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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 757 records · Page 42
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Benchmarking the Performance of Neuromorphic and Spiking Neural Network Simulators

Software simulators play a critical role in the development of new algorithms and system architectures in any field of engineering. Neuromorphic computing, which has shown potential in building brain-inspired energy-efficient hardware, suffers a slow-down in the development cycle due to a lack of flexible and easy-to-use simulators of either neuromorphic hardware itself or of spiking neural networks (SNNs), the type of neural network computation executed on most neuromorphic systems. While there are several openly available neuromorphic or SNN simulation packages developed by a variety of research groups, they have mostly targeted computational neuroscience simulations, and only a few have targeted small-scale machine learning tasks with SNNs. Evaluations or comparisons of these simulators have often targeted computational neuroscience-style workloads. In this work, we seek to evaluate the performance of several publicly available SNN simulators with respect to non-computational neuroscience workloads, in terms of speed, flexibility, and scalability. We evaluate the performance of the NEST, Brian2, Brian2GeNN, BindsNET and Nengo packages under a common front-end neuromorphic framework. Our evaluation tasks include a variety of different network architectures and workload types to mimic the computation common in different algorithms, including feed-forward network inference, genetic algorithms, and reservoir computing. We also study the scalability of each of these simulators when running on different computing hardware, from single core CPU workstations to multi-node supercomputers. Our results show that the BindsNET simulator has the best speed and scalability for most of the SNN workloads (sparse, dense, and layered SNN architectures) on a single core CPU. However, when comparing the simulators leveraging the GPU capabilities, Brian2GeNN outperforms the others for these workloads in terms of scalability. NEST performs the best for small sparse networks and is also the most flexible simulator in terms of reconfiguration capability NEST shows a speedup of at least 2x compared to the other packages when running evolutionary algorithms for SNNs. The multi-node and multi-thread capabilities of NEST show at least 2x speedup compared to the rest of the simulators (single core CPU or GPU based simulators) for large and sparse networks. We conclude our work by providing a set of recommendations on the suitability of employing these simulators for different tasks and scales of operations. We also present the characteristics for a future generic ideal SNN simulator for different neuromorphic computing workloads.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Fourier-DeepONet: Fourier-enhanced deep operator networks for full waveform inversion with improved accuracy, generalizability, and robustness

In this article, full waveform inversion (FWI) infers the subsurface structure information from seismic waveform data by solving a non-convex optimization problem. Data-driven FWI has been increasingly studied with various neural network architectures to improve accuracy and computational efficiency. Nevertheless, the applicability of pre-trained neural networks is severely restricted by potential discrepancies between the source function used in the field survey and the one utilized during training. Here, we develop a Fourier-enhanced deep operator network (Fourier-DeepONet) for FWI with the generalization of seismic sources, including the frequencies and locations of sources. Specifically, we employ the Fourier neural operator as the decoder of DeepONet, and we utilize source parameters as one input of Fourier-DeepONet, facilitating the resolution of FWI with variable sources. To test Fourier-DeepONet, we develop three new and realistic FWI benchmark datasets (FWI-F, FWI-L, and FWI-FL) with varying source frequencies, locations, or both. Our experiments demonstrate that compared with existing data-driven FWI methods, Fourier-DeepONet obtains more accurate predictions of subsurface structures in a wide range of source parameters. Moreover, the proposed Fourier-DeepONet exhibits superior robustness when handling data with Gaussian noise or missing traces and sources with Gaussian noise, paving the way for more reliable and accurate subsurface imaging across diverse real conditions.

42 ENGINEERING↗
DOE OSTI2022

Wind turbine gearbox fault prognosis using high-frequency SCADA data

Condition-based maintenance using routinely collected Supervisory Control and Data Acquisition (SCADA) data is a promising strategy to reduce downtime and costs associated with wind farm operations and maintenance. New approaches are continuously being developed to improve the condition monitoring for wind turbines. Development of normal behaviour models is a popular approach in studies using SCADA data. This paper first presents a data-driven framework to apply normal behaviour models using an artificial neural network approach for wind turbine gearbox prognostics. A one-class support vector machine classifier, combining different error parameters, is used to analyse the normal behaviour model error to develop a robust threshold to distinguish anomalous wind turbine operation. A detailed sensitivity study is then conducted to evaluate the potential of using high-frequency SCADA data for wind turbine gearbox prognostics. The results based on operational data from one wind turbine show that, compared to the conventionally used 10-min averaged SCADA data, the use of high-frequency data is valuable as it leads to improved prognostic predictions. High-frequency data provides more insights into the dynamics of the condition of the wind turbine components and can aid in earlier detection of faults.

17 WIND ENERGY↗
DOE OSTI2025

Data-Driven State of Health Estimation for Second-Life Batteries Using Interpolated Synthetic Data and Feature Selection

Accurate estimation of the State of Health (SOH) for second-life batteries (SLBs) is crucial given their increasing use in energy storage applications. Precise SOH prediction is essential for safe operation and robust battery management systems. A major challenge is the limited availability of datasets for building reliable degradation models. To address this, synthetic data generation through linear interpolation is performed to extend the available data, making it more representative of real-world battery operating conditions. By analyzing feature correlation with SOH, the most relevant features are selected for the model. The proposed approach employs a convolutional neural network (CNN) model trained on this interpolated, feature-selected dataset, using time series data of voltage, temperature, and current over a cycle. By focusing on highly correlated features, the model achieves over 95% accuracy, with mean absolute error and root mean squared error up to 2.27% and 2.64%, respectively, in SOH estimation for two battery datasets tested. These results highlight the potential of combining synthetic data generation and feature selection to enhance SOH predictions, showcasing the superior performance of the proposed CNN model for both new batteries and SLBs.

feature selection↗
DOE OSTI2024

Frequency-domain computing using nonlinear acoustic-wave device on lithium niobate

Abstract Multiply-accumulation are crucial computing operations in signal processing, numerical simulations, and machine learning. In recent years, optical analog approaches have demonstrated higher computing performance and better power efficiency than their digital counterparts. However, analog computing chips usually need large areas and complex structures for parallel computing, as a single device element only executes one computing operation at a single time. Here, we demonstrate frequency-domain computing using the nonlinear acoustic-wave devices on lithium niobate, featuring a normalized external second-harmonic generation conversion efficiency of ~ 5.7 × 10-4 W-1. The second-order sum-frequency nonlinear process of lithium niobate enables multiplication of inputs encoded in the frequency domain. Compared to the analog schemes, our device features a notably simpler design, and nanofabrication requires only one lift-off. Using a single acoustic-wave device within an area of 0.03 mm2, we can simultaneously conduct over 130,000 multiply-accumulation operations. Our acoustic-wave device shows applications in real and complex vector convolutions and image processing. This demonstration sets the stage for experimental realizations into frequency-domain integrated nonlinear acoustic computing systems, potentially shaping future developments in acoustic neural networks and quantum computing.

chai, mingzhao (ORCID:0009000466226341)↗
DOE OSTI2024

Quapprox: A Framework for Benchmarking the Approximability of Variational Quantum Circuit

Most of the existing quantum neural network models, such as variational quantum circuits (VQCs), are limited in their ability to explore the non-linear relationships in input data. This gradually becomes the main obstacle for it to tackle realistic applications, such as natural language processing, medical image processing, and wireless communications. Recently, there have emerged research efforts that enable VQCs to perform non-linear operations. However, it is still unclear on the approximability of a given VQC (i.e., the order of non-linearity that can be handled by a specified design). In response to this issue, we developed an automated tool designed to benchmark the approximation of a given VQC. The proposed tool will generate a set of synthetic datasets with different orders of non-linearity and train the given VQC on these datasets to estimate their approximability. Our experiments benchmark VQCs with different designs, where we know their theoretic approximability. We then show that the proposed tool can precisely estimate the approximability, which is consistent with the theoretic value, indicating that the proposed tool can be used for benchmarking the approximability of a given quantum circuit for learning tasks.

artificial intelligence↗
DOE OSTI2025

High‐Resolution National‐Scale Water Modeling Is Enhanced by Multiscale Differentiable Physics‐Informed Machine Learning

Abstract The National Water Model (NWM) is a key tool for flood forecasting, planning, and water management. Key challenges facing the NWM include calibration and parameter regionalization when confronted with big data. We present two novel versions of high‐resolution (∼37 km 2 ) differentiable models (a type of hybrid model): one with implicit, unit‐hydrograph‐style routing and another with explicit Muskingum‐Cunge routing in the river network. The former predicts streamflow at basin outlets whereas the latter presents a discretized product that seamlessly covers rivers in the conterminous United States (CONUS). Both versions use neural networks to provide a multiscale parameterization and process‐based equations to provide a structural backbone, which were trained simultaneously (“end‐to‐end”) on 2,807 basins across the CONUS and evaluated on 4,997 basins. Both versions show great potential to elevate future NWM performance for extensively calibrated as well as ungauged sites: the median daily Nash‐Sutcliffe efficiency of all 4,997 basins is improved to around 0.68 from 0.48 of NWM3.0. As they resolve spatial heterogeneity, both versions greatly improved simulations in the western CONUS and also in the Prairie Pothole Region, a long‐standing modeling challenge. The Muskingum‐Cunge version further improved performance for basins >10,000 km 2 . Overall, our results show how neural‐network‐based parameterizations can improve NWM performance for providing operational flood predictions while maintaining interpretability and multivariate outputs. The modeling system supports the Basic Model Interface (BMI), which allows seamless integration with the next‐generation NWM. We also provide a CONUS‐scale hydrologic data set for further evaluation and use.

Song, Yalan [Civil and Environmental Engineering T↗
DOE OSTI2024

Machine Learning-Assisted Stability Boundary Determination of Multiport Autonomous Reconfigurable Solar Power Plants

The multiport autonomous reconfigurable solar (MARS) power plant is a promising solution to integrate renewable resources and energy storage systems into the alternating current (ac) power grid and an high-voltage direct current (HVdc) link. In the MARS system, various input power sources are connected to the individual submodules (SMs) through direct current (dc)–dc converters. However, the presence of external power sources can result in unbalanced capacitor voltages of SMs, thereby violating stability constraints under multiple/diverse operating conditions. This article aims to address the gap by accurately determining the stability boundary of the MARS system. As such, a novel machine learning (ML)-assisted energy balancing control (EBC) criterion is proposed. Further, in conjunction with a refined EBC, this approach ensures balanced capacitor voltages across various types of SMs, significantly enhancing the overall system efficiency. The proposed EBC criterion effectively controls EBC activation and deactivation, achieving remarkable accuracy. Both power systems computer aided design (PSCAD)/electromagnetic transients including direct current (EMTDC) simulations and control hardware-in-the-loop (cHIL) tests are conducted to validate the feasibility and efficiency of the proposed method. By combining the EBC and ML-assisted EBC criterion, efficient energy management is achieved for systems featuring multiple input power sources, such as MARS. This approach enables the system to fully exploit its potential across an expanded operational range while upholding high-efficiency standards.

14 SOLAR ENERGY↗
DOE OSTI2025

Application of a Physics-Informed Convolutional Neural Network for Monitoring the Temperature Fields in High-Temperature Gas Reactors

Here, this work presents current advances in applying a physics-informed convolutional neural network (CNN) to evaluate temperature distributions in advanced reactors. Our goal is to demonstrate that the CNN can reconstruct temperature fields within the solid region of a prismatic fuel assembly in a high-temperature gas reactor (HTGR) with sensor data available in only a few cooling channels. Before that, we showcase the superior performance of the physics-informed CNN in comparison to a purely data-driven multilayer perceptron (MLP), considering a canonical heated channel setup. This analysis shows the advantages of our approach and justifies its choice. The datasets employed here are obtained upon numerical simulations performed with codes under the Nuclear Energy Advanced Modeling and Simulation program. This work is important, as industry experience indicates that the assembly material in HTGR concepts is prone to large thermal-mechanical loads nearing operational limits. This makes it crucial to characterize peak temperatures and their distributions near hot spots. Modern thermocouples are unreliable in these types of harsh environments because of the high neutron fluxes and elevated temperatures involved. The CNN-based field reconstruction represents an attractive solution, enabling sensor arrays in less aggressive locations and augmenting indirect predictions for less accessible regions. The results show that the CNN reduces prediction errors by orders of magnitude in comparison to the MLP, considering the simple yet well-representative heated channel case. In the case of the HTGR fuel assembly, the CNN can successfully reconstruct temperature fields over various cooling regimes. Furthermore, we also explore the algorithm’s ability to detect abnormalities. Interestingly, the CNN proves it has the capacity to detect blockage in one of the noninstrumented cooling channels.

Machine learning↗
NASA NTRS2010

Multi-Stage System for Automatic Target Recognition

A multi-stage automated target recognition (ATR) system has been designed to perform computer vision tasks with adequate proficiency in mimicking human vision. The system is able to detect, identify, and track targets of interest. Potential regions of interest (ROIs) are first identified by the detection stage using an Optimum Trade-off Maximum Average Correlation Height (OT-MACH) filter combined with a wavelet transform. False positives are then eliminated by the verification stage using feature extraction methods in conjunction with neural networks. Feature extraction transforms the ROIs using filtering and binning algorithms to create feature vectors. A feedforward back-propagation neural network (NN) is then trained to classify each feature vector and to remove false positives. The system parameter optimizations process has been developed to adapt to various targets and datasets. The objective was to design an efficient computer vision system that can learn to detect multiple targets in large images with unknown backgrounds. Because the target size is small relative to the image size in this problem, there are many regions of the image that could potentially contain the target. A cursory analysis of every region can be computationally efficient, but may yield too many false positives. On the other hand, a detailed analysis of every region can yield better results, but may be computationally inefficient. The multi-stage ATR system was designed to achieve an optimal balance between accuracy and computational efficiency by incorporating both models. The detection stage first identifies potential ROIs where the target may be present by performing a fast Fourier domain OT-MACH filter-based correlation. Because threshold for this stage is chosen with the goal of detecting all true positives, a number of false positives are also detected as ROIs. The verification stage then transforms the regions of interest into feature space, and eliminates false positives using an artificial neural network classifier. The multi-stage system allows tuning the detection sensitivity and the identification specificity individually in each stage. It is easier to achieve optimized ATR operation based on its specific goal. The test results show that the system was successful in substantially reducing the false positive rate when tested on a sonar and video image datasets.

Chao, Tien-Hsin↗
DOE OSTI2022

Numerical modeling based machine learning approach for the optimization of falling - film evaporator in thermal desalination application

Scale formation that drastically increases thermal resistance and reduces freshwater production remains a critical challenge in thermal desalination. Novel designs of falling film evaporator and optimal operating condition hold great promise to mitigate scale formation, and increase heat transfer performance and fresh water production. In this work, CFD simulation based machine learning and multi-objective optimization are performed to identify optimal conditions and tube arrangement for evaporator. Non-dominated sorting genetic algorithm is adopted to determine and analyze the optimal pareto front for multiple objectives in desalination criteria. The errors of training, validation, and testing set are computed to identify an optimal hyperparameter set. For performance ratio, fouling resistance, and water production rate, the average relative error is 2.26%, 3.67%, and 3.24%. At pareto front, both performance ratio and water production rate increase at high temperature with fouling resistance (thermal resistance of the fouling layer) increasing as well. Tradeoffs between mitigating scale formation and enhancing desalination performance are evaluated in optimizations for different objectives. Finally, potential optima are identified and can be applied as guidelines to determine evaporator design and system operating conditions.

42 ENGINEERING↗
DOE OSTI2025

Passband Signal Detection at the Edge

Algorithms for radio frequency (RF) spectrum awareness need to be compatible with edge hardware to be practical for many applications. We developed a signal detection and classification model for the ZCU111 RF System-on-a-Chip (RFSoC) that operates on the fast Fourier transform of passband RF data. The system can detect and classify multiple signals of interest and display the predictions in real-time. The model consists of a modified ConvNeXt backbone and YOLOv3 head to operate on the Deep Learning Processing Unit on the RFSoC. We gathered datasets for training and testing by using a software defined radio to transmit example signals of Wi-Fi 802.11 b/g, Wi-Fi 802.11 n, FM Radio, LTE and LTE-M. By leveraging multiple inputs on the RFSoC frontend, the datasets span up to 4 GHz of bandwidth. The models showed high performance in classification accuracy, center frequency error, bandwidth error, and detection accuracy for both single and multi-signal datasets.

42 ENGINEERING↗
NASA NTRSDate not supplied

Early Information Parameter-Set Analysis for Satellite Close Approaches using Machine Learning

Understanding orbital mechanics is essential in space flight and navigation applications, and leveraging modern force models for flight path projection remains an important aspect in space mission design and operation. However, force models do not capture all the dynamics or perturbations in the space environment and thus are subject to errors in predicting the state vectors. The further out the predicted miss distance between spacecraft is from the time of closest approach (TCA), the larger the propagated errors in the predicted miss distance at TCA is. The dependency on these force models for spacecraft flight state prediction calls for a more reliable method that can quantify, or even reduce, these propagated errors. With recent advances in the field artificial intelligence, specifically in machine and deep learning algorithms, a model that implements these approaches can improve on the modern force model approach. The goal for this work is to provide an early-information decision-making threshold, in order to prioritize risk assessment implementation, given the ongoing increase of space objects. In analyzing the relationship of several parameters from conjunction data messages(CDMs) and solar information, early information becomes viable in miss distance prediction with unsupervised learning techniques, which learn the parameters that are linked together with miss distance and probability of collision (Pc) variables. Another approach implemented for identifying relationships within CDMs is supervised learning, in which a shallow neural network binary classifier learns to distinguish events with Pc values¡108. These parameters detected in the unsupervised process are then applied to a regression neural network, which predicts the miss distance at TCA for a specific event within a given uncertainty bound. For the regression neural network, a Long Short Term Memory (LSTM) neural network is implemented, which yields memory about each time step in an event. Using an LSTM network, the model learns to predict miss distance within 0.2km of the value measured at TCA. Although there is a limited amount of "close miss" data to train a network, the network learns to associate parameters, like large energy dissipation rates with the secondary object, with an elevated Pc

Brianna I. Robertson↗
DOE OSTI2020

Co-optimizing water-alternating-carbon dioxide injection projects using a machine learning assisted computational framework

In this report, a robust machine-learning-based computational framework that couples multi-layer neural network (MLNN) proxies and a multi-objective particle swarm optimizer (MOPSO) to design water-alternating-carbon dioxide injection (CO 2 -WAG) projects is presented. The proposed optimization protocol considers various objectives, including oil recovery and CO 2 storage volume. Expert MLNN systems are trained and employed as surrogate models of the high-fidelity compositional simulator in the optimization workflow. When multiple objective functions are considered, two approaches are employed to treat the objectives: the weighted sum method and the Pareto-front-based scheme. A field-scale implementation focusing on tertiary recovery in the Morrow B formation at Farnsworth Unit (FWU) is presented. The developed Pareto-optimal solutions indicate the maximal available oil production can be 1.64 × 10 7 barrels and maximal carbon storage can achieve 2.35 × 10 7 tons. Trade-offs factor is defined to divide the constructed Pareto front into 4 sections with the trade-off factors’ value ranges from 0.35 to 49.9. This work also compares the optimum solution found by the aggregative objective function and the solution repository covered by the Pareto front that considers the physical and operational constraints and reduces uncertainties involved by the multi-objective optimization process. Our comparison indicates multiple solutions exist to satisfy the objective criteria of the WAG design, and these results cannot be found using the traditional weighted sum method. The Pareto front solution can provide more options for project designers, but decisions regarding necessary trade-offs must be made using the solution repository to balance the project economics and CO 2 storage amount.

10 SYNTHETIC FUELS↗
DOE OSTI2022

Deep Learning Based Frequency Stability Assessment in Power Grid with High Renewables

Frequency stability assessment is one critical aspect of power system security assessment. Traditional N-1 screening method is based on the simulations of a few typical daily and seasonal operation scenarios. However, the increasing integration of inverter-based renewables and the retirement of conventional synchronous generators result in decreasing system inertia and growing complexity of system operating conditions. Selecting a few typical operation scenarios cannot cover all operating conditions, and the time-domain simulation of all operation conditions requires tremendous time. This paper proposes a more efficient frequency stability assessment method based on deep learning. The affinity propagation clustering algorithm is used to divide the dataset into different clusters, so the selected dataset for training can cover the diversified operating conditions as much as possible. Also, feature normalization is applied to both the training dataset and testing dataset in order to remove any unnecessary bias. Especially, trained model based on full dataset normalization has bounded error in the prediction. The case study on the reduced 240-bus WECC system demonstrates that the proposed method can predict accurate frequency nadir with limited training dataset. The deep learning model using the revised feature normalization can predict more accurate frequency nadir than that using the traditional feature normalization and has very small maximum prediction error.

affinity propagation↗
DOE OSTI2024

Graph neural networks for CO 2 solubility predictions in Deep Eutectic Solvents

Deep Eutectic Solvents (DESs) are a promising class of solvents for CO 2 capture. DESs are complex mixtures that can be designed to optimize CO solubility and overall capture process efficiency. However, the vast design landscape of DES mixtures makes experimental investigation prohibitive; as such, there is a need for computational models that can quickly and efficiently navigate the design space and inform data collection efforts. In this work, we propose Graph Neural Network (GNN) models for predicting CO 2 solubility for DESs; the GNN leverages a mixture graph representation that captures the molecular structure of the DES components as well as their intermolecular interactions. Here, we compare the GNN framework against alternative architectures (neural networks, graph convolution networks, and random forests) and data representations (molecular fingerprints, sigma profiles, and graphs). We show that the proposed approach offers superior predictive performance; specifically, we show that solubility can be predicted reliably directly from molecular structure (without the need of using sigma profiles as proposed in previous studies). This result is important, as obtaining sigma profiles requires expensive density functional theory computations. We also explored the ability of GNNs to predict solubility for new DES mixtures and operating conditions. We found that the model extrapolates across temperature reliably. However, we also found deficiencies in the ability of the model to predict solubility for DES mixtures, pressures, and molar ratio not included in the training sets; we show that this is due to an inherent lack of chemical diversity in datasets available in the literature. The proposed computational capabilities can thus help navigate the design space of DES and inform data collection efforts. Our models, data, and benchmarks are shared as Python code implemented in Jupyter notebooks.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

Out-of-distribution detection with non-parametric density estimation for models predicting processing history of uranium ore concentrates

The rapid advancement in machine learning (ML) and computer vision (CV) coincides with the growth of interest in deploying these ML/CV models in numerous fields from medicine to social science. Similar to those areas, we have witnessed a great number of works in materials science employing ML/CV models – neural networks in particular – in their studies in recent years. These models have proven to obtain accurate performance in various tasks. However, these models struggle to attain a similar performance when encountering test samples coming from a distribution that is different from the training set. More importantly, they fail without providing any warning to the users. Therefore, we propose a framework for detecting out-of-distribution (OOD) samples to alert users when a human intervention might be necessary in this work. Specifically, we explore the use of a non-parametric density estimation method to detect OOD samples. Here, we assess OOD detection capability of the proposed framework on ML models developed for categorizing precipitation routes of U 3 O 8 when encountering OOD datasets that contain samples (1) undergone different imaging acquisition process, (2) undergone different material synthesis process, and (3) different materials than ID set. Through those experiments, we achieve an average area under the receiver operating characteristic (AUROC) of at least 91% on average in detecting OOD samples. With minimal overhead cost and superior performance, the proposed framework enables a reliable and safe system when deploying in real-world scenarios.

Convolutional neural networks↗
DOE OSTI2024

Data-driven models of nonautonomous systems

Nonautonomous dynamical systems are characterized by time-dependent inputs, which complicates the discovery of predictive models describing the spatiotemporal evolution of the state variables of quantities of interest from their temporal snapshots. When dynamic mode decomposition (DMD) is used to infer a linear model, this difficulty manifests itself in the need to approximate the time-dependent Koopman operators. Our approach is to approximate the original nonautonomous system with a modified system derived via a local parameterization of the time-dependent inputs. The modified system comprises a sequence of local parametric systems, which are subsequently approximated by a parametric surrogate model using the DRIPS (dimension reduction and interpolation in parameter space) framework. The offline step of DRIPS relies on DMD to build a linear surrogate model, endowed with reduced-order bases for the observables mapped from training data. The online step interpolates on suitable manifolds to construct a sequence of iterative parametric surrogate models; the target/test parameter points on these manifolds are specified by a local parameterization of the test time-dependent inputs. Here, we use numerical experimentation to demonstrate the robustness of our method and compare its performance with that of deep neural networks.

97 MATHEMATICS AND COMPUTING↗