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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
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Tensor-train WENO scheme for compressible flows
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Tensor Train Methods in Quantum Algorithms for Molecular Dynamics
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First Efforts at Tensor-Train for Space-Time DPG Problems
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Three-Dimensional Compressible Chemically Reacting Computational Fluid Dynamics with Tensor Trains
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Automated synthesis of low-rank control systems from sc-LTL specifications using tensor-train decompositions.
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Sampling Polynomial Chaos and Functional Tensor Train Multilevel/Multifidelity Strategies for Forward UQ.
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Approximating Data with Stochastic and Physical Dependence using the Functional Tensor Train Models.
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Low-Rank Functional Tensor Train Representations for High-Dimensional Computational Models.
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A tensor train-based isogeometric solver for large-scale 3D poisson problems
We introduce a three-dimensional (3D), fully tensor train (TT) assembled isogeometric analysis (IGA) framework, TT-IGA, for solving partial differential equations (PDEs). Our method reformulates IGA discrete operators into TT format, enabling efficient compression and computation. Geometry evaluations use the original NURBS description at sampling points and TT approximation is applied to geometry-derived coefficient fields and discrete operators. We demonstrate the effectiveness of the proposed TT-IGA framework on the three-dimensional Poisson equation, achieving substantial reductions in memory and computational cost without compromising solution quality.
Space-Time Finite Element Tensor Network Approach for the Time-Dependent Convection–Diffusion–Reaction Equation with Variable Coefficients
In this paper, we present a new space-time Galerkin-like method, where we treat the discretization of spatial and temporal domains simultaneously. This method utilizes a mixed formulation of the tensor-train (TT) and quantized tensor-train (QTT) (please see Section Tensor-Train Decomposition), designed for the finite element discretization (Q1-FEM) of the time-dependent convection–diffusion–reaction (CDR) equation. We reformulate the assembly process of the finite element discretized CDR to enhance its compatibility with tensor operations and introduce a low-rank tensor structure for the finite element operators. Recognizing the banded structure inherent in the finite element framework’s discrete operators, we further exploit the QTT format of the CDR to achieve greater speed and compression. Additionally, we present a comprehensive approach for integrating variable coefficients of CDR into the global discrete operators within the TT/QTT framework. The effectiveness of the proposed method, in terms of memory efficiency and computational complexity, is demonstrated through a series of numerical experiments, including a semi-linear example.
Georgia Tech Accelerated, Compressed, and Regularized Compute of Kinetic-based PDEs (Final Report)
This report summarizes the collaborative effort between Lawrence Livermore National Laboratory and Georgia Tech to enhance the BoBa library for tensor train computation in PDE solvers, with a target on kinetic equations and their continuum limits. We aimed to reduce computational cost and memory usage by replacing traditional array-based computations with tensor trains. We examined the compressibility of time-evolving solutions to the Euler equations with discontinuities. We also explored using the first invsicid and linear regularization of the compressible flow equations via the information geometric regularization (IGR). We explored this in a tensor train formulation. To identify that inverse terms in the IGR equations pose problems for tensor train formulations and investigate efficient methods for batched inversion of tensor trains.
Collocation methods for nonlinear differential equations on low-rank manifolds
We introduce new methods for integrating nonlinear differential equations on low-rank manifolds. These methods rely on interpolatory projections onto the tangent space, enabling low-rank time integration of vector fields that can be evaluated entry-wise. A key advantage of our approach is that it does not require the vector field to exhibit low-rank structure, thereby overcoming significant limitations of traditional dynamical low-rank methods based on orthogonal projection. To construct the interpolatory projectors, we develop a sparse tensor sampling algorithm based on the discrete empirical interpolation method (DEIM) that parameterizes tensor train manifolds and their tangent spaces with cross interpolation. Using these projectors, we propose two time integration schemes on low-rank tensor train manifolds. The first scheme integrates the solution at selected interpolation indices and constructs the solution with cross interpolation. The second scheme generalizes the well-known orthogonal projector-splitting integrator to interpolatory projectors. We demonstrate the proposed methods with applications to several tensor differential equations arising from the discretization of partial differential equations.
Towards Compact Neural Networks via End-to-End Training: A Bayesian Tensor Approach with Automatic Rank Determination
Post-training model compression can reduce the inference costs of deep neural networks, but uncompressed training still consumes enormous hardware resources and energy. To enable low-energy training on edge devices, it is highly desirable to directly train a compact neural network from scratch with a low memory cost. Low-rank tensor decomposition is an effective approach to reduce the memory and computing costs of large neural networks. However, directly training low-rank tensorized neural networks is a very challenging task because it is hard to determine a proper tensor rank a priori, and the tensor rank controls both model complexity and accuracy. Here, this paper presents a novel end-to-end framework for low-rank tensorized training. We first develop a Bayesian model that supports various low-rank tensor formats (e.g., CANDECOMP/PARAFAC, Tucker, tensor-train, and tensor-train matrix) and reduces neural network parameters with automatic rank determination during training. Then we develop a customized Bayesian solver to train large-scale tensorized neural networks. Our training methods shows orders-of-magnitude parameter reduction and little accuracy loss (or even better accuracy) in the experiments. On a very large deep learning recommendation system with over 4.2 ×10 9 model parameters, our method can reduce the parameter number to 1.6 ×10 5 automatically in the training process (i.e., by 2.6 ×10 4 times) while achieving almost the same accuracy. Code is available at https://github.com/colehawkins/bayesian-tensor-rank-determination.
Neutron (and other Particle) Transport at LANL: An Overview [Presentation]
For decades, Los Alamos National Laboratory has been at the forefront of neutron transport methods research and code development. One such code is PARTISN, the LANL parallel time-dependent discrete ordinate neutron transport code. In this presentation, we describe the various research efforts currently underway by the PARTISN and other code teams. Some examples of current research are a block automated mesh refinement scheme, the application of tensor trains to the discretized neutron transport equation, and GPU code porting. The block automated mesh refinement scheme uses cross section information to refine and coarsen the solution mesh to improve time to solution and reduce memory. The tensor train approach expresses discretized transport operators as tensor products of vectors and matrices to compress the size of linear systems being solved by transport codes. Rather than relying on matrix-free methods such as the transport sweep, we have access to an operator that can be inverted, reshaped, or manipulated algebraically. Finally, we describe how PARTISN is used, what problems we are looking to solve, and what the future holds for neutron transport at LANL. In addition to this, we briefly describe the various research efforts in other particle transport teams using both deterministic and Monte Carlo methods. In the presentation, we list possible opportunities for collaboration between the laboratory and faculty and students.
A sequential DNN based Baseline Energy Prediction Framework with Long term Error Mitigation
In this paper,we aim to present a novel sequential framework containing two deep network architectures for baseline energy prediction, applied to a simulated commercial building dataset. We have observed that this sequential framework helps to improve long term prediction accuracy, there by mitigating prediction error accumulation over time. The proposed framework utilizes convolution layers to extract features from the input data space without changing spatial relations between variables. These features are memorized by tensor train based gated recurrent units for an accurate long-term prediction. Although, architecture comprising the amalgamation of convolution layers and memory cell have shown promising results in domains such as social media analysis, language modeling, video frame prediction and image recognition, this paper extends its scope to the context of energy applications. Furthermore, the addition of tensor train based gated recurrent units is motivated by the necessity of computational time reduction during the training process, there by making this framework more suitable for future in-field deployment. Lastly, results show the current framework outperforms other existing machine learning based methods, in both short-term and long-term prediction categories.
A Low-Rank QTT-based Finite Element Method for Elasticity Problems
We present an efficient and robust numerical algorithm for solving the linear elasticity problem that combines the Quantized Tensor Train format and a domain partitioning strategy. This approach makes it possible to solve the linear elasticity problem on a computational domain that is more general than a square. By integrating Z-ordering and subdomain concatenation, our method substantially decreases memory usage and achieves a notable reduction in rank compared to established Finite Element implementations like the FEniCS platform. This efficiency is maintained while still guaranteeing exponential convergence with respect to the number of degrees of freedom. This performance gain, however, requires a fundamental rethinking of how core finite element operations are implemented. This includes changes to mesh discretization, node and degree of freedom ordering, stiffness matrix and internal nodal force assembly, and the execution of algebraic matrix-vector operations. In this work, we discuss all these aspects in detail and assess the method’s performance in the numerical approximation of three representative test cases.