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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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At least 55 records · Page 3
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Advanced modeling and simulation of research reactors using dynamic mode decomposition

Full text of publication follows. Due to the ever-increasing safety requirements, the current trend of nuclear reactor analysis is shifting towards high-fidelity multi-physics models, which have a very high computational cost and modelling complexity. As the cost of even a single model run makes it impossible to analyse the behaviour and performance of these models on large-scale commercial plants, it has become even more significant to provide suitable benchmarks to validate and test them extensively. In this sense, research reactors offer a promising solution for the initial validation of high-fidelity models, as they are significantly smaller than commercial reactors and their characteristics are well known. In particular, the reactors of the TRIGA family have been used to assess and validate models and methods for Generation-IV designs, as they have some similar features (such as the dominance of natural convection as cooling mechanism and the difficulties in performing sub-channel analysis using standard codes). Still, the computational requirements of high-fidelity models make them unsuitable for real-time analysis, even following their assessment on research reactors. In this sense, Model Order Reduction (MOR) techniques give an additional strategy to reduce the computational cost of high-fidelity models (whilst preserving sufficient accuracy). In particular, this work focuses on Dynamic Mode Decomposition (DMD), a non-intrusive MOR technique that aims at representing models with explicit temporal dynamics by extracting the time-varying characteristics and the governing structures based only on a set of available data, thus without needing any underlying knowledge of the governing equations. In addition, DMD also computes a low-dimensional surrogate of the dynamic matrix of the system, making it suited for stability analysis and real-time evaluations. This work focuses on the application and validation of the DMD method on the Computational Fluid-Dynamics (CFD) model TRIGA Mark II reactor, also discussing in detail the potentiality of this algorithm as an advanced modelling tool for nuclear reactor analysis. (author)

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2024

One-dimensional temperature measurement of supersonic jet flow by resonantly ionized photoemission thermometry of molecular nitrogen

As the field of fluid dynamics progresses, the demand for sophisticated diagnostic methods to accurately assess flow conditions rises. In this work, resonantly ionized photoemission thermometry (RIPT) has been used to directly target and ionize diatomic nitrogen (N 2 ) to measure one-dimensional (1D) temperature profiles in a supersonic jet flow. This technique can be considered non-intrusive as the premise uses resonantly enhanced multiphoton ionization (REMPI) to target molecular nitrogen. This resonance excites N 2 into absorption bands of the P, Q, and R rotational branches of N 2 ( b 1 Π u ). The ideal (3 + 1) REMPI scheme excites from the ground state and ionizes N 2 ( b 1 Π u ← X 1 Σ g + ) where de-excitation results in photoemission from the first negative band of ionized N 2 + ( B 2 Σ u + → X 2 Σ g + ) as nitrogen returns to the ground state. The resulting emission can be observed using an intensified camera, thus permitting inference of the rotational temperature of ground-state molecular nitrogen. A linearly regressive Boltzmann distribution is applied based on previous calibration data for this technique to quantify the temperature along the ionized line. This work applies this technique to a pure N 2 supersonic jet in cross-flow and counter-flow orientations to demonstrate N 2 RIPT’s applications in a supersonic flow. Temperature variations are observed at different locations downstream of the exit in cross-flow, and axisymmetric in counter-flow, to generate profiles characterizing the flow dynamics. Due to the collisional effects resulting from the number density of N 2 at higher pressures, a (3 + 2) REMPI scheme is observed throughout this text.

Clark, Aleksander (ORCID:0009000855600042)↗
DOE OSTI2025

Heliostat optical error inspection with polarimetric imaging drone

On a Concentrated Solar Power (CSP) field, optical errors have significant impacts on the collection efficiency of heliostats. Fast, cost-effective, labor-efficient, and non-intrusive autonomous field inspection remains a challenge. Approaches using imaging drone, i.e., Unmanned Aerial Vehicle (UAV) system integrated with high resolution visible imaging sensors, have been developed to address these challenges; however, these approaches are often limited by insufficient imaging contrast. Here, in this study, we report a polarimetry-based method with a polarization imaging system integrated on UAV to enhance imaging contrast for in-situ detection of heliostat mirrors without interrupting field operation. We developed an optical model for skylight polarization pattern to simulate the polarization images of heliostat mirrors and obtained optimized waypoints for polarimetric imaging drone flight path to capture images with enhanced contrast. The polarimetric imaging-based method improved the success rate of edge detections in scenarios which were challenging for mirror edge detection with conventional imaging sensors. We have performed field tests to achieve significantly enhanced heliostat edge detection success rate and investigate the feasibility of integrating polarimetric imaging method with existing imaging-based heliostat inspection methods, i.e., Polarimetric Imaging Heliostat Inspection Method (PIHIM). Our preliminary field test results suggest that the PIHIM hold the promise to enable sufficient imaging contrast for real-time autonomous imaging and detection of heliostat field, thus suitable for non-interruptive fast CSP field inspection during its operation.

CSP Field↗
DOE OSTI2022

Non-intrusive reduced order modeling of natural convection in porous media using convolutional autoencoders: Comparison with linear subspace techniques

Natural convection in porous media is a highly nonlinear multiphysical problem relevant to many engineering applications (e.g., the process of CO 2 sequestration). Here, we extend and present a non-intrusive reduced order model of natural convection in porous media employing deep convolutional autoencoders for the compression and reconstruction and either radial basis function (RBF) interpolation or artificial neural networks (ANNs) for mapping parameters of partial differential equations (PDEs) on the corresponding nonlinear manifolds. To benchmark our approach, we also describe linear compression and reconstruction processes relying on proper orthogonal decomposition (POD) and ANNs. Further, we present comprehensive comparisons among different models through three benchmark problems. The reduced order models, linear and nonlinear approaches, are much faster than the finite element model, obtaining a maximum speed-up of 7 × 10 6 because our framework is not bound by the Courant–Friedrichs–Lewy condition; hence, it could deliver quantities of interest at any given time contrary to the finite element model. Our model’s accuracy still lies within a relative error of 7% in the worst-case scenario. We illustrate that, in specific settings, the nonlinear approach outperforms its linear counterpart and vice versa. We hypothesize that a visual comparison between principal component analysis (PCA) and t-Distributed Stochastic Neighbor Embedding (t-SNE) could indicate which method will perform better prior to employing any specific compression strategy.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

A Contextually Supervised Optimization-Based HVAC Load Disaggregation Methodology

This paper presents a novel contextually supervised optimization-based approach for disaggregating heating, ventilation, and air-conditioning (HVAC) loads using smart meter or Supervisory Control and Data Acquisition data. To disaggregate the load into HVAC loads, large and infrequently used loads (LIUL), and base loads, we formulate an optimization problem to minimize a set of five loss terms, consisting of the reconstruction errors of the overall load profile, the ramp rate losses, and three distinct loss functions linked with the HVAC load, base load, and LIUL, respectively. To enhance accuracy, we incorporate two forms of contextual information into the problem formulation. First, we utilize mutual information to estimate HVAC energy consumption. Second, we employ a base load dictionary to constrain HVAC load estimation errors. The obtained HVAC load profiles are fine-tuned by abnormal ramp detection followed by binary hypothesis testing. Here, the proposed method is developed and tested using sub-metered residential and commercial building data. Simulation results show that the proposed method outperforms existing methods across various data resolutions and load aggregation levels, showing excellent transferability and generalizability.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2024

A mixed formulation of the plane-stress problem to facilitate reuse of constitutive models in finite-element programs

Here, the plane-stress assumption can be challenging to support in a finite element program because it traditionally requires separate implementations of constitutive models than those intended for three-dimensional or two-dimensional plane-strain simulations. As a solution to this issue, this paper presents a method to solve the plane-stress problem using a mixed formulation. In this formulation, the out-of-plane strain is treated as a field variable that is solved for in addition to the standard in-plane displacement variables, in a manner that weakly enforces the condition that the out-of-plane stress is zero. The proposed formulation is non-intrusive, requiring no modifications to the constitutive models in contrast to the conventional plane-stress formulation. The proposed mixed formulation has been benchmarked against analytical solutions and numerical solutions, with good performance and accuracy.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Detecting thermal crack growth on a large additive manufactured structure using acoustic emission

Large format additive manufacturing (LFAM) proved to have a great potential to become an adjacent technology to traditional manufacturing methods. One of the sectors LFAM is targeting is rapid tool/mold development for composites. This includes large mold structures used for high-temperature molding techniques (in-oven or autoclave). Although, these large printed structures (reaching hundreds of pounds) develop thermal-residual stress during cool-down and can eventually crack, turning the structure into waste. Acoustic emission (AE), a passive non-intrusive global nondestructive evaluation (NDE) technique, was used to monitor crack growth and can provide the right tools that can be used for feedback loop for corrective action. This research performs thermal testing on a large AM mold with preexisting cracks, in an attempt to monitor crack growth using AE. AE was able to detect, identify and locate the crack source by means of acoustic features, waveform characteristics, spectrum analysis, and difference in arrival times.

Spencer, Ryan↗
DOE OSTI2021

Non-intrusive nonlinear model reduction via machine learning approximations to low-dimensional operators

Abstract Although projection-based reduced-order models (ROMs) for parameterized nonlinear dynamical systems have demonstrated exciting results across a range of applications, their broad adoption has been limited by their intrusivity: implementing such a reduced-order model typically requires significant modifications to the underlying simulation code. To address this, we propose a method that enables traditionally intrusive reduced-order models to be accurately approximated in a non-intrusive manner. Specifically, the approach approximates the low-dimensional operators associated with projection-based reduced-order models (ROMs) using modern machine-learning regression techniques. The only requirement of the simulation code is the ability to export the velocity given the state and parameters; this functionality is used to train the approximated low-dimensional operators. In addition to enabling nonintrusivity, we demonstrate that the approach also leads to very low computational complexity, achieving up to $$10^3{\times }$$ 10 3 × in run time. We demonstrate the effectiveness of the proposed technique on two types of PDEs. The domain of applications include both parabolic and hyperbolic PDEs, regardless of the dimension of full-order models (FOMs).

42 ENGINEERING↗
DOE OSTI2026

Advancing Industry 4.0: Multimodal Sensor Fusion for AI-Based Fault Detection in 3D Printing

Additive manufacturing, particularly fused deposition modeling, is transforming modern production by enabling rapid prototyping and complex part fabrication. However, its layer-by-layer process remains vulnerable to faults such as nozzle clogging, filament runout, and layer misalignment, which compromise print quality and reliability. Traditional inspection methods are costly, time-intensive, and often limited to post-process analysis, making them unsuitable for real-time intervention. In this current study, the authors developed a novel, low-cost, and portable faultdetection system that leverages multimodal sensor fusion and artificial intelligence for real-time monitoring in FDM-based 3D printing. The system integrates acoustic, vibration, and thermal sensing into a non-intrusive architecture, capturing complementary data streams that reflect both mechanical and process-related anomalies. Acoustic and thermal sensors operate in a fully contactless manner, while the vibration sensor requires minimal attachment such that it will not interfere with printer hardware, thereby preserving portability and ease of deployment. The multimodal signals are processed into spectrograms and time-frequency features, which are classified using convolutional neural networks for intelligent fault detection. The proposed system advances Industry 4.0 objectives by offering an affordable, scalable, and practical monitoring solution that improves faultdetection accuracy, reduces waste, and supports sustainable, adaptive manufacturing.

42 ENGINEERING↗
DOE OSTI2023

Semi-Supervised, Non-Intrusive Disaggregation of Nodal Load Profiles With Significant Behind-the-Meter Solar Generation

It is of imperative interests for regional transmission organizations (RTOs) to effectively extract actual load profiles at transmission nodes with significant behind-the-meter solar generation, which remains a gap in the existing technology paradigm. This paper proposes an explicit yet efficient linear estimator to disaggregate actual load profiles at transmission buses with significant behind-the-meter (BTM) solar generations. The proposed estimator is based on disaggregating (i.e., extracting) at locations close to transmission buses under consideration. Further, to overcome the lack of “ground truth” and validate the performance of the proposed algorithms, we first propose semi-supervised mechanisms with parameter tuning as well as unsupervised clustering and leverage the unique characteristics of zero-crossing points in BTM solar peaking behaviors, which we refer to as “Zone-to-Node (Z2N)” methods. Next, we further propose a bi-level Node-to-Node (N2N) framework that improves the overall disaggregation performances compared to Z2N. Numerical results are presented using real-world data at PJM Interconnection.

14 SOLAR ENERGY↗
DOE OSTI2026

Uncertainty quantification for Joule heating processes in fibrous pore-resolved media

Joule heating (JH) is an energy-efficient and sustainable technique for heating materials. Its application for industrial heating, particularly, has been gaining attention due to its potential for increasing the yield of various chemical products. The process involves the use of heating elements (materials that are highly conductive electrically and thermally) to heat up other materials or substances. These conductors, however, can exhbit varying degrees of uncertainty due to non-linearities in their temperature-dependent properties, which could result in variable material behavior. In this work, we carry out uncertainty quantification (UQ) at the pore scale to describe the uncertainty of such materials. In so doing, we applied the non-intrusive polynomial chaos expansion (PCE) technique to quantify the uncertainty within the system. The steady state Joule heating equation was solved numerically at the pore scale mimicking conditions within a heating chamber for propane dehydrogenation, and various electro-thermal profiles were obtained. We also examined the effect of the number of sampling points (20 – 100) and order of the PCE coefficients (2 – 5) on the accuracy of the temperature evaluations. The results were then benchmarked with the standard Monte Carlo (MC) method. The average temperature of the 4th-order global PCE showed good agreement with the MC results (which were positively skewed). Orders greater than 4 gave an underestimation of the temperatures while predictions for the peak temperature improved as the number of sampling points increased.

Fagbemi, Samuel [ORNL] (ORCID:0000000236995025)↗
DOE OSTI2022

WISP: Watching grid Infrastructure Stealthily through Proxies (Final Technical Report)

The complex interdependencies of cyber systems (sensors and communications), physical grids and associated electricity market operations make protecting electric power grids a significant challenge. The energy sector is constantly under new, targeted, advanced and dangerous cyber-attacks that have the potential to result in the loss of human life. These threats are further exacerbated by our need to modernize the grid. One focus of cyber security research in smart grids is the securing of the SCADA system through advanced intrusion detection systems (IDS) and bad data detection algorithms in state estimation. These methods either require full knowledge of the system topology and parameters or fail to understand the physical behaviors under attack. WISP (Watching grid Infrastructure Stealthily through Proxies) is designed to provide additional protection to the power grid using only publicly available data. In particular, WISP exploits the spatio-temporal nature of the real time locational marginal prices (LMPs), in conjunction with other information such as bids, weather, outages and load data to analyze anomalous power pricing behaviors and then correlate those observations to localize regions of interest and identify potential cyber events. WISP is non-intrusive as the tool is deployed as a service in the Cloud or on premise and provides reliable information to system operators for enhanced situational awareness, without impeding energy delivery functions. The WISP technology comprises three modules: the data-driven anomaly detection core, the vulnerability and risk analysis and the root cause analysis. The data-driven anomaly detection core performs the tasks of feature selection, anomaly detection and attack region localization. The vulnerability and risk analysis module provides system level information of the vulnerable variables and times, assisting the operators in selecting monitoring and protection nodes. The root cause analysis module takes the detection results and identifies potential operational conditions that contribute to the detected anomalies. In Phase I, we have demonstrated the feasibility and effectiveness of WISP. We developed a realistic electricity market simulator capable of generating normal and attack market data under various operational conditions. We developed a series of cyber-attack detection and analysis algorithms and evaluated them under multiple data sources. Finally, we integrated all modules into an end-to-end software, providing functions for data management, data analytics and visualization. Specifically, we have achieved: (i) real-time data acceptance from external utility interfaces with >99% acceptance rate; (ii) high performance anomaly detection algorithms with >98% detection accuracy and <0.1% false alarm rate; and (iii) ultra-low computing delay <50 milliseconds. Additionally, our team developed algorithms to identify the vulnerable variables in electricity market operations and root cause analysis functions to identify major contributors to the price spikes. These ancillary modules are necessary when deploying WISP in real world industry environment. In Phase II, we have demonstrated the effectiveness of WISP software on realistic largescale power systems. We performed red team testing for the Phase I WISP software and identified software vulnerabilities and implemented corresponding mitigation solutions. We adapted the electricity market simulator for the Texas synthetic 2000-bus system and generated datasets for the false data injection attacks. We created database and visualization interfaces for the Texas system and the ISO New England system. We performed software optimization in terms of operation efficiency, computing speed and detection accuracy. Finally, we tested the software on the Texas system and the ISO New England system and evaluated the detection performance. Overall, we achieved above 89% detection rate, below 3% false alarm rate and below 37 seconds of end-to-end detection delay.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2024

Mathematical methods for optimal polynomial recovery of high-dimensional systems from noisy data

The goal of our Early Career Research Project (ECRP) is to establish a modern mathematical foundation that will enable next-generation computational methods for polynomial approximation of high-dimensional systems, having a certain set of constraints, from a limited amount of noisy data. Such a foundation is critical to realizing the future potential of the DOE user facilities, and will ultimately empower scientists to address a fundamental question, namely, “how many realizations of a nonlinear manifold are required to recover the entire high-dimensional solution map, with optimal approximation guarantees and minimal computational cost?” The central theme of this effort aims to conquer this challenge by pioneering the development of extraordinarily innovative theoretical analysis and transformational non-intrusive computational methodologies. Such approaches will enable the reconstruction of the entire high-dimensional solution map, with accuracy comparable to the best approximation, while utilizing an optimal number of samples. During this reporting period we have made significant progress on four thrusts.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Improvements in Reflected Target Non-intrusive Assessment (ReTNA)

Accurate optical surfaces are a primary driver of concentrated solar power plant performance. Errors in pointing and tracking mirrors, the canting of individual mirror facets, and the surface slope of the mirror itself can be caused by errors during assembly, transportation, wind loading, gravity, and many other sources. The tools that exist to measure these error sources today largely rely on fringe deflectometry (SOFAST, QDec, others), or photogrammetry with targets attached to the mirror surface. In 2022, NREL presented a new indoor optical measurement system to measure fully assembled heliostats during development and on the assembly line for quality assurance purposes, which has undergone significant development in the last year, including the installation of a new, larger laboratory setup. This system, called ReTNA (Reflected Target Non-intrusive Assessment), uses a printed pattern as a deflectometry target. Using printed pattern features as deflectometry targets instead of fringes allows the software to measure the surface slope in two dimensions, eliminating the need to for changing, projected targets. In the last year, NREL's ReTNA tool has been adapted to use a tall, thin target, that only spans one dimension of the mirror, and a camera moving on an automatic track system. We will present these new developments, including the new setup, target patterns, software methodology for fast, automatic optical analysis, and important lessons learned throughout the process. ReTNA results will be compared to established tools, to establish the uncertainty of this new method. ReTNA leverages recent improvements in computer vision and photogrammetry to remove as much careful calibration from the setup process as possible, making the tool low cost, simple, and portable. While it will not achieve the resolution of a fringe deflectometry system, its advantages make it well suited for CSP evaluation, and allow it to fill a different industry need.

concentrating solar power↗
DOE OSTI2021

On the Limits of EM Based Detection of Control Logic Injection Attacks In Noisy Environments

The difficulty in applying traditional security mechanisms in Industrial Control System (ICS) environments makes a large portion of these mission-critical assets vulnerable to cyber attacks. Therefore, there is a dire need for the development of novel security mechanisms specifically designed to protect such critical systems. Recently a lot of attention has been given to mechanisms that exploit the EM emanations of devices for defense purposes. Such practices may lead to the development of robust external and non-intrusive anomaly detection systems. Nevertheless, the majority of current work in the area neglects to consider the implications of real-life environments, particularly environmental noise. In this work, we explore the limits of EM-based anomaly detection towards identifying injection attacks in control logic software in noisy environments. Our study conducted upon both synthetically generated and real signals identified that indeed environmental noise might significantly degrade the accuracy of the anomaly detection process. Experiments done upon synthetic data indicated that assuming that signals are captured with high sampling rates, even minor code injections can be detected with above-90% accuracy in noisy environments where SNR is up to -2dB. This is true even if naive detection methods are considered. Moreover, experiments done using a real-life testbed attest that even single-instruction injections can be detected with near-perfect accuracy in relatively clean environments. Finally, noise-elimination techniques can drastically improve the reliability of the detection mechanism even in noisy environments.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Predicting critical heat flux using localized sensing at invisible vapor-liquid interfaces

Predicting critical heat flux (CHF) in two-phase electronics cooling systems remains a significant challenge due to the sudden onset of boiling crisis and the difficulty in directly visualizing vapor-liquid interfaces. Existing sensing methods rely on lagging temperature measurements, optically accessible systems, or spatially averaged signals that cannot pinpoint CHF initiation at localized high-heat-flux regions. Here, we report a planar capacitive sensing approach that enables real-time, localized detection of vapor-liquid interface dynamics for CHF prediction in boiling heat transfer. The capacitive sensor exploits the dielectric constant difference between liquid and vapor phases to capture bubble nucleation, growth, and departure dynamics with a temporal resolution down to 2 ms. The capacitive sensing reveals distinct signals across boiling regimes: from high-frequency fluctuations during strong nucleate boiling to low-frequency fluctuations with increased amplitudes when approaching CHF. The multi-sensor array experiments demonstrate real-time localized sensing, where each sensor responds exclusively to boiling in its immediate vicinity without crosstalk from neighboring regions. This non-intrusive sensing approach provides predictive rather than lagging sensing signals of CHF occurrence, offering predictive diagnosis of two-phase liquid cooling for the thermal management of high-power-density electronics.

CHF↗
DOE OSTI2022

CSP Optical Facilities at the National Renewable Energy Laboratory

The concentrating Solar Power (CSP) Optical Facilities at NREL comprise unique optical tools and facilities for testing and characterizing concentrating solar optical components. These facilities are maintained to aid researchers and industry efforts to reduce market barriers, improve the performance, reduce the cost, and improve the lifetime and reliability of CSP materials, components, subsystems, and integrated concepts, where accurate and timely characterization methods are needed. To reach this goal, NREL is committed to collaborating in ongoing and projected research needs for industry, academia, and government entities, in the development and characterization of low-cost, high-performance reflector and absorber materials in addition to their components-a key element in the conversion of concentrated sunlight into energy. To accomplish this mission, the core laboratory includes the Advanced Optical Materials Laboratory, indoor and outdoor accelerated weather testing tools, "Distant Observer" for parabolic troughs, the High Flux Solar Furnace, and the ultra-accelerated weather testing tools. This equipment enables to rapidly respond to new challenges with novel solutions that align with the goals of the Solar Energy Technology Office (SETO) at the U.S. Department of Energy. The goal of this paper is to present the state-of art-optical facilities available at NREL that can help support the research needs of the CSP community. Therefore, we review here the different components of the CSP Optical Facilities at NREL and highlight their characteristics and importance for the CSP community.

accelerated weather testing↗
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↗