimentation, we establish a comprehensive model for moisture transfer. Strikingly, our model accurately predicts the evolution of bound water's spatial distribution for a wide range of sample porosities, as verified through magnetic resonance imaging. Showing that bound water transport can be dominant over vapor transport, this work offers a change of paradigm and unprecedented control over humidity-related processes.

Identification of diffusion coefficient in a semi-linear parabolic equation with incomplete initial condition: a no-regret control method

This paper addresses the problem of identifying the unknown diffusion coefficient in a semi-linear parabolic equation with incomplete initial condition. We propose an optimal control approach using the no-regret control method and the adapted low-regret control. Our approach provides a full characterisation of the unknown diffusion coefficient independent of the missing initial condition. We also present an optimality system that describes the adapted low-regret control and use it to find a full description of the no-regret control by taking the limit of the sequence of adapted low-regret controls.

Hybrid neural-network FEM approximation of diffusion coefficient in elliptic and parabolic Problems

Abstract In this work we investigate the numerical identification of the diffusion coefficient in elliptic and parabolic problems using neural networks (NNs). The numerical scheme is based on the standard output least-squares formulation where the Galerkin finite element method (FEM) is employed to approximate the state and NNs act as a smoothness prior to approximate the unknown diffusion coefficient. A projection operation is applied to the NN approximation in order to preserve the physical box constraint on the unknown coefficient. The hybrid approach enjoys both rigorous mathematical foundation of the FEM and inductive bias/approximation properties of NNs. We derive a priori error estimates in the standard $L^2(\varOmega )$ norm for the numerical reconstruction, under a positivity condition which can be verified for a large class of problem data. The error bounds depend explicitly on the noise level, regularization parameter and discretization parameters (e.g., spatial mesh size, time step size and depth, upper bound and number of nonzero parameters of NNs). We also provide extensive numerical experiments, indicating that the hybrid method is very robust for large noise when compared with the pure FEM approximation.

Utilising physics-informed neural networks for optimisation of diffusion coefficients in pseudo-binary diffusion couples

ABSTRACT We propose a numerical inverse method based on physics-informed neural networks (PINN) for calculating composition-dependent diffusion coefficients in pseudo-binary (PB) diffusion couples in multicomponent alloys. Traditional methods rely solely on experimental diffusion profiles as design targets, which can lead to unreliable estimates. In contrast, PINN uses a combination of available data and physics-based constraints to obtain optimised design parameters and exact solutions for constrained governing differential equations. The constraints in PINN include governing partial differential equations, initial and boundary conditions, and any other equality/inequality relations obeyed by physical parameters. Our study shows the necessity of experimentally estimated intrinsic diffusion coefficients for the prediction of reliable composition-dependent mobility data. In the absence of such data, different combinations of unknown intrinsic diffusion coefficients can also produce reasonable approximations of diffusion profiles and interdiffusion coefficients while wrongly predicting more fundamental quantities (i.e. intrinsic diffusivities). Our method utilises PINN to simultaneously obtain optimised design parameters (diffusion coefficients) and exact solutions for governing diffusion equations. The implementation of PINN uses experimentally estimated diffusion coefficients obtained by the PB diffusion couple method, in addition to diffusion profiles, as design targets. The method is extended to non-ideal PB diffusion profiles, such as conventional diffusion couples in which all the components develop diffusion profiles, by incorporating additional constraints such as zero composition gradient of certain component(s) and experimentally estimated interdiffusion coefficients at extrema in the composition profiles. PINN is found to be a promising approach for obtaining reliable estimates of diffusion coefficients in multicomponent alloys.

A physics-informed neural network based simulation tool for reacting flow with multicomponent reactants

The reacting flow involves nonlinear partial differential equations which are mostly solved numerically for autocatalytic reaction between multi reactants. One-dimensional convection-diffusion-reaction equations are widely used for modeling reacting flows. This paper developed a method to solve the convection-diffusion-reaction equations using a neural network model, called the physics-informed neural network. It was applied to solve the nonlinear reacting flow for forward as well as inverse problems. To enhance accuracy and to eliminate non-physical oscillations, a derivative constraint of the spatial dimension was employed. In order to avoid the slow training speed caused by the derivative constraint, a new neural network connection structure is proposed. For validation, three different forward reacting flow problems involving steep fronts were analyzed, including the convection reaction, and steady and unsteady convection-diffusion-reaction systems. For the inverse problem, except for the kinetic parameters, the unknown convection velocity and diffusion coefficients were correctly predicted, demonstrating the capability of the proposed method for identifying reacting flow parameters. Moreover, under the guidance of the improved algorithm, the sensitivity of the reacting flow to certain kinetic parameter was investigated.

Identifying an unknown coefficient in the fractional parabolic differential equation

In this study, we considered a fractional parabolic equation for identifying the unknown diffusion coefficient from the noisy measurement of the ultimate time solution. It is an inverse problem involving a nonlocal operator that is nonlinear and poorly formulated. By demonstrating the existence of this inverse problem singular solution with regard to the final observed data, we demonstrate the identifiability of this problem. The inverse problem is expressed as a regularized optimization problem that minimizes a cost function of the least-squares kind. We have covered various theoretical and practical difficulties pertaining to the problem under consideration. It has been demonstrated that a unique stable solution to the optimization problem exists. The Morozov discrepancy principle and the conjugate gradient approach are used to develop an iterative reconstruction procedure. The accuracy and efficiency of the suggested method are illustrated using a few numerical examples.

Spatiotemporal adaptive state feedback control of a linear parabolic partial differential equation

AbstractThis article deals with the issue of asymptotic stabilization for a linear parabolic partial differential equation (PDE) with an unknown space‐varying reaction coefficient and multiple local piecewise uniform control. Clearly, the unknown reaction coefficient belongs to a function space. Hence, the fundamental difficulty for such issue lies in the lack of a conceptually simple but effective parameter identification technique in a function space. By the Lyapunov technique combined with a variant of Poincaré‐Wirtinger inequality, an update law is derived for estimate of the unknown reaction coefficient in a function space. Then a spatiotemporal adaptive state feedback control law is constructed such that the estimate of the unknown coefficient is bounded and the closed‐loop PDE is asymptotically stable in the sense of spatial norm if a sufficient condition given in terms of space‐time varying linear matrix inequalities (LMIs) is fulfilled for the estimated coefficient and the control gains. Both analytical and numerical approaches are proposed to construct a feasible solution to the space‐time varying LMI problem. With the aid of the semigroup theory, the well‐posedness and regularity of the closed‐loop PDE is also analyzed. Moreover, two extensions of the proposed adaptive control scheme are discussed: the PDE in ‐D space and the PDE with unknown diffusion and reaction coefficients. Finally, numerical simulation results are presented to support the proposed spatiotemporal adaptive control design.

Identification of a time-dependent diffusivity coefficient in heat-like space-time fractional differential equations

The goal of this research is to reveal the unknown time dependent diffusion coefficient in space-time fractional differential equations by means of fractional Taylor series method. Unlike most methods used in inverse problems, using no over-measured data is a substantial advantage of this method.
 As a result, the unknown diffusion coefficient could be determined with high precision. Illustrative examples shows that the retrieved unknown coefficient and the solution of the problem are in a high agreement with the exact solution of the corresponding the inverse problems.

Convergence rate analysis of the coefficient identification problems in a Kuramoto–Sivashinsky equation

The convergence rate results of the mixed total variation (TV) and L p regularization to the identification of the unknown anti-diffusion coefficient, and the mixed H 2-L p regularization for the identification of the unknown diffusion coefficient in a Kuramoto–Sivashinsky (KS) equation are investigated with some additional observation. Under suitable source conditions, the convergence rates to regularized solutions for the TV-L p regularization of the anti-diffusion coefficient are obtained. Analogously, for the identification of diffusion coefficient with H 2-L p regularization, the convergence rates are derived with proper source conditions. In addition, simple and easily interpretable source conditions are also established to deduce the same convergence rates under some assumptions to the sought coefficients. For the source conditions in both two coefficient identification problems, no restrictive smallness condition to the source functions is required.

Robust nonparametric estimation for the volatility of financial market

The occurrence of a macroeconomic policy would lead to a jump of financial data and the presence of jump behaviors might make the statistical methods for high-frequency sampling data to face new challenges. This paper will use the threshold function technique to disentangle the continuous part and the jump part from the high frequency financial data. Moreover, in the financial practices, the abnormal observations contained in the data could cause bias from nonparametric estimation based on least squares. The paper will employ the local M estimation to provide a robust estimator for the unknown diffusion coefficient of the diffusion model with jumps under high frequency sampling data. Under certain conditions for the initial values, this paper further considers one-step local M estimation for the unknown diffusion coefficient which can reduce the calculation quantity under the estimation efficiency. The Monte Carlo numerical simulation results verify that compared with the local linear threshold estimator, the threshold one-step local M estimator is more accurate and more robust. Finally, the threshold one-step local M estimator in this paper is applied to the Shanghai composite index of 2015 and 2020 in China and the Nasdaq index of 2020 in USA, which illustrates the method considered in this paper possesses good finite sample properties.

Recovery of a space-time-dependent diffusion coefficient in subdiffusion: stability, approximation and error analysis

Abstract In this work we study an inverse problem of recovering a space-time-dependent diffusion coefficient in the subdiffusion model from the distributed observation, where the mathematical model involves a Djrbashian–Caputo fractional derivative of order $\alpha \in (0,1)$ in time. The main technical challenges of both theoretical and numerical analyses lie in the limited smoothing properties due to the fractional differential operator and high degree of nonlinearity of the forward map from the unknown diffusion coefficient to the distributed observation. We establish two conditional stability results using a novel test function, which leads to a stability bound in $L^2(0,T;L^2(\varOmega ))$ under a suitable positivity condition. The positivity condition is verified for a large class of problem data. Numerically, we develop a rigorous procedure for recovering the diffusion coefficient based on a regularized least-squares formulation, which is then discretized by the standard Galerkin method with continuous piecewise linear elements in space and backward Euler convolution quadrature in time. We provide a complete error analysis of the fully discrete formulation, by combining several new error estimates for the direct problem (optimal in terms of data regularity), a discrete version of fractional maximal $L^p$ regularity and a nonstandard energy argument. Under the positivity condition, we obtain a standard $\ell ^2(L^2(\varOmega ))$ error estimate consistent with the conditional stability. Further, we illustrate the analysis with some numerical examples.

Physics-informed Karhunen-Loéve and neural network approximations for solving inverse differential equation problems

We present the PI-CKL-NN method for parameter estimation in differential equation (DE) models given sparse measurements of the parameters and states. In the proposed approach, the space- or time-dependent parameters are approximated by Karhunen-Loéve (KL) expansions that are conditioned on the parameters' measurements, and the states are approximated by deep neural networks (DNNs). The unknown weights in the KL expansions and DNNs are found by minimizing the cost function that enforces the measurements of the states and the DE constraint. Regularization is achieved by adding the l2 norm of the conditional KL coefficients into the loss function. Our approach assumes that the parameter fields are correlated in space or time and enforces the statistical knowledge (the mean and the covariance function) in addition to the DE constraints and measurements as opposed to the physics-informed neural network (PINN) and other similar physics-informed machine learning methods where only DE constraints and data are used for parameter estimation. We use the PI-CKL-NN method for parameter estimation in an ordinary differential equation with an unknown time-dependent parameter and the one- and two-dimensional partial differential diffusion equations with unknown space-dependent diffusion coefficients. We also demonstrate that PI-CKL-NN is more accurate than the PINN method, especially when the observations of the parameters are very sparse.

Time-averaged mean squared displacement ratio test for Gaussian processes with unknown diffusion coefficient.

The time-averaged mean squared displacement (TAMSD) is one of the most common statistics used for the analysis of anomalous diffusion processes. Anomalous diffusion is manifested by non-linear (mostly power-law) characteristics of the process in contrast to normal diffusion where linear characteristics are expected. One can distinguish between sub- and super-diffusive processes. We consider Gaussian anomalous diffusion models and propose a new approach used for their testing. This approach is based on the TAMSD ratio statistic for different time lags. Similar to the TAMSD, this statistic exhibits a specific behavior in the anomalous diffusion regime. Through its structure, it is independent of the diffusion coefficient, which, in general, does not influence anomalous diffusion behavior. Thus, the TAMSD ratio-based approach does not require preliminary knowledge of the diffusion coefficient's value, in contrast to the TAMSD-approach, where this value is crucial in the testing procedure. Based on the quadratic form representation of the TAMSD ratio, we calculate its main characteristics and propose a step-by-step testing procedure that can be applied for any Gaussian process. For the anomalous diffusion model used here, namely, the fractional Brownian motion, we demonstrate the effectiveness of the proposed methodology. We show that the new approach outperforms the TAMSD-based one, especially for small sample sizes. Finally, the methodology is applied to the real data from the financial market.

Threshold Estimation for a Spectrally Negative Lévy Process

Consider a spectrally negative Lévy process with unknown diffusion coefficient and Lévy measure and suppose that the high frequency trading data is given. We use the techniques of threshold estimation and regularized Laplace inversion to obtain the estimator of survival probability for a spectrally negative Lévy process. The asymptotic properties are given for the proposed estimator. Simulation studies are also given to show the finite sample performance of our estimator.

A total variation regularization method for an inverse problem of recovering an unknown diffusion coefficient in a parabolic equation

This paper studies an inverse problem of recovering an unknown diffusion coefficient in a parabolic equation. We adopt a total variation regularization method to deal with the ill-posedness. This method has the advantage to solve problems that the solution is non-smooth or discontinuous. By transforming the problem into an optimal control problem, we derive a necessary condition of the control functional. Through some prior estimates of the direct problem, the uniqueness and stability of the minimizer are obtained. In the numerical part, a Gauss–Jacobi iteration scheme is used to deal with the non-linear term. Some numerical examples are presented to illustrate the performance of the proposed algorithm.

Photo-acoustic tomography in the rotating setting

Photo-acoustic tomography is a coupled-physics (hybrid) medical imaging modality that aims to reconstruct optical parameters in biological tissues from ultrasound measurements. As propagating light gets partially absorbed, the resulting thermal expansion generates minute ultrasonic signals (the photo-acoustic effect) that are measured at the boundary of a domain of interest. Standard inversion procedures first reconstruct the source of radiation by an inverse ultrasound (boundary) problem and second describe the optical parameters from internal information obtained in the first step.This paper considers the rotating experimental setting. Light emission and ultrasound measurements are fixed on a rotating gantry, resulting in a rotation-dependent source of ultrasound. The two-step procedure we just mentioned does not apply. Instead, we propose an inversion that directly aims to reconstruct the optical parameters quantitatively. The mapping from the unknown (absorption and diffusion) coefficients to the ultrasound measurement via the unknown ultrasound source is modeled as a composition of a pseudo-differential operator and a Fourier integral operator. We show that for appropriate choices of optical illuminations, the above composition is an elliptic Fourier integral operator. Under the assumption that the coefficients are unknown on a sufficiently small domain, we derive from this a (global) injectivity result (measurements uniquely characterize our coefficients) combined with an optimal stability estimate. The latter is the same as that obtained in the standard (non-rotating experimental) setting.

Modelling Forward Osmosis Treatment of Automobile Wastewaters.

Forward osmosis (FO) has rarely been investigated as a treatment technology for industrial wastewaters. Within this study, common FO model equations were applied to simulate forward osmosis treatment of industrial wastewaters from the automobile industry. Three different models from literature were used and compared. Permeate and reverse solute flux modelling was implemented using MS Excel with a Generalized Reduced Gradient (GRG) Nonlinear Solver. For the industrial effluents, the unknown diffusion coefficients were calibrated and the influences of the membrane parameters were investigated. Experimental data was used to evaluate the models. It could be proven that common model equations can describe FO treatment of industrial effluents from the automobile industry. Even with few known solution properties, it was possible to determine permeate fluxes and draw conclusions about mass transport. However, the membrane parameters, which are apparently not solution independent and seem to differ for each industrial effluent, are critical values. Fouling was not included in the model equations although it is a crucial point in FO treatment of industrial wastewaters. But precisely for this reason, modelling is a good complement to laboratory experiments since the difference between the results allows conclusions to be drawn about fouling.

Evaluation of a Diffusion Coefficient Based on Proper Solution Space

Abstract This note is concerned with the evaluation of the unknown diffusion coefficient in a steady-state heat conduction problem. The proposed method is iterative and, starting with an initial guess, updates the assumed value at every iteration. The updating stage is achieved by generating a set of functions that satisfy some of the required boundary conditions. The correction to the assumed value is then computed by imposing the remaining boundary conditions. Numerical examples are used to study the applicability of this method.

Tracking a Diffusing Three-Dimensional Source via Nonholonomic Extremum Seeking

One method for autonomously exploring unknown environments is to treat the environment as a scalar potential field. In this scenario, one or more robots acquire measurements of the field (e.g., radiation, temperature, or carbon dioxide concentrations) and use those measurements to guide their path as they explore. An important behavior robots may need is that of extremum seeking—that is, the ability to autonomously drive toward a region containing a maximum or a minimum of the field and reliably stay with this extremum even if its location is moving. In this paper, we describe an extremum seeking control law for three-dimensional scalar potential fields built upon a previous two-dimensional law. We derive an equilibrium trajectory, prove its local stability for the case of a radial scalar field, and characterize the stability as a function of parameters by numerically evaluating its Floquet multipliers. Additionally, we consider the case where the field is constant except for a small range around the source and allow the source to be diffusing with an unknown diffusion coefficient. We derive an approximate expected first passage time and use numerical simulations to show how to optimally select parameters so that the expected tracking time is maximized.