Efficient nonlinear manifold reduced order model
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Mathematics and Computing↗
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Results for “Manifold model”
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Operator inference for non-intrusive model reduction with quadratic manifolds
This paper proposes a novel approach for learning a data-driven quadratic manifold from high-dimensional data, then employing this quadratic manifold to derive efficient physics-based reduced-order models. The key ingredient of the approach is a polynomial mapping between high-dimensional states and a low-dimensional embedding. This mapping consists of two parts: a representation in a linear subspace (computed in this work using the proper orthogonal decomposition) and a quadratic component. The approach can be viewed as a form of data-driven closure modeling, since the quadratic component introduces directions into the approximation that lie in the orthogonal complement of the linear subspace, but without introducing any additional degrees of freedom to the low-dimensional representation. Combining the quadratic manifold approximation with the operator inference method for projection-based model reduction leads to a scalable non-intrusive approach for learning reduced-order models of dynamical systems. Further, applying the new approach to transport-dominated systems of partial differential equations illustrates the gains in efficiency that can be achieved over approximation in a linear subspace.
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Learning physics-based reduced-order models from data using nonlinear manifolds
Here we present a novel method for learning reduced-order models of dynamical systems using nonlinear manifolds. First, we learn the manifold by identifying nonlinear structure in the data through a general representation learning problem. The proposed approach is driven by embeddings of low-order polynomial form. A projection onto the nonlinear manifold reveals the algebraic structure of the reduced-space system that governs the problem of interest. The matrix operators of the reduced-order model are then inferred from the data using operator inference. Numerical experiments on a number of nonlinear problems demonstrate the generalizability of the methodology and the increase in accuracy that can be obtained over reduced-order modeling methods that employ a linear subspace approximation.
97 MATHEMATICS AND COMPUTING↗
Nonlinear manifold-based reduced order model
Traditional linear subspace reduced order models (LS-ROMs) are able to accelerate physical simulations, in which the intrinsic solution space falls into a subspace with a small dimension, i.e., the solution space has a small Kolmogorov n-width. However, for physical phenomena not of this type, e.g., any advection-dominated flow phenomena such as in traffic flow, atmospheric flows, and air flow over vehicles, a low-dimensional linear subspace poorly approximates the solution. To address cases such as these, we have developed a fast and accurate physicsinformed neural network ROM, namely nonlinear manifold ROM (NM-ROM), which can better approximate high-fidelity model solutions with a smaller latent space dimension than the LSROMs. Our method takes advantage of the existing numerical methods that are used to solve the corresponding full order models. The efficiency is achieved by developing a hyper-reduction technique in the context of the NM-ROM. Numerical results show that neural networks can learn a more efficient latent space representation on advection-dominated data from 1D and 2D Burgers’ equations. A speedup of up to 2.6 for 1D Burgers’ and a speedup of 11.7 for 2D Burgers’ equations are achieved with an appropriate treatment of the nonlinear terms through a hyper-reduction technique. Finally, a posteriori error bounds for the NM-ROMs are derived that take account of the hyper-reduced operators.
97 MATHEMATICS AND COMPUTING↗
State Estimation Model Reduction Through the Manifold Boundary Approximation Method
Not provided.
Engineering↗
Manifold traversing as a model for learning control of autonomous robots
This paper describes a recipe for the construction of control systems that support complex machines such as multi-limbed/multi-fingered robots. The robot has to execute a task under varying environmental conditions and it has to react reasonably when previously unknown conditions are encountered. Its behavior should be learned and/or trained as opposed to being programmed. The paper describes one possible method for organizing the data that the robot has learned by various means. This framework can accept useful operator input even if it does not fully specify what to do, and can combine knowledge from autonomous, operator assisted and programmed experiences.
Szakaly, Zoltan F.↗
Domain decomposition-based reduced-order models using nonlinear-manifolds and interpolatory projections
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Mathematics and Computing↗
Data-driven surrogates for high dimensional models using Gaussian process regression on the Grassmann manifold
This paper introduces a surrogate modeling scheme based on Grassmannian manifold learning to be used for cost-efficient predictions of high-dimensional stochastic systems. The method exploits subspace-structured features of each solution by projecting it onto a Grassmann manifold. This point-wise linear dimensionality reduction harnesses the structural information to assess the similarity between solutions at different points in the input parameter space. The method utilizes a solution clustering approach in order to identify regions of the parameter space over which solutions are sufficiently similarly such that they can be interpolated on the Grassmannian. In this clustering, the reduced-order solutions are partitioned into disjoint clusters on the Grassmann manifold using the eigen-structure of properly defined Grassmannian kernels and, the Karcher mean of each cluster is estimated. Then, the points in each cluster are projected onto the tangent space with origin at the corresponding Karcher mean using the exponential mapping. For each cluster, a Gaussian process regression model is trained that maps the input parameters of the system to the reduced solution points of the corresponding cluster projected onto the tangent space. Using this Gaussian process model, the full-field solution can be efficiently predicted at any new point in the parameter space. In certain cases, the solution clusters will span disjoint regions of the parameter space. In such cases, for each of the solution clusters we utilize a second, density-based spatial clustering to group their corresponding input parameter points in the Euclidean space. The proposed method is applied to two numerical examples. Here, the first is a nonlinear stochastic ordinary differential equation with uncertain initial conditions where the surrogate is used to predict the time history solution. The second involves modeling of plastic deformation in a model amorphous solid using the Shear Transformation Zone theory of plasticity, where the proposed surrogate is used to predict the full strain field of a material specimen under large shear strains.
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Microscale mechanical modeling of deformable geomaterials with dynamic contacts based on the numerical manifold method
Abstract Micromechanical modeling of geomaterials is challenging because of the complex geometry of discontinuities and potentially large number of deformable material bodies that contact each other dynamically. In this study, we have developed a numerical approach for micromechanical analysis of deformable geomaterials with dynamic contacts. In our approach, we detect contacts among multiple blocks with arbitrary shapes, enforce different contact constraints for three different contact states of separated, bonded, and sliding, and iterate within each time step to ensure convergence of contact states. With these features, we are able to simulate the dynamic contact evolution at the microscale for realistic geomaterials having arbitrary shapes of grains and interfaces. We demonstrate the capability with several examples, including a rough fracture with different geometric surface asperity characteristics, settling of clay aggregates, compaction of a loosely packed sand, and failure of an intact marble sample. With our model, we are able to accurately analyze (1) large displacements and/or deformation, (2) the process of high stress accumulated at contact areas, (3) the failure of a mineral cemented rock samples under high stress, and (4) post-failure fragmentation. The analysis highlights the importance of accurately capturing (1) the sequential evolution of geomaterials responding to stress as motion, deformation, and high stress; (2) large geometric features outside the norms (such as large asperities and sharp corners) as such features can dominate the micromechanical behavior; and (3) different mechanical behavior between loosely packed and tightly packed granular systems.
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Extended Finite Element Based Approach in Additive Manufacturing Modeling for Optimizing Highly Complex Manifold in Protonic Ceramic Electrochemical Cells
The objective of this project is to develop a novel numerical approach based on the extended finite element method (XFEM) for modeling moving boundaries of material deposited in additive manufacturing (AM) process with improved accuracy and reduced computational cost. One of the major challenges in simulating AM processes is that the boundaries of the computational domain need to change as material is deposited. Previous approaches have achieved this by activating new finite elements, but this requires a high level of mesh refinement to capture small movement of the boundary. XFEM allows the solution boundary to move independently of the mesh, permitting smooth representation of the evolution of the boundary as material is deposited. As a result, this new method will significantly improve the simulation accuracy while reducing the computational cost compared with existing approaches. Furthermore, this project will bring significant improvements for the design and AM process optimization in an iterative fashion among numerical simuation, parameter optimization, and experimental validation. Specifically, this project will support the protonic ceramic electrochemical cells (PCEC) stack development at INL by providing insights into the density, residual stress, thermal and mechanical properties of the PCEC manifold and interconnect, which is critical in enhancing the system lifetime and reducing the overall cost. Upon the success of proposed development and validation of the proposed approach, simulation will be applied to determine an optimal set of AM process parameters for the PCEC manifold production. This project will consolidate Idaho National Laboratory (INL)’s simulation capabilities provided by the MOOSE framework and the Valhalla AM simulation application, with encouraging expansion to emerging PCEC applications and AM technologies at INL and beyond.
97 MATHEMATICS AND COMPUTING↗
Signature analysis of high-throughput transcriptomics screening data for mechanistic inference and chemical grouping
Abstract High-throughput transcriptomics (HTTr) uses gene expression profiling to characterize the biological activity of chemicals in in vitro cell-based test systems. As an extension of a previous study testing 44 chemicals, HTTr was used to screen an additional 1,751 unique chemicals from the EPA’s ToxCast collection in MCF7 cells using 8 concentrations and an exposure duration of 6 h. We hypothesized that concentration-response modeling of signature scores could be used to identify putative molecular targets and cluster chemicals with similar bioactivity. Clustering and enrichment analyses were conducted based on signature catalog annotations and ToxPrint chemotypes to facilitate molecular target prediction and grouping of chemicals with similar bioactivity profiles. Enrichment analysis based on signature catalog annotation identified known mechanisms of action (MeOAs) associated with well-studied chemicals and generated putative MeOAs for other active chemicals. Chemicals with predicted MeOAs included those targeting estrogen receptor (ER), glucocorticoid receptor (GR), retinoic acid receptor (RAR), the NRF2/KEAP/ARE pathway, AP-1 activation, and others. Using reference chemicals for ER modulation, the study demonstrated that HTTr in MCF7 cells was able to stratify chemicals in terms of agonist potency, distinguish ER agonists from antagonists, and cluster chemicals with similar activities as predicted by the ToxCast ER Pathway model. Uniform manifold approximation and projection (UMAP) embedding of signature-level results identified novel ER modulators with no ToxCast ER Pathway model predictions. Finally, UMAP combined with ToxPrint chemotype enrichment was used to explore the biological activity of structurally related chemicals. The study demonstrates that HTTr can be used to inform chemical risk assessment by determining in vitro points of departure, predicting chemicals’ MeOA and grouping chemicals with similar bioactivity profiles.
Toxicology↗
Double Diffusion Maps and their Latent Harmonics for scientific computations in latent space
In this work, we introduce a data-driven approach to building reduced dynamical models through manifold learning; the reduced latent space is discovered using Diffusion Maps (a manifold learning technique) on time series data. A second round of Diffusion Maps on those latent coordinates allows the approximation of the reduced dynamical models. This second round enables mapping the latent space coordinates back to the full ambient space (what is called lifting); it also enables the approximation of full state functions of interest in terms of the reduced coordinates. In our work, we develop and test three different reduced numerical simulation methodologies, either through pre-tabulation in the latent space and integration on the fly or by going back and forth between the ambient space and the latent space. The data-driven latent space simulation results, based on the three different approaches, are validated through (a) the latent space observation of the full simulation through the Nyström Extension formula, or through (b) lifting the reduced trajectory back to the full ambient space, via Latent Harmonics. Latent space modeling often involves additional regularization to favor certain properties of the space over others, and the mapping back to the ambient space is then constructed mostly independently from these properties; here, we use the same data-driven approach to construct the latent space and then map back to the ambient space.
97 MATHEMATICS AND COMPUTING↗
Data-driven prediction of α IIb β 3 integrin activation paths using manifold learning and deep generative modeling
The integrin heterodimer is a transmembrane protein critical for driving cellular process and is a therapeutic target in the treatment of multiple diseases linked to its malfunction. Activation of integrin involves conformational transitions between bent and extended states. Some of the conformations that are intermediate between bent and extended states of the heterodimer have been experimentally characterized, but the full activation pathways remain unresolved both experimentally due to their transient nature and computationally due to the challenges in simulating rare barrier crossing events in these large molecular systems. An understanding of the activation pathways can provide new fundamental understanding of the biophysical processes associated with the dynamic interconversions between bent and extended states and unveil new putative therapeutic targets. In this work, we apply nonlinear manifold learning to coarse-grained molecular dynamics simulations of bent, extended, and two intermediate states of αI I b β3 integrin to learn a low-dimensional embedding of the configurational phase space. We then train deep generative models to learn an inverse mapping between the low-dimensional embedding and high-dimensional molecular space and use these models to interpolate the molecular configurations constituting the activation pathways between the experimentally characterized states. Furthermore, this work furnishes plausible predictions of integrin activation pathways and reports a generic and transferable multi-scale technique to predict transition pathways for biomolecular systems.
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Gradient-based constrained optimization using a database of linear reduced-order models
A methodology grounded in model reduction is presented for accelerating the gradient-based solution of a family of linear or nonlinear constrained optimization problems where the constraints include at least one linear Partial Differential Equation (PDE). A key component of this methodology is the construction, during an offline phase, of a database of pointwise, linear, Projection-based Reduced-Order Models (PROM)s associated with a design parameter space and the linear PDE(s). A parameter sampling procedure based on an appropriate saturation assumption is proposed to maximize the efficiency of such a database of PROMs. A real-time method is also presented for interpolating at any queried but unsampled parameter vector in the design parameter space the relevant sensitivities of a PROM. The practical feasibility, computational advantages, and performance of the proposed methodology are demonstrated for several realistic, nonlinear, aerodynamic shape optimization problems governed by linear aeroelastic constraints.
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G2Aero: A Python package for separable shape tensors
G2Aero is a Python package for the design and deformation of discrete planar curves and tubular surfaces using a geometric data-driven approach. G2Aero utilizes a topology of product manifolds: the Grassmannian, $\mathcal{G}$($\mathcal{n}$, 2) - the set of 2-dimensional subspaces in $\mathbb{R}$ $\mathcal{n}$ - and the symmetric positive-definite (SPD) manifold, $\mathcal{S}^{2}_{++}$ - the set of 2x2 SPD matrices. The package provides a novel framework for representing separable deformations to shapes, which consist of stretching, scaling, rotating, and translating - also known as affine deformations - and a set of complementary deformations - which we refer to as undulation-type deformations. We focus on airfoil and blade design applications to emphasize the utility of the methods in an environment where the separation of affine and undulation-type deformations is critical. Notable functionalities of the framework for blade design include: 1) generating novel 2D (airfoil) shapes informed by a database of physically relevant airfoils, 2) building 3D blades by interpolating sequences of 2D airfoil cross-sections, and 3) generating blades with consistent perturbations along the blade span. We discuss the framework and provide examples in the context of wind energy applications, specifically wind turbine blade design. Figure 1 shows the wire frame obtained by interpolating airfoils defining the IEA 15-MW wind turbine blade and applying affine transformations corresponding to twist, chordal scaling, and bending. This, and all other figures in the paper, can be reproduced following examples and referencing supporting documentation provided in the G2Aero package.
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Deep nonparametric estimation of intrinsic data structures by chart autoencoders: Generalization error and robustness
Autoencoders have demonstrated remarkable success in learning low-dimensional latent features of high-dimensional data across various applications. Assuming that data are sampled near a low-dimensional manifold, we employ chart autoencoders, which encode data into low-dimensional latent features on a collection of charts, preserving the topology and geometry of the data manifold. Our paper establishes statistical guarantees on the generalization error of chart autoencoders, and we demonstrate their denoising capabilities by considering n noisy training samples, along with their noise-free counterparts, on a d-dimensional manifold. By training autoencoders, we show that chart autoencoders can effectively denoise the input data with normal noise. We prove that, under proper network architectures, chart autoencoders achieve a squared generalization error in the order of n–$\frac{2}{d+2}$log 4 n, which depends on the intrinsic dimension of the manifold and only weakly depends on the ambient dimension and noise level. We further extend our theory on data with noise containing both normal and tangential components, where chart autoencoders still exhibit a denoising effect for the normal component. As a special case, our theory also applies to classical autoencoders, as long as the data manifold has a global parametrization. Furthermore, our results provide a solid theoretical foundation for the effectiveness of autoencoders, which is further validated through several numerical experiments.
97 MATHEMATICS AND COMPUTING↗
A transfer learning approach for acoustic emission zonal localization on steel plate-like structure using numerical simulation and unsupervised domain adaptation
The detection and localization of damage in metallic structures using acoustic emission (AE) monitoring and artificial intelligence technology such as deep learning has been widely studied. However, a current challenge of this approach is the difficulty of obtaining sufficient labeled historical AE signals for the training process of deep learning models. This problem can be approached through the implementation of transfer learning. The innovation of this paper lies in the development of a transfer learning approach for AE source localization on a stainless-steel structure when no historical labeled AE signals are available for training. A finite element model is developed to generate numerical AE signals for the training. Unsupervised domain adaptation (UDA) technology is utilized to reduce the distribution difference between the numerical and the realistic AE signals and to derive the localization results of the unlabeled realistic AE signals. Finally, the results suggest that the proposed approach is capable of localizing AE signals with high accuracy in the absence of labeled training data.
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Separable shape tensors for aerodynamic design
Airfoil shape design is a classical problem in engineering and manufacturing. In this work, we combine principled physics-based considerations for the shape design problem with modern computational techniques using a data-driven approach. Modern and traditional analyses of two-dimensional (2D) and three-dimensional (3D) aerodynamic shapes reveal a flow-based sensitivity to specific deformations that can be represented generally by affine transformations (rotation, scaling, shearing, and translation). We present a novel representation of shapes that decouples affine-style deformations over a submanifold and a product submanifold principally of the Grassmannian. As an analytic generative model, the separable representation, informed by a database of physically relevant airfoils, offers: (i) a rich set of novel 2D airfoil deformations not previously captured in the data, (ii) an improved low-dimensional parameter domain for inferential statistics informing design/manufacturing, and (iii) consistent 3D blade representation and perturbation over a sequence of nominal 2D shapes.
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