Search NASA⌕ Search

SEARCH · Search NASA

Results for “Basis functions”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Adaptive Methods for Radial Basis Functions

Radial basis functions (RBFs) are a powerful tool for constructing high-order accurate reduced representations of scattered data in arbitrary dimension and on manifolds. We present a method of constructing data approximations in which we utilize a functional tail to capture a global background profile and a RBF neural network (NN) to capture the smaller-scale features. In the RBF NN the RBF centers, matrix shape parameters were selected adaptively for each RBF. We also utilized a geodesic notion of distance on the manifold on which the data lies, e.g., the spherical geodesic for data on the sphere. Although each of these ideas have been been investigated separately in previous works, their combination into a single algorithm is novel. We defined a machine learning problem in which these properties are learned to minimize the data reduction error. We demonstrate the algorithm for applications of scattered data reduction in the plane and on the sphere.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Relativistic corrections to the correlated basis function effective nuclear Hamiltonian

We discuss the inclusion of relativistic boost corrections into the correlated basis function effective nuclear Hamiltonian, derived from a realistic model of two- and three-nucleon interactions using the formalism of correlated basis functions and the cluster expansion technique. Different procedures to take into account the effects of boost interactions are compared on the basis of the ability to reproduce the nuclear matter equation of state obtained from accurate quantum many-body calculations. Furthermore, the results of our study show that the repulsive contribution of the boost interaction significantly depends on the underlying model of the nonrelativistic potential. On the other hand, the dominant relativistic correction turns out to be the corresponding reduction of the strength of repulsive three-nucleon interactions, leading to a significant softening of the equation of state of nuclear matter at supranuclear densities.

Neutron stars & pulsars↗
DOE OSTI2024

A new computational framework for spinor-based relativistic exact two-component calculations using contracted basis functions

Here, a new computational framework for spinor-based relativistic exact two-component (X2C) calculations is developed using contracted basis sets with a spin–orbit contraction scheme. Generally contracted, j-adapted basis sets of p-block elements using primitive functions in the correlation-consistent basis sets are constructed for the X2C Hamiltonian with atomic mean-field spin–orbit integrals (the X2CAMF scheme). The contraction coefficients are taken from atomic X2CAMF Hartree–Fock spinors, thereby following the simple concept of a linear combination of atomic orbitals. Benchmark calculations of spin–orbit splittings, equilibrium bond lengths, and harmonic vibrational frequencies demonstrate the accuracy and efficacy of the j-adapted spin–orbit contraction scheme.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2024

A flexible class of priors for orthonormal matrices with basis function-specific structure

Statistical modeling of high-dimensional matrix-valued data motivates the use of a low-rank representation that simultaneously summarizes key characteristics of the data and enables dimension reduction. Low-rank representations commonly factor the original data into the product of orthonormal basis functions and weights, where each basis function represents an independent feature of the data. However, the basis functions in these factorizations are typically computed using algorithmic methods that cannot quantify uncertainty or account for basis function correlation structure a priori. While there exist Bayesian methods that allow for a common correlation structure across basis functions, empirical examples motivate the need for basis function-specific dependence structure. We propose a prior distribution for orthonormal matrices that can explicitly model basis function-specific structure. The prior is used within a general probabilistic model for singular value decomposition to conduct posterior inference on the basis functions while accounting for measurement error and fixed effects. We discuss how the prior specification can be used for various scenarios and demonstrate favorable model properties through synthetic data examples. Finally, we apply our method to two-meter air temperature data from the Pacific Northwest, enhancing our understanding of the Earth system’s internal variability.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Neural chaos: A spectral stochastic neural operator

Building surrogate models for operators with uncertainty quantification capabilities is essential for many engineering applications where randomness–such as variability in material properties, boundary conditions, and initial conditions–is unavoidable. Polynomial Chaos Expansion (PCE) is widely recognized as a go-to method for constructing stochastic surrogates in both intrusive and non-intrusive ways, and it has recently been used in the context of operator learning. However, its application becomes challenging for complex or high-dimensional processes, as achieving accuracy requires higher-order polynomials, which can increase computational demand and/or the risk of overfitting. Furthermore, PCE requires specialized treatments to manage random variables that are not independent, and these treatments may be problem-dependent or may fail with increasing complexity. Here, in this work, we adopt the same formalism as the spectral expansion used in PCE; however, we replace the classical polynomial basis functions with neural network (NN) basis functions to leverage their expressivity. To achieve this, we propose an algorithm that identifies NN-parameterized basis functions in a purely data-driven manner, without any prior assumptions about the joint distribution of the random variables involved, whether independent or dependent, or about their marginal distributions. The proposed algorithm identifies each NN-parameterized basis function sequentially, ensuring they are orthogonal with respect to the data distribution. The basis functions are constructed directly on the joint stochastic variables without requiring a tensor product structure or assuming independence of the random variables. This approach may offer greater flexibility for complex stochastic models, while simplifying implementation compared to the tensor product structures typically used in PCE to handle random vectors. This is particularly advantageous given the current state of open-source packages, where building and training neural networks can be done with just a few lines of code and extensive community support. We demonstrate the effectiveness of the proposed scheme through several numerical examples of varying complexity and provide comparisons with classical PCE.

Polynomial chaos expansion↗
DOE OSTI2024

A Characteristics Approach to the Finite Element Method

Herein, we present a new method for solving the linear Boltzmann transport equation. Two commonly used and well-understood methods for solving partial differential equations are the method of characteristics (MOC) and the finite element method (FEM). We propose a new method that combines the fundamental concept of the FEM with the analytic solution from the MOC to obtain coefficients for the FEM basis function expansion. Traditionally, coefficients for the FEM basis function expansion are obtained via matrix inversion. Instead, we solve for the coefficients with the MOC and represent the underlying fields with the basis function expansion using these coefficients. We provide a convergence study for our method with results from two sets of FEM basis functions: Gauss-Legendre and Gauss-Lobatto sets. We also compare two different variations of our method categorized as short characteristics and intermediate characteristics.

42 ENGINEERING↗
DOE OSTI2025

A resolution independent neural operator

The Deep operator network (DeepONet) is a powerful yet simple neural operator architecture that utilizes two deep neural networks to learn mappings between infinite-dimensional function spaces. This architecture is highly flexible, allowing the evaluation of the solution field at any location within the desired domain. However, it imposes a strict constraint on the input space, requiring all input functions to be discretized at the same locations; this limits its practical applications. Here, in this work, we introduce a general framework for operator learning from input–output data with arbitrary number and locations of sensors. This begins by introducing a resolution-independent DeepONet (RI-DeepONet), enabling it to handle input functions that are arbitrarily, but sufficiently finely, discretized. To this end, we propose two dictionary learning algorithms to adaptively learn a set of appropriate continuous basis functions, parameterized as implicit neural representations (INRs), from correlated signals defined on arbitrary point cloud data. These basis functions are then used to project arbitrary input function data as a point cloud onto an embedding space (i.e., a vector space of finite dimensions) with dimensionality equal to the dictionary size, which can be directly used by DeepONet without any architectural changes. In particular, we utilize sinusoidal representation networks (SIRENs) as trainable INR basis functions. The introduced dictionary learning algorithms are then used in a similar way to learn an appropriate dictionary of basis functions for the output function data, which defines a new neural operator architecture referred to as the R esolution I ndependent N eural O perator (RINO). In the RINO, the operator learning task simplifies to learning a mapping from the coefficients of input basis functions to the coefficients of output basis functions. We demonstrate the robustness and applicability of RINO in handling arbitrarily (but sufficiently richly) sampled input and output functions during both training and inference through several numerical examples.

Deep operator network (DeepONet)↗
DOE OSTI2024

Augmenting machine learning of Grad–Shafranov equilibrium reconstruction with Green's functions

This work presents a method for predicting plasma equilibria in tokamak fusion experiments and reactors. The approach involves representing the plasma current as a linear combination of basis functions using principal component analysis of plasma toroidal current densities (J t ) from the EFIT-AI equilibrium database. Then utilizing EFIT's Green's function tables, basis functions are created for the poloidal flux (ψ) and diagnostics generated from the toroidal current (J t ). Similar to the idea of a physics-informed neural network (NN), this physically enforces consistency between ψ, J t , and the synthetic diagnostics. First, the predictive capability of a least squares technique to minimize the error on the synthetic diagnostics is employed. The results show that the method achieves high accuracy in predicting ψ and moderate accuracy in predicting J t with median R 2 = 0.9993 and R 2 = 0.978, respectively. A comprehensive NN using a network architecture search is also employed to predict the coefficients of the basis functions. The NN demonstrates significantly better performance compared to the least squares method with median R 2 = 0.9997 and 0.9916 for J t and ψ, respectively. The robustness of the method is evaluated by handling missing or incorrect data through the least squares filling of missing data, which shows that the NN prediction remains strong even with a reduced number of diagnostics. Additionally, the method is tested on plasmas outside of the training range showing reasonable results.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2025

Energy conservation in real-time nuclear–electronic orbital Ehrenfest dynamics

Real-time nuclear–electronic orbital Ehrenfest (RT-NEO-Ehrenfest) dynamics methods provide a first-principles approach for describing nonadiabatic molecular processes with nuclear quantum effects. For an efficient description of proton transfer within RT-NEO-Ehrenfest dynamics, the basis function center associated with the quantum proton can be allowed to move classically. Here, this traveling proton basis (TPB) approach effectively captures proton quantum dynamics, although its energy conservation behavior is not yet fully satisfactory. Two recently proposed TPB approaches, in principle, conserve the extended energy, which includes both the system energy and the kinetic energy associated with the proton basis function center. Herein, a thermostatted TPB approach is proposed to improve the conservation of the system energy, excluding the kinetic energy associated with the proton basis function center. In this approach, the quantum proton dynamics are modulated by dynamically rescaling the proton momentum operator to maintain the system energy conservation. With the excited-state intramolecular proton transfer of o-hydroxybenzaldehyde as an example, this approach is shown to significantly improve the system energy conservation while preserving the accuracy of the quantum proton dynamics as achieved in the original TPB approach.

Ehrenfest dynamics↗