Nonlinear Manifold Research Articles

Non-linear Manifold Reduced-Order Models with Convolutional Autoencoders and Reduced Over-Collocation Method

Non-affine parametric dependencies, nonlinearities and advection-dominated regimes of the model of interest can result in a slow Kolmogorov n-width decay, which precludes the realization of efficient reduced-order models based on linear subspace approximations. Among the possible solutions, there are purely data-driven methods that leverage autoencoders and their variants to learn a latent representation of the dynamical system, and then evolve it in time with another architecture. Despite their success in many applications where standard linear techniques fail, more has to be done to increase the interpretability of the results, especially outside the training range and not in regimes characterized by an abundance of data. Not to mention that none of the knowledge on the physics of the model is exploited during the predictive phase. In order to overcome these weaknesses, we implement the non-linear manifold method introduced by Lee and Carlberg (J Comput Phys 404:108973, 2020) with hyper-reduction achieved through reduced over-collocation and teacher–student training of a reduced decoder. We test the methodology on a 2d non-linear conservation law and a 2d shallow water models, and compare the results obtained with a purely data-driven method for which the dynamics is evolved in time with a long-short term memory network.

4D Atlas: Statistical Analysis of the Spatiotemporal Variability in Longitudinal 3D Shape Data.

We propose a novel framework to learn the spatiotemporal variability in longitudinal 3D shape data sets, which contain observations of objects that evolve and deform over time. This problem is challenging since surfaces come with arbitrary parameterizations and thus, they need to be spatially registered. Also, different deforming objects, hereinafter referred to as 4D surfaces, evolve at different speeds and thus they need to be temporally aligned. We solve this spatiotemporal registration problem using a Riemannian approach. We treat a 3D surface as a point in a shape space equipped with an elastic Riemannian metric that measures the amount of bending and stretching that the surfaces undergo. A 4D surface can then be seen as a trajectory in this space. With this formulation, the statistical analysis of 4D surfaces can be cast as the problem of analyzing trajectories embedded in a nonlinear Riemannian manifold. However, performing the spatiotemporal registration, and subsequently computing statistics, on such nonlinear spaces is not straightforward as they rely on complex nonlinear optimizations. Our core contribution is the mapping of the surfaces to the space of Square-Root Normal Fields (SRNF) where the [Formula: see text] metric is equivalent to the partial elastic metric in the space of surfaces. Thus, by solving the spatial registration in the SRNF space, the problem of analyzing 4D surfaces becomes the problem of analyzing trajectories embedded in the SRNF space, which has a euclidean structure. In this paper, we develop the building blocks that enable such analysis. These include: (1) the spatiotemporal registration of arbitrarily parameterized 4D surfaces even in the presence of large elastic deformations and large variations in their execution rates; (2) the computation of geodesics between 4D surfaces; (3) the computation of statistical summaries, such as means and modes of variation, of collections of 4D surfaces; and (4) the synthesis of random 4D surfaces. We demonstrate the performance of the proposed framework using 4D facial surfaces and 4D human body shapes.

Data-driven control of agent-based models: An Equation/Variable-free machine learning approach

We present an Equation/Variable free machine learning (EVFML) framework for the control of the collective dynamics of complex/multiscale systems modeled via microscopic/agent-based simulators. The approach obviates the need for construction of surrogate, reduced-order models. The proposed implementation consists of three steps: (A) from high-dimensional agent-based simulations, machine learning, in particular, non-linear manifold learning (Diffusion Maps (DMs)) identifies a set of coarse-grained variables that parametrize the low-dimensional manifold on which the emergent/collective dynamics evolve. The out-of-sample extension and pre-image problems, i.e. the construction of non-linear mappings from the high-dimensional input space to the low-dimensional manifold and back, are solved by coupling DMs with the Nyström extension and Geometric Harmonics, respectively; (B) having identified the manifold and its coordinates, we exploit the Equation-free approach to perform numerical bifurcation analysis of the emergent dynamics; then (C) based on the previous steps, we design data-driven embedded wash-out controllers that drive the agent-based simulators to their intrinsic, imprecisely known, emergent open-loop unstable steady-states, thus demonstrating that the scheme is robust against numerical approximation errors and modeling uncertainty. The efficiency of the framework is illustrated by controlling emergent unstable (i) traveling waves of a deterministic agent-based model of traffic dynamics, and (ii) equilibria of a stochastic financial market model with mimesis.

From snapshots to manifolds – a tale of shear flows

We propose a novel nonlinear manifold learning from snapshot data and demonstrate its superiority over proper orthogonal decomposition (POD) for shedding-dominated shear flows. Key enablers are isometric feature mapping, Isomap, as encoder and,$K$-nearest neighbours ($K$NN) algorithm as decoder. The proposed technique is applied to numerical and experimental datasets including the fluidic pinball, a swirling jet and the wake behind a couple of tandem cylinders. Analysing the fluidic pinball, the manifold is able to describe the pitchfork bifurcation and the chaotic regime with only three feature coordinates. These coordinates are linked to the vortex-shedding phases and the force coefficients. The manifold coordinates of the swirling jet are comparable to the POD mode amplitudes, yet allow for a more distinct and less noise-sensitive manifold identification. A similar observation is made for the wake of two tandem cylinders. The tandem cylinders are aligned and located at a streamwise distance which corresponds to the transition between the single bluff body and the reattachment regimes of vortex shedding. Isomap unveils these two shedding regimes while the Lissajous plot of the first two POD mode amplitudes features a single circle. The reconstruction error of the manifold model is small compared with the fluctuation level, indicating that the low embedding dimensions contain the coherent structure dynamics. The proposed Isomap–$K$NN manifold learner is expected to be of great importance in estimation, dynamic modelling and control for a large range of configurations with dominant coherent structures.

Model reduction for the material point method via an implicit neural representation of the deformation map

This work proposes a model-reduction approach for the material point method on nonlinear manifolds. Our technique approximates the kinematics by approximating the deformation map using an implicit neural representation that restricts deformation trajectories to reside on a low-dimensional manifold. By explicitly approximating the deformation map, its spatiotemporal gradients—in particular the deformation gradient and the velocity—can be computed via analytical differentiation. In contrast to typical model-reduction techniques that construct a linear or nonlinear manifold to approximate the (finite number of) degrees of freedom characterizing a given spatial discretization, the use of an implicit neural representation enables the proposed method to approximate the continuous deformation map. This allows the kinematic approximation to remain agnostic to the discretization. Consequently, the technique supports dynamic discretizations—including resolution changes—during the course of the online reduced-order-model simulation.To generate dynamics for the generalized coordinates, we propose a family of projection techniques. At each time step, these techniques: (1) Calculate full-space kinematics at quadrature points, (2) Calculate the full-space dynamics for a subset of ‘sample’ material points, and (3) Calculate the reduced-space dynamics by projecting the updated full-space position and velocity onto the low-dimensional manifold and tangent space, respectively. We achieve significant computational speedup via hyper-reduction that ensures all three steps execute on only a small subset of the problem's spatial domain. Large-scale numerical examples with millions of material points illustrate the method's ability to gain an order of magnitude computational-cost saving—indeed real-time simulations—with negligible errors.

Parametric control of flexible timing through low-dimensional neural manifolds

Biological brains possess an unparalleled ability to adapt behavioral responses to changing stimuli and environments. How neural processes enable this capacity is a fundamental open question. Previous works have identified two candidate mechanisms: a low-dimensional organization of neural activity and a modulation by contextual inputs. We hypothesized that combining the two might facilitate generalization and adaptation in complex tasks. We tested this hypothesis in flexible timing tasks where dynamics play a key role. Examining trained recurrent neural networks, we found that confining the dynamics to a low-dimensional subspace allowed tonic inputs to parametrically control the overall input-output transform, enabling generalization to novel inputs and adaptation to changing conditions. Reverse-engineering and theoretical analyses demonstrated that this parametric control relies on a mechanism where tonic inputs modulate the dynamics along non-linear manifolds while preserving their geometry. Comparisons with data from behaving monkeys confirmed the behavioral and neural signatures of this mechanism.

Distance-preserving manifold denoising for data-driven mechanics

This article introduces an isometric manifold embedding data-driven paradigm designed to enable model-free simulations with noisy data sampled from a constitutive manifold. The proposed data-driven approach iterates between a global optimization problem that seeks admissible solutions for the balance principle and a local optimization problem that finds the closest point projection of the Euclidean space that isometrically embeds a nonlinear constitutive manifold. To de-noise the database, a geometric autoencoder is introduced such that the encoder first learns to create an approximated embedding that maps the underlying low-dimensional structure of the high-dimensional constitutive manifold onto a flattened manifold with less curvature. We then obtain the noise-free constitutive responses by projecting data onto a denoised latent space that is completely flat by assuming that the noise and the underlying constitutive signal are orthogonal to each other. Consequently, a projection from the conservative manifold onto this de-noised constitutive latent space enables us to complete the local optimization step of the data-driven paradigm. Finally, to decode the data expressed in the latent space without reintroducing noise, we impose a set of isometry constraints while training the autoencoder such that the nonlinear mapping from the latent space to the reconstructed constituent manifold is distance-preserving. Numerical examples are used to both validate the implementation and demonstrate the accuracy, robustness, and limitations of the proposed paradigm.

Learning Structure Aware Deep Spectral Embedding.

Spectral Embedding (SE) has often been used to map data points from non-linear manifolds to linear subspaces for the purpose of classification and clustering. Despite significant advantages, the subspace structure of data in the original space is not preserved in the embedding space. To address this issue subspace clustering has been proposed by replacing the SE graph affinity with a self-expression matrix. It works well if the data lies in a union of linear subspaces however, the performance may degrade in real-world applications where data often spans non-linear manifolds. To address this problem we propose a novel structure-aware deep spectral embedding by combining a spectral embedding loss and a structure preservation loss. To this end, a deep neural network architecture is proposed that simultaneously encodes both types of information and aims to generate structure-aware spectral embedding. The subspace structure of the input data is encoded by using attention-based self-expression learning. The proposed algorithm is evaluated on six publicly available real-world datasets. The results demonstrate the excellent clustering performance of the proposed algorithm compared to the existing state-of-the-art methods. The proposed algorithm has also exhibited better generalization to unseen data points and it is scalable to larger datasets without requiring significant computational resources.

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.

Analytical Study of Face Recognition Methods Using Machine Learning

Many facial recognition algorithms, as well as their adaptations, have previously been devised during the last decades. A number is indicated for standard algorithms. They are divided into two categories: appearance-based and model-based. Three distinct linear subspace analysis strategies for appearance-based processes are presented, along with a brief examination of numerous non-linear manifold analysis approaches for face recognition. Elastic Bunch Graph matching, Active Appearance Model, and 3D Model processes are discussed. A vast variety of publicly available face datasets and performance evaluation ratings are examined. Based on previous recognition outcomes, more study areas are recommended. Face recognition has piqued the interest of academics in biometrics, pattern recognition, and computer vision. Previous research in this field had a basic fault in that it was designed to recognize just single faces, resulting in a long detection time and additional time for annotation. The goal of this study is to create a face recognition system that makes use of a machine learning algorithm and principal component analysis (PCA). The study makes use of linear discriminant analysis, multilayer perceptrons, Naive Bayes, and support vector machines. Furthermore, it obtained 97% and 100% identification accuracy, respectively, using PCA and linear discriminant analysis. The extensive comprehension CNNs are the most frequent form of machine learning algorithm used for facial recognition. CNNs are a sort of artificial neural network that performs exceptionally well at image categorization. Face recognition is a popular topic in artificial intelligence. This program was widely used in our daily life. Several face recognition devices were installed on phones to protect private information and utilized on Facebook to quickly recognize when Facebook users appeared in photographs. Several face recognition algorithms have previously been developed; yet, it remains incredibly difficult in real-world situations. A basic technique for differentiating people is influenced by factors such as partial facial occlusion, illumination, and posture variance.

Cross-Subject Emotion Recognition Based on Domain Similarity of EEG Signal Transfer Learning.

For solving the problem of the inevitable decline in the accuracy of cross-subject emotion recognition via Electroencephalograph (EEG) signal transfer learning due to the negative transfer of data in the source domain, this paper offers a new method to dynamically select the data suitable for transfer learning and eliminate the data that may lead to negative transfer. The method which is called cross-subject source domain selection (CSDS) consists of the next three parts. 1) First, a Frank-copula model is established according to Copula function theory to study the correlation between the source domain and the target domain, which is described by the Kendall correlation coefficient. 2) The calculation method for the Maximum Mean Discrepancy is improved to determine the distance between classes in a single source. After normalization, the Kendall correlation coefficient is superimposed, and the threshold is set to identify the source-domain data most suitable for transfer learning. 3) In the process of transfer learning, on the basis of Manifold Embedded Distribution Alignment, the Local Tangent Space Alignment method is used to provide a low-dimensional linear estimation of the local geometry of nonlinear manifolds, which maintains the local characteristics of the sample data after dimensionality reduction. Experimental results show that compared with the traditional methods, the CSDS increases the accuracy of emotion classification by approximately 2.8% and reduces the runtime by approximately 65%.

Data-driven discovery of intrinsic dynamics

Dynamical models underpin our ability to understand and predict the behaviour of natural systems. Whether dynamical models are developed from first-principles derivations or from observational data, they are predicated on our choice of state variables. The choice of state variables is driven by convenience and intuition, and, in data-driven cases, the observed variables are often chosen to be the state variables. The dimensionality of these variables (and consequently the dynamical models) can be arbitrarily large, obscuring the underlying behaviour of the system. In truth these variables are often highly redundant and the system is driven by a much smaller set of latent intrinsic variables. In this study we combine the mathematical theory of manifolds with the representational capacity of neural networks to develop a method that learns a system’s intrinsic state variables directly from time-series data, as well as predictive models for their dynamics. What distinguishes our method is its ability to reduce data to the intrinsic dimensionality of the nonlinear manifold they live on. This ability is enabled by the concepts of charts and atlases from the theory of manifolds, whereby a manifold is represented by a collection of patches that are sewn together—a necessary representation to attain intrinsic dimensionality. We demonstrate this approach on several high-dimensional systems with low-dimensional behaviour. The resulting framework provides the ability to develop dynamical models of the lowest possible dimension, capturing the essence of a system. Learning minimal representations of dynamical systems is essential for mathematical modelling and prediction in science and engineering. Floryan and Graham propose a deep learning framework able to estimate accurate global dynamical models by sewing together multiple local representations learnt from high-dimensional time-series data.

Convergence Trajectory Optimization of Supertwisting Sliding-Mode Current Control for Induction Motor Drives

For the induction motor (IM) current loop, the supertwisting sliding-mode control (ST-SMC) has been considered as an effective approach to achieve chattering suppression and steady-state errorless control. However, due to the system disturbance, the conventional ST-SMC suffers from control delay in the convergence trajectory, resulting in decreased antidisturbance capability. To address this problem, this article proposed a ST-SMC with convergence trajectory optimization. First, a nonlinear sliding-mode manifold is designed to achieve the ideal ST-SMC convergence trajectory. Then, a disturbance compensation term is added into the control law to eliminate the system control delay. Compared with the conventional ST-SMC, the studied method can effectively enhance the antidisturbance capability of IM current loop, leading to improved transient performance. Finally, the superiority of the investigated method is verified by the experimental results from a 3.7 kW IM test bench.

Reduced order modeling for flow and transport problems with Barlow Twins self-supervised learning

We propose a unified data-driven reduced order model (ROM) that bridges the performance gap between linear and nonlinear manifold approaches. Deep learning ROM (DL-ROM) using deep-convolutional autoencoders (DC–AE) has been shown to capture nonlinear solution manifolds but fails to perform adequately when linear subspace approaches such as proper orthogonal decomposition (POD) would be optimal. Besides, most DL-ROM models rely on convolutional layers, which might limit its application to only a structured mesh. The proposed framework in this study relies on the combination of an autoencoder (AE) and Barlow Twins (BT) self-supervised learning, where BT maximizes the information content of the embedding with the latent space through a joint embedding architecture. Through a series of benchmark problems of natural convection in porous media, BT–AE performs better than the previous DL-ROM framework by providing comparable results to POD-based approaches for problems where the solution lies within a linear subspace as well as DL-ROM autoencoder-based techniques where the solution lies on a nonlinear manifold; consequently, bridges the gap between linear and nonlinear reduced manifolds. We illustrate that a proficient construction of the latent space is key to achieving these results, enabling us to map these latent spaces using regression models. The proposed framework achieves a relative error of 2% on average and 12% in the worst-case scenario (i.e., the training data is small, but the parameter space is large.). We also show that our framework provides a speed-up of 7 times 10^{6} times, in the best case, and 7 times 10^{3} times on average compared to a finite element solver. Furthermore, this BT–AE framework can operate on unstructured meshes, which provides flexibility in its application to standard numerical solvers, on-site measurements, experimental data, or a combination of these sources.

Kernelized transformed subspace clustering with geometric weights for non-linear manifolds

The naive assumption of subspace clustering is that the data should be separable into separate subspaces. Another consideration of the conventional subspace clustering methods is the linear manifolds. What if, the data doesn’t hold this assumption? We propose a novel subspace clustering framework that works even if the raw data is not separable into separate subspaces. It also generalizes it for non-linear manifolds. To achieve the intended goal, we embed subspace clustering techniques into kernelized transform learning with weighting matrix regularization which accounts for nonlinearity. For the weighting matrix, we use similarity between tangent spaces on data manifolds for local structure and euclidean distances for capturing global geometric structure. The complete optimization problem is solved using alternate minimization. The weighted norm regularization is solved by designing a fixed-point continuation algorithm to obtain an approximate closed solution. To test the performance of the proposed framework, we provide the experimental results on handwritten digits clustering, face image clustering, and motion segmentation. The superiority of the results proves the effectiveness of the weighting matrix in kernelized transformed subspace clustering formulation.

Optimisation of the core subset for the APY approximation of genomic relationships

BackgroundBy entering the era of mega-scale genomics, we are facing many computational issues with standard genomic evaluation models due to their dense data structure and cubic computational complexity. Several scalable approaches have been proposed to address this challenge, such as the Algorithm for Proven and Young (APY). In APY, genotyped animals are partitioned into core and non-core subsets, which induces a sparser inverse of the genomic relationship matrix. This partitioning is often done at random. While APY is a good approximation of the full model, random partitioning can make results unstable, possibly affecting accuracy or even reranking animals. Here we present a stable optimisation of the core subset by choosing animals with the most informative genotype data.MethodsWe derived a novel algorithm for optimising the core subset based on a conditional genomic relationship matrix or a conditional single nucleotide polymorphism (SNP) genotype matrix. We compared the accuracy of genomic predictions with different core subsets for simulated and real pig data sets. The core subsets were constructed (1) at random, (2) based on the diagonal of the genomic relationship matrix, (3) at random with weights from (2), or (4) based on the novel conditional algorithm. To understand the different core subset constructions, we visualise the population structure of the genotyped animals with linear Principal Component Analysis and non-linear Uniform Manifold Approximation and Projection.ResultsAll core subset constructions performed equally well when the number of core animals captured most of the variation in the genomic relationships, both in simulated and real data sets. When the number of core animals was not sufficiently large, there was substantial variability in the results with the random construction but no variability with the conditional construction. Visualisation of the population structure and chosen core animals showed that the conditional construction spreads core animals across the whole domain of genotyped animals in a repeatable manner.ConclusionsOur results confirm that the size of the core subset in APY is critical. Furthermore, the results show that the core subset can be optimised with the conditional algorithm that achieves an optimal and repeatable spread of core animals across the domain of genotyped animals.

Neural Subspace Learning for Surface Defect Detection

Surface defect inspection is a key technique in industrial product assessments. Compared with other visual applications, industrial defect inspection suffers from a small sample problem and a lack of labeled data. Therefore, conventional deep-learning methods depending on huge supervised samples cannot be directly generalized to this task. To deal with the lack of labeled data, unsupervised subspace learning provides more clues for the task of defect inspection. However, conventional subspace learning methods focus on studying the linear subspace structure. In order to explore the nonlinear manifold structure, a novel neural subspace learning algorithm is proposed by substituting linear operators with nonlinear neural networks. The low-rank property of the latent space is approximated by limiting the dimensions of the encoded feature, and the sparse coding property is simulated by quantized autoencoding. To overcome the small sample problem, a novel data augmentation strategy called thin-plate-spline deformation is proposed. Compared with the rigid transformation methods used in previous literature, our strategy could generate more reliable training samples. Experiments on real-world datasets demonstrate that our method achieves state-of-the-art performance compared with unsupervised methods. More importantly, the proposed method is competitive and has a better generalization capability compared with supervised methods based on deep learning techniques.

NIMG-22. AN AI-BASED COORDINATE SYSTEM ELUCIDATES RADIOGENOMIC HETEROGENEITY OF GLIOBLASTOMA VIA DEEP LEARNING AND MANIFOLD EMBEDDINGS

Abstract PURPOSE There is evidence that molecular heterogeneity of glioblastoma is associated with heterogeneity of MR imaging signatures. Modern machine learning models, such as deep neural networks, provide a tool for capturing such complex relationships in high-dimensional datasets. This study leverages recent advances in visualizing neural networks to construct a radiogenomics coordinate system whose axes reflect the expression of imaging signatures of genetic mutations commonly found in glioblastoma. METHODS Multi-parametric MRI (mpMRI) scans (T1, T1-Gd, T2, T2-FLAIR, DSC, DTI) of 254 subjects with glioblastoma were retrospectively collected. Radiomics features, including histograms, morphologic and textural descriptors, were derived. Genetic markers were obtained through next generation sequencing (NGS) panel. A multi-label classification deep neural network was trained for predicting mutation status in key driver genes, EGFR, PTEN, NF1, TP53 and RB1. We utilized a nonlinear manifold learning method called Intensive Principal Component Analysis (InPCA), to visualize the output probability distributions from the trained model. The first three principal components (PCs) were selected for constructing the coordinate system. RESULTS The axes derived from InPCA analysis were associated with molecular pathways known to be implicated in glioblastoma: (1) Increasing values of PC1 were associated with primary involvement of P53 then RB1 then MAPK then RTK/PI3K; (2) Increasing values of PC2 were associated with primary involvement of RTK then RB1/P53/MAPK then PI3K; (3) Increasing values of PC3 were associated with primary involvement of MAPK then RB1/P53/RTK/PI3K. Imaging features significantly associated with each of three PCs (p< 0.05) were identified by Pearson correlation analysis. CONCLUSION Deep learning followed by nonlinear manifold embedding identifies a radiogenomics coordinate system spanned by three components which were associated with different molecular pathways of glioblastoma. The heterogeneity of radiogenomic signatures captured by this coordinate system offers in vivo biomarkers of the molecular heterogeneity of glioblastoma.

Model reduction of convection-dominated partial differential equations via optimization-based implicit feature tracking

This work introduces a new approach to reduce the computational cost of solving partial differential equations (PDEs) with convection-dominated solutions: model reduction with implicit feature tracking. Traditional model reduction techniques use an affine subspace to reduce the dimensionality of the solution manifold and, as a result, yield limited reduction and require extensive training due to the slowly decaying Kolmogorov n-width of convection-dominated problems. The proposed approach circumvents the slowly decaying n-width limitation by using a nonlinear approximation manifold systematically defined by composing a low-dimensional affine space with a space of bijections of the underlying domain. Central to the implicit feature tracking approach is a residual minimization problem over the reduced nonlinear manifold that simultaneously determines the reduced coordinates in the affine space and the domain mapping that minimize the residual of the unreduced PDE discretization. This is analogous to standard minimum-residual reduced-order models, except instead of only minimizing the residual over the affine subspace of PDE states, our method enriches the optimization space to also include admissible domain mappings. The nonlinear trial manifold is constructed by using the proposed residual minimization formulation to determine domain mappings that cause parametrized features to align in a reference domain for a set of training parameters. Because the feature is stationary in the reference domain, i.e., the convective nature of solution removed, the snapshots are effectively compressed to define an affine subspace. The space of domain mappings, originally constructed using high-order finite elements, is also compressed in a way that ensures the boundaries of the original domain are maintained. Several numerical experiments are provided, including transonic and supersonic, inviscid, compressible flows, to demonstrate the potential of the method to yield accurate approximations to convection-dominated problems with limited training.

Study on general criteria for generalized chaotic synchronization with a desired manifold

One of the techniques to improve the underwater acoustic stealth performance of ships is to use the wide-spectrum characteristics of chaotic signals to weaken the line spectrum of radiated underwater acoustic signals. To modulate the signal in an expected way, this paper proposes a criterion for constructing a bidirectionally coupled system based on the original system, which enables the response system and the drive system to achieve generalized chaotic synchronization with the desired manifold and gives a method to determine the reliable interval of coupling coefficients. The simplified criterion is suitable for constructing a unidirectionally coupled system. Taking the Chen system and Chua’s circuit as examples, the reliable interval of coupling coefficient is deduced by using the proposed criterion. Bidirectionally coupled chaotic systems are constructed for the desired linear manifold and nonlinear manifold, and the validity of the construction method and the criterion is verified by numerical simulation.