Transport Partial Differential Equations Research Articles

Delay‐adaptive control of first‐order hyperbolic partial integro‐differential equations

AbstractWe develop a delay‐adaptive controller for a class of first‐order hyperbolic partial integro‐differential equations with an unknown input delay. By employing a transport PDE to represent delayed actuator states, the system is transformed into a transport partial differential equation with unknown propagation speed cascaded with a PIDE. A parameter update law is designed using a Lyapunov argument and the infinite‐dimensional backstepping technique to establish global stability results. Subsequently, the effectiveness of the proposed approach was validated through numerical simulations.

A Lifshitz-Slyozov type model for adipocyte size dynamics: limit from Becker-Döring system and numerical simulation.

Biological data show that the size distribution of adipose cells follows a bimodal distribution. In this work, we introduce a Lifshitz-Slyozov type model, based on a transport partial differential equation, for the dynamics of the size distribution of adipose cells. We prove a new convergence result from the related Becker-Döring model, a system composed of several ordinary differential equations, toward mild solutions of the Lifshitz-Slyozov model using distribution tail techniques. Then, this result allows us to propose a new advective-diffusive model, the second-order diffusive Lifshitz-Slyozov model, which is expected to better fit the experimental data. Numerical simulations of the solutions to the diffusive Lifshitz-Slyozov model are performed using a well-balanced scheme and compared to solutions to the transport model. Those simulations show that both bimodal and unimodal profiles can be reached asymptotically depending on several parameters. We put in evidence that the asymptotic profile for the second-order system does not depend on initial conditions, unlike for the transport Lifshitz-Slyozov model.

Technical note: The Graetz problem at small Peclet number and its application to sizing condensation particle counters

We extend to small Peclet number Pe the familiar high-Pe solution to the Graetz problem of heat/mass transfer in a tube with parabolic flow with a wall temperature discontinuity at an axial position z = 0. Retaining diffusion in the axial direction results only in slight shifts in the eigenfunctions describing the problem. This enables the characterization of the saturation field S by separation of variables, which we implement for Pe = 0, 0.1, 0.25. Calculations based on discretization of the transport partial differential equations are also used to analyze the problem at Pe up to 5. As Pe tends to 0, the heat and mass transfer problems are both described by the same dimensionless function θ(z,r) of the axial and radial coordinates z and r. S then depends on the single quantity θ, and its maximum value Smax is exactly the same along all streamlines. This would result in sharp activation curves in a hypothetical low-Pe condensation particle counter, similarly as in the special circumstance when the ratio Le between heat and mass diffusivities is unity. Calculations are run to determine how the radial uniformity of Smax is degraded at increasing Pe. When accounting for axial diffusion, Smax at the wall is independent of Pe or Le. The commonly considered high Pe limit fails to satisfy this requirement, and therefore predicts an incorrect Smax at distances from the wall discontinuity small compared to Pe−1/2.

Prediction-based control of multi-input linear systems subject to distinct unknown delays

The paper is concerned with the stability of linear time-invariant (LTI) systems with distinct multi-input unknown delays. Our method is based on transport partial differential equations (PDEs) representation of the multi-input delays and infinite-dimensional backstepping transformation. By introducing the new Lyapunov–Krasovskii functional, exponential stabilization of the closed-loop system is achieved, in which fixed constant horizon prediction-based controllers are introduced to robustly compensate for the different delays. There are two control designs proposed: single-constant and multi-constant horizon prediction-based control design. For the latter, we introduce special infinite-dimensional backstepping transformation, backstepping-forwarding transformation, to offset delays from small to large. In particular, constants selection are required to be within a narrow enough range. At last, to illustrate the validity of the proposed results, a numerical example is provided.

Bilateral boundary control of an input delayed 2-D reaction–diffusion equation

In this paper, a delay compensation design method based on PDE backstepping is developed for a two-dimensional reaction–diffusion partial differential equation (PDE) with bilateral input delays. The PDE is defined in a rectangular domain, and the bilateral control is imposed on a pair of opposite sides of the rectangle. To represent the delayed bilateral inputs, we introduce two 2-D transport PDEs that form a cascade system with the original PDE. A novel set of backstepping transformations is proposed for delay compensator design, including one Volterra integral transformation and two affine Volterra integral transformations. Unlike the kernel equation for 1-D PDE systems with delayed boundary input, the resulting kernel equations for the 2-D system have singular initial conditions governed by the Dirac Delta function. Consequently, the kernel solutions are written as a double trigonometric series with singularities. To address the challenge of stability analysis posed by the singularities, we prove a set of inequalities by using the Cauchy–Schwarz inequality, the 2-D Fourier series, and the Parseval’s theorem. A numerical simulation illustrates the effectiveness of the proposed delay-compensation method.

Robust stabilization of 2 × 2 first-order hyperbolic PDEs with uncertain input delay

A backstepping-based compensator design is developed for a system of 2 × 2 first-order linear hyperbolic partial differential equations (PDE) in the presence of an uncertain long input delay at boundary. We introduce a transport PDE to represent the delayed input, which leads to three coupled first-order hyperbolic PDEs. A novel backstepping transformation, composed of two Volterra transformations and an affine Volterra transformation, is introduced for the predictive control design. The resulting kernel equations from the affine Volterra transformation are two coupled first-order PDEs and each with two boundary conditions, which brings challenges to the well-posedness analysis. We solve the challenge by using the method of characteristics and the successive approximation. To analyze the sensitivity of the closed-loop system to uncertain input delay, we introduce a neutral system which captures the control effect resulted from the delay uncertainty. It is proved that the proposed control is robust to small delay variations. Numerical examples illustrate the performance of the proposed compensator.

Augmenting Deep Residual Surrogates with Fourier Neural Operators for Rapid Two-Phase Flow and Transport Simulations

Summary Accurate numerical modeling of multiphase flow and transport mechanisms is essential to study varied, complex physical phenomena including flow in subsurface oil and gas reservoirs and subsurface aquifers subject to CO2 sequestration. State-of-the-art complete physics-based solvers suffer from many computational challenges. High-fidelity data-driven surrogate models that solve the governing partial differential equations (PDEs) have the potential to optimize the time to solution and increase confidence in critical business and engineering decisions through better quantification of solution statistics. We leverage the recently proposed Fourier neural operators (FNOs) with quasilinear time complexity to capture the spectral information from feature maps to solve the coupled porous flow and transport PDEs. Embedding Fourier layers within the residual blocks results in a highly effective structure that, while achieving competitive accuracy, also enables efficient training of deeper networks with a dramatically reduced number of trainable parameters. The resulting novel deep-learning (DL) architecture is coined as FResNet++. FResNet++ uses squeeze and excitation blocks, atrous spatial pyramid pooling (ASPP), and attention blocks to increase its sensitivity to the relevant features and capture multiscale information, and it is specifically tuned to operate optimally to learn from and predict numerically simulated flow (pressure and saturation) fields. We demonstrate the ability of FResNet++ to generalize over multiple high-dimensional input parameter spaces that describe subsurface permeability and porosity heterogeneity. The resulting DL architecture accurately captures the complex interplay between viscous forces and highly heterogeneous permeability and porosity fields. We investigate two-phase flow in porous media, which is the archetypal problem for reservoir simulation giving rise to a system of nonlinearly coupled PDEs with highly heterogeneous coefficients. We show in blind tests that FResNet++ predicts saturation fields more accurately compared to ResU-Net and original FNO with fully connected linear layers. We additionally investigate the effects of using alternative loss functions and an alternative way of utilizing FResNet++ to increase its effectiveness. For the first time in the literature, we show that the spatiotemporal evolution of pressure and saturation fields can be jointly predicted with good accuracy using a single FResNet++ network over long time horizons in response to previously unseen permeability and porosity fields. After a moderate training investment on graphics processing units (GPUs), FResNet++ yields a speedup of at least four orders of magnitude compared to a conventional numerical PDE solver and operates with notably fewer trainable parameters compared to the original FNO. Our numerical experiments validate that FNOs can be utilized in various convolutional neural network-based architectures and can effectively substitute for repetitive physics-based forward simulations for scenario testing.

Finite/fixed-time stabilization of a chain of integrators with input delay via PDE-based nonlinear backstepping approach

In this paper, we present a general approach to studying the problem of finite-time and fixed-time stabilization of a chain of integrators with input delay. To accomplish this, we first reformulate the chain of integrators with input delay as a cascade ODE–PDE system (i.e., a cascade of a linear transport partial differential equation (PDE) with the chain of integrators) where the transport equation models the effect of the delay on the input. Next, we use a nonlinear infinite-dimensional backstepping transformation to convert the cascade system to a suitable target system that is chosen to be finite-time or fixed-time stable. We perform the stability analysis on the target system by means of classical non-asymptotic concepts and tools such as the linear homogeneity and “generalized KL” functions. Then, we use the inverse transformation to transfer back the stability property to the closed-loop system. Finally, we give some characterizations of finite/fixed time predictor-based controllers followed by numerical simulations.

Numerical geometric acoustics: An eikonal-based approach for modeling sound propagation in 3D environments

We present algorithms for solving high-frequency acoustic scattering problems in complex domains. The eikonal and transport partial differential equations from the WKB/geometric optic approximation of the Helmholtz equation are solved recursively to generate boundary conditions for a tree of eikonal/transport equation pairs, describing the phase and amplitude of a geometric optic wave propagating in a complicated domain, including reflection and diffraction. Edge diffraction is modeled using the uniform theory of diffraction. For simplicity, we limit our attention to domains with piecewise linear boundaries and a constant speed of sound. The domain is discretized into a conforming tetrahedron mesh. For the eikonal equation, we extend the jet marching method to tetrahedron meshes. Hermite interpolation enables second order accuracy for the eikonal and its gradient and first order accuracy for its Hessian, computed using cell averaging. To march the eikonal on an unstructured mesh, we introduce a new method of rejecting unphysical updates by considering Lagrange multipliers and local visibility. To handle accuracy degradation near caustics, we introduce several fast Lagrangian initialization algorithms. We store the dynamic programming plan uncovered by the marcher in order to propagate auxiliary quantities along characteristics. We introduce an approximate origin function which is computed using the dynamic programming plan, and whose 1/2-level set approximates the geometric optic shadow and reflection boundaries. We also use it to propagate geometric spreading factors and unit tangent vector fields needed to compute the amplitude and evaluate the high-frequency edge diffraction coefficient. We conduct numerical tests on a semi-infinite planar wedge to evaluate the accuracy of our method. We also show an example with a more realistic building model with challenging architectural features. Finally, we demonstrate a simple approach to extending the method to handle nonconstant speeds of sound by modifying the semi-Lagrangian updates to account for a varying speed.

On convective heat and mass transport of radiative double diffusive Williamson hybrid nanofluid by a Riga surface

Convective heat and mass transport of radiative Williamson hybrid [Formula: see text] nanofluid (NF) by a Riga surface with the novel features of Cattaneo–Christov double-diffusion has been investigated. Thermal contributions of internal heat mechanism and Arrhenius energy in Darcy–Forchheimer medium have also been incorporated in the modeling. Mathematical modeling has been completed by using suitable mathematical expressions for thermophysical features of hybrid nanofluid (HNF). Transport partial differential equations (PDEs) have been transformed into ordinary differential equations (ODEs) by means of similarity variables. Numerical approximation of the transformed system has been obtained by using shooting-based Runge–Kutta–Fehlberg approach. Results have been presented through various graphs and discussed physically in detail. Solution is validated for limited cases. Concentration of the hybrid mixture is reduced for progressive concentration-relaxation parameter. Temperature is alleviated for developing thermal-relaxation parameter. Nusselt number is observed to be higher for Williamson HNF than simple ordinary NF.

Observer Design for n + m Linear Hyperbolic ODE-PDE-ODE Systems

In this letter, an observer design is presented for a system of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n + m$ </tex-math></inline-formula> hetero-directional transport partial differential equations (PDEs) coupled on both boundaries of the domain to ordinary differential equations (ODEs). This class of systems can represent, for instance, actuator and load dynamics at the boundaries of a hyperbolic system. The results in this letter provide a constructive way to reconstruct the state of the ODEs and the PDEs using only available measurements on one of the two ODEs. This observer design completes existing full-state feedback designs for this class of systems and enables, together with previous results, the construction of output-feedback stabilizers. As a complementary result, this letter includes a simple (constructive) method to obtain a stable left-inverse of a specific transfer matrix required in the observer design, which can also be applied (after a transposition) to the computation of the analogous stable right-inverses required for the control design.

Density-Based Stochastic Reachability Computation for Occupancy Prediction in Automated Driving

We propose a stochastic reachability computation framework for occupancy prediction in automated driving by directly solving the underlying transport partial differential equation (PDE) governing the advection of the closed-loop joint density functions. The resulting nonparametric gridless computation is based on integration along the characteristic curves, and it allows online computation of the time-varying collision probabilities. We provide numerical simulations for multi-lane highway driving scenarios to highlight the scope of the proposed method.

Modeling metastatic tumor evolution, numerical resolution and growth prediction

In this paper we consider a generalized metastatic tumor growth model that describes the primary tumor growth by means of an Ordinary Differential Equation (ODE) and the evolution of the metastatic density using a transport Partial Differential Equation (PDE). The numerical method is based on the resolution of a linear Volterra integral equation (VIE) of the second kind, which arises from the reformulation of the ODE–PDE model. The convergence of the method is proved and error estimates are given. The computation of the approximate solution leads to solving well conditioned linear systems. Here we focus our attention on two different case studies: lung and breast cancer. We assume five different tumor growth laws for each of them, different metastatic emission rates between primary and secondary tumors, and lastly that the newborn metastases can be formed by clusters of several cells.

Delay-Adaptive Predictor Feedback Control of Reaction–Advection–Diffusion PDEs With a Delayed Distributed Input

We consider a system of reaction–advection–diffusion partial differential equation (PDE) with a distributed input subject to an unknown and arbitrarily large time delay. Using Lyapunov technique, we derive a delay-adaptive predictor feedback controller to ensure the global stability of the closed-loop system in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L^2$</tex-math></inline-formula> sense. More precisely, we express the input delay as a 1-D transport PDE with a spatial argument leading to the transformation of the time delay into a spatially distributed shift. For the resulting mixed transport and reaction–advection–diffusion PDE system, we employ a PDE backstepping design and certainty equivalence principle to derive the suitable adaptive control law that compensates for the effect of the unknown time delay. Our controller ensures the global stabilization in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L^2$</tex-math></inline-formula> sense. Our result is the first delay-adaptive predictor feedback controller with a PDE plant subject to a delayed distributed input. The feasibility of the proposed approach is illustrated by considering a mobile robot that spread a neutralizer over a polluted surface to achieve efficient decontamination with an unknown actuator delay arising from the noncollocation of the contaminant diffusive process and the moving neutralizer source. Consistent simulation results are presented to prove the effectiveness of the proposed approach.

Predictor-Feedback Prescribed-Time Stabilization of LTI Systems With Input Delay

This article first deals with the problem of prescribed-time stability of linear systems without delay. The analysis and design involve the <i>Bell polynomials</i>, <i>the generalized Laguerre polynomials</i>, <i>the Lah numbers</i>, and a suitable <i>polynomial-based Vandermonde matrix</i>. The results can be used to design a new controller&#x2014;with time-varying gains&#x2014;ensuring prescribed-time stabilization of controllable linear time-invariant (LTI) systems. The approach leads to similar results compared to Holloway <i>et al.</i> 2019, but offers an alternative and compact control design (especially for the choice of the time-varying gains). Based on the preliminary results for the delay-free case, we then study controllable LTI systems with single input and subject to a constant input delay. We design a predictor feedback with time-varying gains. To achieve this, we model the input delay as a transport partial differential equation (PDE) and build on the cascade PDE&#x2013;ordinary differential equation setting (inspired by Krstic 2009) so as the design of the prescribed-time predictor feedback is carried out based on the backstepping approach, which makes use of <i>time-varying kernels</i>. We guarantee the bounded invertibility of the backstepping transformation, and we prove that the closed-loop solution converges to the equilibrium in a prescribed time. A simulation example illustrates the results.

Frame Invariance and Scalability of Neural Operators for Partial Differential Equations

Partial differential equations (PDEs) play a dominant role in the mathematical modeling of many complex dynamical processes. Solving these PDEs often requires prohibitively high computational costs, especially when multiple evaluations must be made for different parameters or conditions. After training, neural operators can provide PDEs solutions significantly faster than traditional PDE solvers. In this work, invariance properties and computational complexity of two neural operators are examined for transport PDE of a scalar quantity. Neural operator based on graph kernel network (GKN) operates on graph-structured data to incorporate nonlocal dependencies. Here we propose a modified formulation of GKN to achieve frame invariance. Vector cloud neural network (VCNN) is an alternate neural operator with embedded frame invariance which operates on point cloud data. GKN-based neural operator demonstrates slightly better predictive performance compared to VCNN. However, GKN requires an excessively high computational cost that increases quadratically with the increasing number of discretized objects as compared to a linear increase for VCNN.

Observer and output feedback control for nonlinear ordinary differential equation coupled to an under‐actuated transport partial differential equation

AbstractIn this article, the under‐actuation problem for a class of first order hyperbolic partial differential equation system coupled to a nonlinear ordinary differential equation is considered. An anticollocated observer design and an appropriate output feedback control is developed based on the backstepping methodology. The stability conditions for the nonlinear coupled system are proposed using the Lyapunov theory and are formulated in terms of the LMI constraints. Further, we have extended the stability results and backstepping control design for the fractional‐order coupled system. The stability conditions and the control law proposed in the work are evaluated considering a numerical simulation.

On Gaussian spin glass with P-wise interactions

The purpose of this paper is to face up the statistical mechanics of dense spin glasses using the well-known Ising case as a prelude for testing the methodologies we develop and then focusing on the Gaussian case as the main subject of our investigation. We tackle the problem of solving for the quenched statistical pressures of these models both at the replica symmetric level and under the first step of replica symmetry breaking by relying upon two techniques: the former is an adaptation of the celebrated Guerra’s interpolation (closer to probability theory in its spirit) and the latter is an adaptation of the transport partial differential equation (closer to mathematical physics in spirit). We recover, in both assumptions, the same expression for quenched statistical pressure and self-consistency equation found with other techniques, including the well-known replica trick technique.

Input Delay Compensation for Neuron Growth by PDE Backstepping

Neurological studies show that injured neurons can regain their functionality with therapeutics such as Chondroitinase ABC (ChABC). These therapeutics promote axon elongation by manipulating the injured neuron and its intercellular space to modify tubulin protein concentration. This work introduces an input delay compensation with state-feedback control law for axon elongation by regulating tubulin concentration. Axon growth dynamics with input delay is modelled as coupled parabolic diffusion-reaction-advection Partial Differential Equations (PDE) with a boundary governed by a nonlinear Ordinary Differential Equations (ODE), associated with a transport PDE. A novel feedback law is proposed by using backstepping method for PDE and input-delay compensation. The gain kernels are provided after transforming the interconnected PDE-ODE-PDE system to a target system. The stability analysis is presented by applying Lyapunov analysis to the target system in the spatial H1-norm, thereby the local exponential stability of the original error system is proved.

Explicit Prediction-Based Control for Linear Difference Equations With Distributed Delays

This paper presents a prediction-based control law for linear difference equations subject to a distributed state delay and a pointwise input delay. We propose to use a prediction-based control to overcome the instability potentially related to the distributed delay. We obtain an explicit formulation of the controller, depending only on the state and input history and involving integral kernels, which are the solutions to recursive Volterra equations. In view of future delay-sensitivity analysis, we develop an alternative approach to prove closed-loop stability, recasting the input delay as a transport Partial Differential Equation. In an analog manner to the stability analysis methodology developed for linear Delay Differential Equations, we propose a backstepping transformation to map the closed-loop system to a distributed-delay free target system. Simulation results underline the efficiency of the proposed control design.