Intrinsic Low-dimensional Manifolds Research Articles

Numerical and computational analysis on a dissipative dynamical system: Slow invariant manifold for complex chemical mechanism

This article presents the notion about Slow Invariant Manifold (SIM) and their fundamental role in model reduction techniques (MRTs) for challenges encountered in mechanical engineering within dissipative systems of chemical kinetics. Focusing on the reaction routes of complex mechanisms, we construct and compare primary approximations of the SIM through MRTs, including the Spectral Quasi Equilibrium Manifold (SQEM) and Intrinsic Low Dimensional Manifold (ILDM). These methods effectively transform high-dimensional complex problems into lower dimensions, solving them without compromising crucial information about the complex systems modified for homogeneous reactive systems. Employing the sensitivity analysis by using the MATLAB's toolbox, we present the numerical findings in a tabular format obtained through MRTs. This study provides the understanding about the accessible exploration of numerical solutions, improving insights of the complex variation within the system.

Simplified modeling of butane dehydrogenation: deeper understanding of the system’s dynamics

The dehydrogenation of butane is a catalytic process that involves the separation of hydrogen atoms from an organic molecule, and it relies on the presence of a catalyst Z, such as Ni, Pt, Pd, Fe, ZnO, Cr 2 O 3, or Fe 2 O 3. In this paper, we introduce two highly effective mathematical techniques known as Lumping Compartments (LC) and Intrinsic Low Dimensional Manifold (ILDM) to simplify the complexity of the dehydrogenation of butane. Initially, the model consists of seven nonlinear differential equations with eight parameters. To streamline the model, we applied LC to ten different cases, considering various initial states and parameters. As a result, the number of compartments significantly decreased from seven to three. The obtained computational results and total differences reveal a remarkable agreement between the complete model and the simplified version, demonstrating a high degree of predictive accuracy. Additionally, ILDM is employed to enhance the initial estimation of the slow invariant manifold. This enables us to easily distinguish between the fast and slow variations at each point, with the variations acting as chemical kinetics attractors. Fast chemical reactions are drawn towards these attractors, while slow chemical reactions occur within the manifolds.

Model simplification and sensitivity analysis of chemical species for oxidation behavior in multi route complex chemical mechanism

This study employs a Model Reduction Technique (MRT) to simplify the four-step catalytic carbon monoxide (CO) oxidation reaction. The C-matrix method identifies key elements, key/non key components, and key reactions, while the Intrinsic Low-Dimensional Manifold (ILDM) pinpoints a Slow-Invariant Manifold (SIM) important for understanding key species behavior. Sensitivity analysis can be considered for measuring the efficiency of the chemical species in detailed mechanism. This systematic approach contributes to optimizing and controlling complex reactions offering broad application potential. In addition to the mathematical proof, the validation of the given chemical model is rectified. The comparison between the slow invariant manifold of both reaction routes is reported and the computational based results performed in this study are obtained through MATLAB.

Analyzing multi-step reaction mechanisms: Unveiling non-linear dynamics with advanced reduction techniques

This study aims to achieve the computational results for the oxidation of carbon monoxide using the spectral quasi-equilibrium manifold and the intrinsic low-dimensional manifold techniques, enabling the identification of the slow invariant manifold and subsequent simulation of the solutions. In addition, the application of Gibbs’ rule facilitates the reduction of the system, thereby effectively transforming higher dimensions into lower dimensions, ultimately revealing the underlying 1D manifold dynamics. This research serves to elucidate the intricacies of phase-shifting behaviors in the context of key species within the overarching and interconnected multi-step reaction mechanisms, contributing significantly to the advancement of understanding in this complex field.

Novel reduction schemes for a dissipative dynamical system: A study on slow invariant manifolds in chemical kinetics

This research explores the intricate concept of the Slow Invariant Manifold (SIM) and its pivotal role in developing model reduction techniques (MRTs) for challenges within dissipative systems in chemical kinetics, specifically in mechanical engineering. Focusing on the multi-step mechanism with two intermediates, primary approximations of the SIM are constructed and compared using two prominent MRTs: The Spectral Quasi Equilibrium Manifold (SQEM) and Intrinsic Low Dimensional Manifold (ILDM). At the given rate coefficient, a special computational experiment was performed in which the efficiency of chemical species has been compared. Noteworthy innovation involves evaluating SIM separately for reduced species, departing from the conventional approach of considering every species within the mechanism. The study employs local sensitivity analysis with MATLAB's Sim-Biology toolbox, presenting quantitative findings in a tabular format for a comprehensive MRT comparison. Beyond contributing to a deeper understanding of model reduction techniques in complex chemical kinetics, this research marks the first computational exploration of dissipative systems. The novel perspective of evaluating SIM for reduced species offers nuanced insights, emphasizing the critical role of SIM in effectively addressing challenges in mechanical engineering applications. In summary, this study introduces computational advancements and novel approaches, advancing model reduction techniques and highlighting the significance of SIM in addressing challenges within mechanical engineering contexts.

COMPUTATIONAL SIMULATION OF THERMAL AND CHEMICAL BEHAVIOR DURING THE COMBUSTION PROCESS IN DIESEL ENGINE

The purpose of this paper is the modeling and simulation of one of the most challenging processes in internal combustion engines, combustion. The first objective was to analyze and study the thermodynamic and fluid dynamic behavior within the combustion chamber. Since detailed chemistry requires a great deal of time and computer resources, our aim is to implement a simple and accurate model to simulate and predict the combustion process. To this end, numerical methods have been developed through the techniques of Computational Fluid Dynamics (CFD). In addition to the basic equations of mass, momentum and energy, the mathematical model developed included equations of state, a turbulence model and a model of the spraying mechanism. For chemical behavior and pollutant formation, we developed a new chemical model based on the Intrinsic Low-dimensional Manifold (ILDM) and homogeneous reaction methods that automatically reduces detailed chemistry. Kiva CFD software is used as the solver for all models developed. The analysis of the results shows the importance of swirling and turbulent motions for spray penetration, ignition, and flame propagation. The results for NO x and soot formation, as the main pollutant emissions from the diesel engine, showed different behavior for their formation conditions. These results also show that there is good agreement within the mechanism used to make their predictions. The credibility of the built model is increased by comparing the pressure results with the experimental ones. Keywords: Combustion, Diesel Engine, Modeling, Computational Fluid Dynamics, Pollutant Formation DOI： https://doi.org/10.35741/issn.0258-2724.58.2.20

Manifold Learning via Linear Tangent Space Alignment (LTSA) for Accelerated Dynamic MRI With Sparse Sampling.

The spatial resolution and temporal frame-rate of dynamic magnetic resonance imaging (MRI) can be improved by reconstructing images from sparsely sampled k -space data with mathematical modeling of the underlying spatiotemporal signals. These models include sparsity models, linear subspace models, and non-linear manifold models. This work presents a novel linear tangent space alignment (LTSA) model-based framework that exploits the intrinsic low-dimensional manifold structure of dynamic images for accelerated dynamic MRI. The performance of the proposed method was evaluated and compared to state-of-the-art methods using numerical simulation studies as well as 2D and 3D in vivo cardiac imaging experiments. The proposed method achieved the best performance in image reconstruction among all the compared methods. The proposed method could prove useful for accelerating many MRI applications, including dynamic MRI, multi-parametric MRI, and MR spectroscopic imaging.

Intrinsic Low-Dimensional Nonlinear Manifold Structure of Radio Frequency Signals

Nonlinear dynamic features are crucial in specific emitter identification (SEI), which works based on the nonlinearity of the emitter. Nevertheless, its underlying principles and stability have not been systematically studied, so these methods are close to a black box, making the recognition performance of the system unstable in applications. In this study, we provide an in-depth analysis of its mechanism and theoretical model, extracting, for the first time, the intrinsic low-dimensional nonlinear manifold structure (IMS) of radio frequency signals based on manifold learning and analyzing its robustness and distinctiveness for SEI. Hardware type and intentional modulation are found to determine the overall shape of IMS, and the device-specific information for SEI is subtle and distributed locally on IMS. Notably, sampling-related random disturbances cause the distortion, zooming, or rotation of IMS, which lowers the identification accuracy by a range of 1.74%–52%, depending on the feature, device type, signal type, and other factors.

Intrinsic low-dimensional manifold (ILDM)-based concept for the coupling of turbulent mixing with manifold-based simplified chemistry for the turbulent flame simulation

Manifold based simplified chemistry is an efficient reduction technique for the chemical kinetics, which aims to reduce the computational effort in numerical simulations. While the concept of reduced chemistry has been used for decades and various models have been developed up to now, their coupling with turbulent physical processes (e.g., mixing processes) has not been investigated extensively. This is attributed to the fact that the turbulent physical processes act as perturbation to the chemistry which pulls the thermo-kinetic states away from the manifold, and these states must relax back onto the manifold again. The present work gives insight into the coupling of reduced kinetic and the turbulent mixing processes. Accordingly, a strategy based on the Intrinsic Low-Dimensional Manifold concept is proposed. This coupling strategy is validated through the well-known Sandia Flame series. It is shown that the numerical results agree very well with those using detailed chemistry (no coupling model required) and experimental measurement. The suggested coupling strategy can be used for any manifold based simplified chemistry.

Multi-manifold discriminant local spline embedding

Manifold learning reveals the intrinsic low-dimensional manifold structure of high-dimensional data and has achieved great success in a wide spectrum of applications. However, traditional manifold learning methods assume that all the data lie on a common manifold, hence fail to capture the complicated geometry structure of the real-world data lying on multiple manifolds. This paper proposes a novel Multi-manifold Discriminant Local Spline Embedding (MDLSE) algorithm for high-dimensional classification, which considers a more realistic scenario where data of the same class lies on the same manifold. On the basis of this assumption, MDLSE seeks to reconstruct multiple manifolds for different classes of data in the embedding and separate them as apart as possible. In order to preserve the geometry structure of all the manifolds, MDLSE employs thin plate splines to align the local patches within each manifold compatibly in the global embedding. Meanwhile, to separate the different manifolds, MDLSE utilizes discriminative information to ensure the neighboring data from different manifolds to be mapped far from each other. Extensive experiments on both synthetic and real-world datasets demonstrate the superiority of MDLSE over the other representative manifold learning algorithms. The advantage of MDLSE is often more obvious on smaller size of training data and in lower embedding dimensions.

Computing the Steady state Dynamics: The Chemical Graphs of Complex Reaction

In a thermodynamically isolated system, in order to obtain numerical approximations of the complex models, different model reduction techniques are applied to reduce the complex chemical reactions from high dimensional to low dimensional manifolds. These techniques not only reduce the system but also provide the complete description of reaction kinetics. These techniques include Quasi Steady State Approximations, Partial Equilibrium Technique, Intrinsic Low Dimensional Manifold Method, etc. Among all these techniques, Spectral Quasi Equilibrium Manifold Method is one of the most convenient way to find the initial approximations of slow invariant manifolds. Initial invariant grids completely describe the slow invariant manifolds which are obtained from the dissipative system. This method is applicable for high dimensional complex chemical reactions.

E2E-LIADE: End-to-End Local Invariant Autoencoding Density Estimation Model for Anomaly Target Detection in Hyperspectral Image.

Hyperspectral anomaly target detection (also known as hyperspectral anomaly detection (HAD)] is a technique aiming to identify samples with atypical spectra. Although some density estimation-based methods have been developed, they may suffer from two issues: 1) separated two-stage optimization with inconsistent objective functions makes the representation learning model fail to dig out characterization customized for HAD and 2) incapability of learning a low-dimensional representation that preserves the inherent information from the original high-dimensional spectral space. To address these problems, we propose a novel end-to-end local invariant autoencoding density estimation (E2E-LIADE) model. To satisfy the assumption on the manifold, the E2E-LIADE introduces a local invariant autoencoder (LIA) to capture the intrinsic low-dimensional manifold embedded in the original space. Augmented low-dimensional representation (ALDR) can be generated by concatenating the local invariant constrained by a graph regularizer and the reconstruction error. In particular, an end-to-end (E2E) multidistance measure, including mean-squared error (MSE) and orthogonal projection divergence (OPD), is imposed on the LIA with respect to hyperspectral data. More important, E2E-LIADE simultaneously optimizes the ALDR of the LIA and a density estimation network in an E2E manner to avoid the model being trapped in a local optimum, resulting in an energy map in which each pixel represents a negative log likelihood for the spectrum. Finally, a postprocessing procedure is conducted on the energy map to suppress the background. The experimental results demonstrate that compared to the state of the art, the proposed E2E-LIADE offers more satisfactory performance.

Spectral quasi equilibrium manifold and intrinsic low dimensional manifold: A multi-step reaction mechanism

In this paper there are the collection of constructive methods to study slow invariant manifold of dynamic system. These techniques reduce the high dimensional complex problem to low dimensional and solve these problem without losing the information about the complex system. According to Gibbs rule we can estimate the degree of freedom (number of key components Nkc), in which we can reduce the system which is: Nkc = Ns − Newhere Ns and Ne are number of steps and number of elements respectively in a complex reaction.Several strategies have been used to reduce the detailed chemical kinetics. The method of Intrinsic low dimensional manifold (ILDM) and Spectral Quasi Equilibrium Manifold (SQEM) both methods are developed for homogeneous reactive system. In the absence of any transport system like diffusion, which can be modeled by system of ODE,s. the behavior of the reactive system is defined by trajectories in composition space starting from initials and approaches towards the chemical equilibrium. These methods are described in detail.

Deep Residual Networks for Flamelet/progress Variable Tabulation with Application to a Piloted Flame with Inhomogeneous Inlet

ABSTRACT In this work, a deep neural network is presented which is trained on flamelet/progress variable (FPV) tables and validated in a combustion large eddy simulation (LES) of the Sydney/Sandia flame with inhomogeneous inlets. Using data scaling and transformation techniques, as well as novel network architectures as the residual skip connection we are able to store all combustion relevant quantities in one single network, thus reducing effectively the memory footprint compared to the FPV tables, while keeping data retrieval times similar to table interpolation. The accuracy of the deep neural networks (DNN) is compared to its FPV counterpart and achieves excellent agreement. The DNN is also able to accurately predict the stiff progress variable source term in the thin reaction zone. Finally, the DNN thermochemistry representation is validated in combustion simulations of a 2D laminar premixed flame and a 3D LES of the Sydney/Sandia piloted jet flame with inhomogeneous inlet. The DNN results are in very good agreement with the conventional tabulated FPV simulations and show a promising way to efficiently reduce the storage size of high dimensional pretabulated thermochemical state space in reactive flow simulations. Abbreviations: ANN: Artificial neural network; API: Application programming interface; CFD: Computational fluid dynamics; CFL: Courant-Friedrichs-Lewy; CPU: Central processing unit; DL: Deep learning; DNN: Deep neural network; FGM: Flamelet generated manifold; FLOPS: Floating point operation per second; FPV: Flamelet/progress variable; GPU: Graphics processing unit; HPC: High-performance computing; ILDM: Intrinsic low-dimensional manifold; LES: Large eddy simulation; MAE: Mean absolute error; MPI: Message passing interface; MSE: Mean squared error; PDF: Probability density function; SGS: Subgrid scale; SOM: Self-organizing map; TCI: Turbulence–chemistry interaction.

Multiscale Superpixelwise Locality Preserving Projection for Hyperspectral Image Classification

Manifold learning is a powerful dimensionality reduction tool for a hyperspectral image (HSI) classification to relieve the curse of dimensionality and to reveal the intrinsic low-dimensional manifold. However, a specific characteristic of HSIs, i.e., irregular spatial dependency, is not taken into consideration in the method design, which can yield many spatially homogenous subregions in an HSI scence. Conventional manifold learning methods, such as a locality preserving projection (LPP), pursue a unified projection on the entire HSI, while neglecting the local homogeneities on the HSI manifold caused by those spatially homogenous subregions. In this work, we propose a novel multiscale superpixelwise LPP (MSuperLPP) for HSI classification to overcome the challenge. First, we partition an HSI into homogeneous subregions with a multiscale superpixel segmentation. Then, on each scale, subregion specific LPPs and the associated preliminary classifications are performed. Finally, we aggregate the classification results from all scales using a decision fusion strategy to achieve the final result. Experimental results on three real hyperspectral data sets validate the effectiveness of our method.

The reaction routes comparison with respect to slow invariant manifold and equilibrium points

Mathematically, complex chemical reactions can be simplified by “model reduction,” that is, the rigorous way of approximating and representing a complex model in simplified form. Furthermore, to reduce the dimension of the reaction mechanism there are different available model reduction techniques (MRT). Two MRT Spectral Quasi Equilibrium Manifold (SQEM) and Intrinsic Low Dimensional Manifold (ILDM) are applied here. Both techniques are good approximation techniques but not completely analytical. While for a case study, the complex behavior of the two-route reaction mechanism allied graphically. The characteristic properties of nodes and trees of reaction mechanism are addressed with the graph theory. The invariant solution curves and their comparison for two-route reaction mechanism have never been reported so far. Therefore, the aim of this article is to study such facts. The possible approaches to their solution and key problems are based on the Horiuti’s rules and their slow invariants. Both model-reduction techniques are applied to solve the high dimensional complex problem and comparison between invariant curves of both routes are presented graphically. The efficiency of ILDM with respect to time is compared with the SQEM graphically and in tabulated form by using MATLAB.

Assessment of spray combustion models in large-eddy simulations of a polydispersed acetone spray flame

Spray combustion is of practical importance to various applications. To study spray flames, benchmark cases were investigated both experimentally and numerically. Previous numerical studies of these flames have identified sensitivities to gas-phase combustion models and the description of the droplet evaporation. The objective of this work is to examine effects of combustion models on the structure of an acetone spray flame. To this end, three different combustion models are examined, namely a finite-rate chemistry model, a flamelet/progress variable model, and a flame prolongation of intrinsic low dimensional manifold model. In the two flamelet approaches, effects of spray evaporation are considered in the limit of small Stokes number. By examining radial profiles and employing a doubly-conditioned analysis on gaseous mixture fraction and liquid-to-gas mass ratio, it is shown that both flamelet models show differences in the evaporation and subsequent gas phase combustion and temperature field. The use of a finite-rate combustion model in conjunction with a reduced chemical mechanism provide improved predictions of heat release and spray-flame structure.

Large-eddy simulation of direct injection natural gas combustion in a heavy-duty truck engine using modified conditional moment closure model with low-dimensional manifold method

This article describes the application of a modified first-order conditional moment closure model used in conjunction with the trajectory-generated low-dimensional manifold method in large-eddy simulation of pilot ignited high-pressure direct injection natural gas combustion in a heavy-duty diesel engine. The article starts with a review of the intrinsic low-dimensional manifold method for reducing detailed chemistry and various formulations for the construction of such manifolds. It is followed by a brief review of the conditional moment closure method for modelling the interaction between turbulence and combustion chemistry. The high computational cost associated with the direct implementation of the basic conditional moment closure model was discussed. The article then describes the formulation of a modified approach to solve the conditional moment closure equation, whose reaction source terms for the conditional mass fractions for species were obtained by projecting the turbulent perturbation onto the reaction manifold. The main model assumptions were explained and the resulting limitations were discussed. A numerical experiment was conducted to examine the validity the model assumptions. The model was then implemented in a combustion computational fluid dynamics solver developed on an open-source computational fluid dynamics platform. Non-reactive jet simulations were first conducted and the results were compared to the experimental measurement from a high-pressure visualization chamber to verify that the jet penetration under engine relevant conditions was correctly predicted. The model was then used to simulate natural gas combustion in a heavy-duty diesel engine equipped with a high-pressure direct injection system. The simulation results were compared with the experimental measurement from a research engine to verify the accuracy of the model for both the combustion rate and engine-out emissions.

FGM with REDx: chemically reactive dimensionality extension

We propose a new approach to improve the accuracy of flamelet-generated manifolds (FGMs) method by extending the manifolds with additional chemically reactive degrees of freedom. Following the ideas of intrinsic low-dimensional manifold, the dimensionality of the FGM is increased by performing a local time-scale analysis of the chemical source term. A few slow characteristic directions of the reaction kinetics are used to extend the FGM, while the remaining reaction groups, characterised by fast time-scales, are assumed in steady state. The introduced method for FGM REactive Dimensionality extension is abbreviated as FGM-REDx. It is tested in one-dimensional simulations reproducing an expansion of burnt gases in an aero-engine stator. This process is characterised by a rapid change of enthalpy and pressure, altering, among others, the chemistry of pollutants CO and NO. The primary focus was on the assessment of the FGM's capability to predict the pollutants emissions. The rates of physical/thermodynamic perturbations turned out to be severe enough for the chemical species composition to go off the flamelet. The FGM extended with one additional chemically reactive dimension has been generated and successfully applied to the test cases, yielding a high accuracy gain over the standard FGM.

Analysis of the approximate slow invariant manifold method for reactive flow equations

The Approximate Slow Invariant Manifold method of Singh, Powers and Paolucci is a useful method for addressing model reduction in systems of reactive flow equations. It exploits separations of time scales between slow and fast species, and it generalizes the Intrinsic Low-Dimensional Manifold method, which was developed for model reduction in the context of reaction kinetics, to systems with diffusive and active transport. In this article, we present a mathematical analysis of the Approximate Slow Invariant Manifold method in the context of systems of reaction–diffusion equations with slow and fast reaction kinetics. Beginning with systems of two species (one slow and one fast), and then treating general systems with multiple slow and fast species, we explicitly determine the accuracy of the Approximate Slow Invariant Manifold method. We find that it is correct up to and including the terms of first order in the small parameter that measures the separation of the kinetics time scales, and that it captures many of the terms at second order, as well. Our analysis includes precise statements of the errors at second order, and we find that these are proportional to the slow components of the reaction–diffusion equation, as well as to the curvature of the critical manifold. We illustrate the results analytically on two prototypical examples, the Michaelis–Menten–Henri model with diffusion of the slow species and the Davis–Skodje model in which both the slow and fast species diffuse.