Low-dimensional Manifold Method Research Articles

Generative interpolation via a diffusion probabilistic model

Seismic data interpolation is essential in a seismic data processing workflow, recovering data from sparse sampling. Traditional and deep-learning-based methods have been widely used in the seismic data interpolation field and have achieved remarkable results. In this paper, we develop a seismic data interpolation method through the novel application of diffusion probabilistic models (DPMs). DPM transforms the complex end-to-end mapping problem into a progressive denoising problem, enhancing the ability to reconstruct complex situations of missing data, such as large proportions and large-gap missing data. The interpolation process begins with a standard Gaussian distribution and seismic data with missing traces and then removes noise iteratively with a UNet trained for different noise levels. Our DPM-based interpolation method allows interpolation for various missing cases, including regularly missing, irregularly missing, consecutively missing, noisy missing, and different ratios of missing cases. The ability to generalize to different seismic data sets is also discussed in this paper. Numerical results of synthetic and field data indicate satisfactory interpolation performance of the DPM-based interpolation method in comparison with the f- x prediction filtering method, the curvelet transform method, the low-dimensional manifold method (LDMM), and the coordinate attention-based UNet method, particularly in cases with large proportions and large-gap missing data. Diffusion is all we need for seismic data interpolation.

Curvature-regularized manifold for seismic data interpolation

Seismic data can be described in a low-dimensional manifold. Thus, the low dimensionality of the seismic data patch manifold can serve as a good regularizer for seismic data interpolation. However, we have found that having only low-dimensional manifold regularization is not sufficient for interpolating seismic data with large data gaps or spectral aliasing. Therefore, we propose the application of a curvature-regularized low-dimensional manifold method for seismic data interpolation. A windowed version of the method is proposed, which provides adaptability and efficiency for seismic data interpolation. Numerical experiments on the synthetic and field data indicate that the proposed method outperforms the f- x, curvelet, and low-dimensional manifold methods in many missing data cases, especially in the presence of large gaps or spectral aliasing.

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.

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.

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.

A Geometric Method for Model Reduction of Biochemical Networks with Polynomial Rate Functions

Model reduction of biochemical networks relies on the knowledge of slow and fast variables. We provide a geometric method, based on the Newton polytope, to identify slow variables of a biochemical network with polynomial rate functions. The gist of the method is the notion of tropical equilibration that provides approximate descriptions of slow invariant manifolds. Compared to extant numerical algorithms such as the intrinsic low-dimensional manifold method, our approach is symbolic and utilizes orders of magnitude instead of precise values of the model parameters. Application of this method to a large collection of biochemical network models supports the idea that the number of dynamical variables in minimal models of cell physiology can be small, in spite of the large number of molecular regulatory actors.

RANS simulation of a turbulent premixed bluff body flame using conditional source-term estimation

In the present work, simulations are performed in order to study a turbulent premixed flame stabilised behind a bluff body burner. Conditional Source-term Estimation (CSE) is the combustion model adopted for this purpose and it is coupled with a trajectory gene- rated low-dimensional manifold method for chemistry reduction: conditional-averaged chemical source terms are closed by conditional-averaged scalars, which are obtained by inverting an integral equation. Two regularisation methods are implemented. The optimal regularisation parameter is determined and a sensitivity analysis is performed related to the effect of the value of the regularisation parameter. In the present study, only small differences in the conditional and unconditional averages are noticed for values of the regularisation parameter located in the optimal range determined by the L-curve. Two-equation k–ε and Shear Stress Transport (SST) turbulence models are used to solve the flow field in either the reactive or non-reactive case. Different values of the constant in the standard k–ε model and of γ2 in the SST k–ω model are tested. Mean axial velocity and progress variable profiles are compared with experimental data. It is found that SST k–ω gives better results for the velocity profile in both the reactive and non-reactive cases. A value of γ2 equal to 0.65 provides best predictions in the reactive case. The progress variable prediction exhibits similar profiles in the case of CSE and the flamelet combustion model approach. Very good agreement is achieved between the CSE predictions and experimental data.

Investigation of conditional source-term estimation applied to a non-premixed turbulent flame

A turbulent combustion model, Conditional Source-term Estimation (CSE) is applied to a non-premixed turbulent jet methane flame. The conditional chemical source terms are determined on the basis of first order closure and the conditional averaged species concentrations are obtained by inverting an integral equation. The Tikhonov method is implemented for regularisation. Detailed chemistry is tabulated using the trajectory generated low-dimensional manifold method. Radiation due to the gaseous species is included. Reynolds Averaged Navier–Stokes calculations are performed using two different turbulence models. The objectives of the paper are (i) assessment of the impact of the main numerical parameters in CSE and (ii) comparison of the CSE numerical predictions with available experimental data and results from previous simulations for the selected flame. The number of CSE domains and the number of points in each CSE domain are shown to have a significant impact on the results if not selected appropriately. The present CSE calculations always converge to unique and stable predictions. The corrected k–ε model yields mixture fraction profiles in good agreement with the experimental data values for axial locations in the first half of the flame. Farther downstream, the RNG k–ε model performs better. Overall, the current predictions for the mixture fraction are in good agreement with the experimental data. The predicted temperatures using CSE and the k–ε turbulence model with a modified value of Cε1 = 1.47 are found to be in very good agreement with the experimental data. Further, the current CSE results are of comparable quality with previous simulations using the flamelet model and conditional moment closure. Future work may include further investigation on optimal determination of the regularisation parameter and alternative regularisation techniques, soot modelling within the CSE formulation, and improved formulation of radiation.

Classical and Dynamic Analysis of Gas Phase Reactivity: Influence of Carbon Precursor in the CVD of SiC

The intrinsic low dimensional manifold method (ILDM), which is a reduction method of large gaseous chemical systems, is applied to the gas-phase chemistry used to deposit 4H or 6H silicon carbide (SiC). The 1-dimensional (1-D) manifold corresponding to the relaxation of all the fast reaction modes of the complex gaseous chemical system is calculated and represented according to different projections in the state space. The ILDM method is also used to compare trajectories corresponding to various initial gas-phase compositions. These initial gas-phase mixtures, which include different carbon precursors, all converge towards the same thermodynamic equilibrium. Long before thermodynamic equilibrium is reached, the trajectories are attracted by the 1-D manifold. © 2004 The Electrochemical Society. All rights reserved.

Intrinsic low-dimensional manifold method extended with diffusion

It is necessary to use chemical reduction techniques for the modeling of complex burner systems. The intrinsic low-dimensional manifold (ILDM) method has proved to be a useful method to simplify detailed reaction mechanisms. However, diffusion is neglected in the construction of the manifold, which results in less accurate manifolds in regions where both chemistry and diffusion are significant, which is generally the case in a large part of the reaction zone. This paper presents a new reduction method, which uses the basic principles of intrinsic low-dimensional manifolds. A manifold is constructed in the composition phase space instead of the composition space, so that the gradients of species concentrations and enthalpy can be used to include diffusion within the manifold. The composition phase space is spanned by the species mass fractions, the enthalpy, and their corresponding diffusive fluxes. The second-order conservation equations are transformed into first-order equations by introducing flux terms. The resulting set of first-order equations are used to construct the manifold. The procedure used to define the manifold is equivalent to the ILDM method. The new method, called phase space ILDM (PS-ILDM), is applied to a simple but illustrative example, and finally a manifold for a CO/H2 mechanism is presented. The results show that the new method gives more accurate manifolds than the ILDM method for parts of the flame where both reaction and diffusion are important. In addition, fluctuations of enthalpy and element fractions due to preferential diffusion are included in the manifold. This may result in a database with a lower dimension and therefore a more effective reduction.

Application of repro-modeling for the reduction of combustion mechanisms

The basic problem of mechanism reduction methods is to find functional relationships between selectedstate variables (e.g., some concentrations and temperature) and their rates. However, this information is present during the simulations with the full chemical model. As a new application of the repro-modeling approach, information for rates is extracted from detailedchemical calculations and stored in the form of high-order multivariate, polynomials. For an efficient utilization of the polynomials, a computer program was written that rearranges them to the form of multivariate Horner equations. The repro-modeling method is an alternative to the application of the quasi-steady-state approximation (QSSA) and of the low-dimensional manifold method. Pros and cons of these three methods are discussed in detail considering the preparations required, the accuracy attainable, the yield in computer time, and the limitations of the techniques. Simulations of the combustion of wet CO using two-variable and three-variable repro-models were24,000 and 11,700 times faster, respectively, than the SENKIN calculation using the full model. These calculations represent the first use of repro-modeling for combustion mechanism reduction.