Search NASA⌕ Search

SEARCH · Search NASA

Results for “quantitative image analysis”

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 325 records · Page 18

Core Mass Estimates in Strong Lensing Galaxy Clusters Using a Single-halo Lens Model

The core mass of galaxy clusters is an important probe of structure formation. Here we evaluate the use of a single-halo model (SHM) as an efficient method to estimate the strong lensing cluster core mass, testing it with ray-traced images from the Outer Rim simulation. Unlike detailed lens models, the SHM represents the cluster mass distribution with a single halo and can be automatically generated from the measured lensing constraints. In this work we find that the projected core mass estimated with this method, $M_{\text{SHM}}$, has a scatter of 8.52% and a bias of 0.90% compared to the "true" mass within the same aperture. Our analysis shows no systematic correlation between the scatter or bias and the lens-source system properties. The bias and scatter can be reduced to 3.26% and 0.34%, respectively, by excluding models that fail a visual inspection test. We find that the SHM success depends on the lensing geometry, with single giant arc configurations accounting for most of the failed cases due to their limiting constraining power. When excluding such cases, we measure a scatter and bias of 3.88% and 0.84%, respectively. Finally, we find that when the source redshift is unknown, the model-predicted redshifts are overestimated, and the $M_{\text{SHM}}$ is underestimated by a few percent, highlighting the importance of securing spectroscopic redshifts of background sources. Our analysis provides a quantitative characterization of $M_{\text{SHM}}$, enabling its efficient use as a tool to estimate the strong lensing cluster core masses in the large samples, expected from current and future surveys.

79 ASTRONOMY AND ASTROPHYSICS↗

Experimental distinction of the molecularly induced Balmer emission contribution and its application for inferring molecular divertor density with 2D filtered camera measurements during detachment in JET L-mode plasmas

A previously presented model for generating 2D estimates of the divertor plasma conditions at JET from deuterium Balmer line intensity ratios, obtained from tomographic reconstructions of divertor camera images, was amended to consider also the Balmer emission arising from molecular processes. Utilizing the AMJUEL and H2VIBR atomic and molecular databases of EIRENE enabled also inference of the molecular divertor density from the distinguished molecularly induced emission. Analysis of a JET L-mode density scan suggests the molecularly induced emission accounting for up to 60%–70% and 10%–20% of the Balmer Dα and Dγ intensities, respectively, at the onset of detachment, while electron-ion recombination becomes increasingly dominant with deepening detachment. Similar observations were made by post-processing EDGE2D-EIRENE simulations, which indicated significant roles of molecular D$^{+}_{2}$ ions and vibrational excitation of the D 2 molecules as precursors for the molecularly induced emission. The experimentally inferred molecular density at the outer strike point was found to increase monotonously with decreasing strike point temperature, reaching approximately 30%–50% of the local electron density at n mol, osp = 1–2 X 10 20 m -3 at T e,osp ≈ 0.7 eV. A further steep increase by a factor of 3–5 was observed with decrease of T e,osp to 0.5 eV. The observations are in qualitative and reasonable quantitative agreement with EDGE2D-EIRENE predictions of n mol, osp within the uncertainties of the experimental data.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Spectro-Microscopy Studies of Atmospheric Particles

Our project investigated composition and physical properties of individual atmospheric particles collected in field campaigns organized by the DOE Atmospheric Systems Research and Atmospheric Radiation Measurement (ASR/ARM) programs, such as the Green Ocean Amazon (GoAmazon, 2014/5), the Holistic Interactions of Shallow Clouds, Aerosols, and Land-Ecosystems (HI-SCALE, 2016/7), the Aerosol and Cloud Experiment in the Eastern North Atlantic (ACE-ENA, 2017/8), the Aerosol–Ice Formation Closure Pilot Study (AEROICESTUDY, 2019) and in additional small-scale supporting field experiments focused on the investigation of the light-absorbing (aka brown carbon) atmospheric particles. In these studies, we investigated the contribution of natural and anthropogenic sources to populations of atmospheric particles during representative atmospheric conditions; characterized particle composition and evaluated their hygroscopic properties and propensity to act as cloud condensation nuclei (CCN) and ice nuclei (IN); correlated typical particle-type and mixing state characteristics with real-time measurements in the areas of studies. We used multi-modal spectro-microscopy techniques to examine elemental and molecular composition, mixing state, size, and higher order morphology for individual particles. Specifically, we used computer-controlled scanning electron microscopy and X-ray microanalysis for quantitative analysis of particle elemental composition, and we used transmission electron microscopy to provide additional information on the distribution of different components within individual particles (particle heterogeneity). We used synchrotron-based scanning transmission X-ray microscopy for quantitative description of the mixing state and molecular bonding of carbon in particles. The particle composition, mixing state, and morphology from analyzed periods was then combined with the real-time ARM measurements of aerosol size distribution, CCN concentration, hygroscopicity, and aerosol bulk composition. Combined together, results of our studies allowed us to assess the major particle sources (biogenic vs. anthropogenic, primary vs secondary) and gain insight into the atmospheric processing of aerosol particles resulting from condensation, coagulation, oxidative aging, and cloud processing. In collaboration with other scientists involved in field campaigns, we carried out statistical analysis of particle properties and their source-specific contributions to regional aerosol loading under representative air mass, meteorological, and cloud conditions.

scanning transmission X-ray microscopy↗

Phase Segmentation in Atom-Probe Tomography Using Deep Learning-Based Edge Detection

Abstract Atom-probe tomography (APT) facilitates nano- and atomic-scale characterization and analysis of microstructural features. Specifically, APT is well suited to study the interfacial properties of granular or heterophase systems. Traditionally, the identification of the interface between, for precipitate and matrix phases, in APT data has been obtained either by extracting iso-concentration surfaces based on a user-supplied concentration value or by manually perturbing the concentration value until the iso-concentration surface qualitatively matches the interface. These approaches are subjective, not scalable, and may lead to inconsistencies due to local composition inhomogeneities. We introduce a digital image segmentation approach based on deep neural networks that transfer learned knowledge from natural images to automatically segment the data obtained from APT into different phases. This approach not only provides an efficient way to segment the data and extract interfacial properties but does so without the need for expensive interface labeling for training the segmentation model. We consider here a system with a precipitate phase in a matrix and with three different interface modalities—layered, isolated, and interconnected—that are obtained for different relative geometries of the precipitate phase. We demonstrate the accuracy of our segmentation approach through qualitative visualization of the interfaces, as well as through quantitative comparisons with proximity histograms obtained by using more traditional approaches.

47 OTHER INSTRUMENTATION↗

Segmentation and Classification of Fission as Pores in Reactor Irradiated Annular U–10Zr Metallic Fuel Using Machine Learning Models

Metallic fuels, particularly U—10Zr, are promising candidates for next-generation sodium-cooled fast reactors. Irradiation of nuclear fuels in reactors can lead to the formation of solid and gas fission product which subsequently forms microstructural pores, deteriorating fuel performance. Due to the massive amount of pores and complex phases formed, a quantitative description of fission gas pores is not yet available, preventing the development of microstructure-informed fuel performance modeling for fuel qualification. This paper applied a pre-trained deep learning model to ~10,260 high magnification scanning electron microscopy images. This method increased the accuracy of fission gas pore segmentation and allows statistical features to be extracted which cannot be achieved manually. A pre-trained decision tree model worked on the segemenation results and further classified the pores into different categories to produce a correlation between the pores, movement of lanthanides, and temperature gradient during irradiation. Finally, this paper emphasizes the potentials of machine learning models to accelerate fuel research, development, and qualification for advanced reactors.

36 MATERIALS SCIENCE↗

Particle image velocimetry analysis with simultaneous uncertainty quantification using Bayesian neural networks

Particle image velocimetry (PIV) is an effective tool in experimental fluid mechanics for extracting flow fields from images. Recently, convolutional neural networks (CNNs) have been used to perform PIV analysis with accuracy on par with classical methods. Here we extend the use of CNNs to analyze PIV data while providing simultaneous uncertainty quantification on the inferred flow field. The method we apply in this paper is a Bayesian convolutional neural network (BCNN) which learns distributions of the CNN weights through variational Bayes. In order to demonstrate the utility of BCNNs for the PIV task, we compare the performance of three distinct BCNN models with simple architectures. The first network estimates flow velocity from image interrogation regions only. Our second model learns to infer velocity from both the image interrogation regions and interrogation region cross-correlation maps. Finally, our best performing network infers velocities from interrogation region cross-correlation maps only. We find that BCNNs using interrogation region cross-correlation maps as inputs perform better than those using interrogation windows only as inputs and discuss reasons why this may be the case. Additionally, we test the best performing BCNN on a full synthetic test image pair and a real image pair from the 1st International PIV Challenge. We show that ~98% of true particle displacements from the full synthetic image pair can be captured within the BCNN's 95% confidence intervals, and that the BCNN's performance on the real image pair is quantitatively similar to that of algorithms tested in the 1st International PIV Challenge. Finally, we show that BCNNs can be generalized to be used with multi-pass PIV algorithms with a moderate loss in accuracy, which may be overcome by future work on finetuning and training schemes. So to our knowledge, this is the first use of Bayesian neural networks to perform PIV.

47 OTHER INSTRUMENTATION↗

Deep Nonparametric Estimation of Operators between Infinite Dimensional Spaces

Learning operators between infinitely dimensional spaces is an important learning task arising in machine learning, imaging science, mathematical modeling and simulations, etc. This paper studies the nonparametric estimation of Lipschitz operators using deep neural networks. Non-asymptotic upper bounds are derived for the generalization error of the empirical risk minimizer over a properly chosen network class. Under the assumption that the target operator exhibits a low dimensional structure, our error bounds decay as the training sample size increases, with an attractive fast rate depending on the intrinsic dimension in our estimation. Our assumptions cover most scenarios in real applications and our results give rise to fast rates by exploiting low dimensional structures of data in operator estimation. We also investigate the influence of network structures (e.g., network width, depth, and sparsity) on the generalization error of the neural network estimator and propose a general suggestion on the choice of network structures to maximize the learning efficiency quantitatively.

97 MATHEMATICS AND COMPUTING↗

Corresponding Standard Reference Material Data used in Partial Least Squares Regression Models for Sugar Composition Estimates in Biomass in: Economic Impact of Yield and Composition Variation in Bioenergy Crops: Populus trichocarpa

Corresponding Standard Reference Material Data used in Partial Least Squares Regression Models for Sugar Composition Estimates in Biomass in: Economic Impact of Yield and Composition Variation in Bioenergy Crops: Populus trichocarpa (for corresponding manuscript: DOI: 10.1002/bbb.2148) PDF Files: Images of 1H NMR spectra for neutralized 2-stage acid hydrolysates of 4 NIST Standard Reference Material biomass samples (Monterey Pine 8493, Sugarcane Bagasse 8491, Wheat Straw 8494, and Eastern Cottonwood/Poplar 8492) and 2 Center for Bioenergy Innovation reference biomass samples (Poplar - Populus trichocarpa and Switchgrass - Panicum Virgatum). Suppression of the water peak was achieved using a NOESY-1D with presaturation, a recycle delay of 5 s, and a total of 64 scans. Spectra were acquired at 298 K and processed with automatic phase correction, baseline correction, and chemical shift referencing to TSP-d4. Images show all 1H data from 10 to 1ppm with inset spectra of region of interest (4.0 to 3.1 ppm). Text Files: Spectra for neutralized 2-stage acid hydrolysates of 4 NIST Standard Reference Material biomass samples (Monterey Pine 8493, Sugarcane Bagasse 8491, Wheat Straw 8494, and Eastern Cottonwood/Poplar 8492) and 2 Center for Bioenergy Innovation reference biomass samples (Poplar - Populus trichocarpa and Switchgrass - Panicum Virgatum) were converted into text files for plotting. Files contain 8192 points of raw spectral data from 12.23 to -2.78 ppm. The text file contains 4 columns of data and includes: Point number, Intensity, Hz, and ppm. Xcel Spreadsheet: HPLC measured monomeric sugar concentrations and bucketed 1H NMR data used to build monomeric sugar composition prediction models. Sugar composition in biomass determined from HPLC analyses are given in mg sugar/mg of biomass. Spectral bucketing was performed using Bruker’s AMIX software. Spectra were divided into 0.005 ppm buckets in the region of 3.10– 4.15 ppm for a total of 210 buckets. Headers for the bucketed data are the chemical shift in ppm of the center of the bucket. Bucketed data was used to build partial least squares models for subsequent predictions in The Unscrambler v. 10.5(CAMO A/S, Trondheim, Norway). The formation of methanol during hydrolysis interferes with the quantitative NMR analysis of sugars, so the methanol peak centered at 3.37 ppm and spanning four buckets (3.2925 – 3.2775 ppm) was set to zero for all spectra.

09 BIOMASS FUELS↗

Predicting defects in laser powder bed fusion using in-situ thermal imaging data and machine learning

Variation in the local thermal history during the Laser Powder Bed Fusion (LPBF) process in Additive Manufacturing (AM) can cause micropore defects, which add to the uncertainty of the mechanical properties (e.g., fatigue life, tensile strength) of the built materials. In-situ sensing has been proposed for monitoring the AM process to minimize defects, but successful minimization requires establishing a quantitative relationship between the sensing data and the porosity, which is particularly challenging with a large number of variables (e.g., laser speed, power, scan path, powder property). Physics-based modeling can simulate such an in-situ sensing-porosity relationship, but it is computationally costly. In this work, we develop Machine Learning (ML) models that can use in-situ thermographic data to predict the micropore of LPBF stainless steel materials. This work considers two identified key features from the thermal histories: the time above the apparent melting threshold ($\tau$) and the maximum radiance ( T max ). These features are computed, stored for each voxel in the built material, and then used as inputs. The binary state of each voxel, either defective or normal, is the output. Different ML models are trained and tested for the binary classification task. In addition to using the thermal features of each voxel to predict its own state, the thermal features of neighboring voxels are also included as inputs. This is shown to improve the prediction accuracy, which is consistent with thermal transport physics around each voxel contributing to its final state. Among the models trained, the F1 scores on test sets reach above 0.96 for Random Forests. Feature importance analysis based on the ML models shows that T max is more important to the voxel state than $\tau$ . The analysis also finds that the thermal history of the voxels above the present voxel is more influential than those beneath it. Our study significantly extends the capability of using in-situ thermographic data to predict porosity in LPBF materials. Finally, since ML models are fast, they may play integral roles in the optimization and control of such AM technologies.

36 MATERIALS SCIENCE↗

Quantitative Insight to Fission Gas Pores Distribution in Irradiated Annular U-10Zr Metallic Fuel Using Machine Learning

Metallic fuels, particularly U-10Zr and its performance in reactor irradiation conditions, have been thoroughly investigated and are a promising candidate for next-generation sodium-cooled fast spectrum nuclear reactors. Irradiation in reactors can lead to the formation of fission gas and increased pore formation which can significantly impact fuel performance. Due to the large number of pores and various phases formed in metallic fuel during irradiation, a quantitative description of fission gas pores as a function of irradiation conditions is not yet available, undermining the fidelity of fuel performance modeling to support fuel qualification. It has been difficult to clearly detect pore boundaries and distinguish matrix phases from fission gas pores using optical microscopy by using simple threshold methods working with low magnification images. The pre-trained deep learning model for fission gas pore detection was applied to ~10,260 high magnification scanning electron microscopy images. The model increased the accuracy of fission gas pore segmentation to obtain statistical features, which cannot be processed manually. A pre-trained decision tree model was used to classify pores as isolated or connected pores, providing new insight into the correlation between the movement of lanthanides, solid fission products, and the radial temperature gradient developed in fuel irradiation conditions. This paper emphasizes the potential that artificial intelligence-based machine learning models have to accelerate qualification and support nuclear fuel development.

36 MATERIALS SCIENCE↗

Water-Soluble Flexible Organic Frameworks That Include and Deliver Proteins

Four water-soluble hydrazone-based three-dimensional (3D) flexible organic frameworks FOF-1-4 have been synthesized from a semirigid tetracationic tetraaldehyde and four flexible dihydrazides. 1 H NMR spectroscopy indicated the quantitative formation of FOF-1-4 in D 2 O, while dynamic light scattering experiments revealed that, depending on the concentration, these porous frameworks display hydrodynamic diameters ranging from 50 to 120 nm. The porosity of the frameworks is confirmed by ethanol vapor adsorption experiments of the solid samples as well as the high loading capacity for a 2.3 nm porphyrin guest in water. The new water-soluble frameworks exhibit low cytotoxicity and form inherent pores with diameters of 5.3 or 6.7 nm, allowing rapid inclusion of proteins such as bovine serum albumin and green and orange fluorescent proteins, and efficient delivery of the proteins into normal and cancer cells. Flow cytometric analysis reveals percentages of the delivered cells up to 99.8%.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Quantifying Moss Response to Metal Contaminant Exposure Using Laser-Induced Fluorescence

Tracing sources of contamination, including potentially toxic elements (PTEs), has historically been achieved through sampling and analysis of soil or biota, which are labor-intensive, costly, and destructive methods. Thus, availability of a non-destructive in situ remote sensing method for monitoring metals deposited in biota is of great interest. Laser-induced fluorescence (LIF) is an emerging spectroscopic and imaging technique that documents changes in molecular energy level in plants as a biological response to metal contamination. For a proof-of-concept study and preliminary experiment, moss was selected for experimentation due to its long history of use in tracing atmospheric deposition of PTEs. Consecutive treatments of copper chloride (CuCl2) were administered to three moss samples, simulating wet deposition every 48 h over 10 days until reaching cumulative Cu concentrations of 2.690 to 8.075 μmol/cm2. While these Cu amounts are above environmentally relevant concentrations, they allowed the best conditions for testing and fine tuning of the imaging and data processing protocols presented in this paper. Moss fluorescence was induced using both 532 nm green and 355 nm UV lasers. A CMOS camera captured images of the LIF response, and red–green–blue (RGB) decimal code values were extracted for each pixel in the images, and pixel densities of color channels from treated and untreated moss samples were compared. Results show a shift towards lower color decimal codes corresponding to increased Cu concentration. We developed and contrasted multiple quantitative analyses of color distributions and demonstrated that LIF shows great promise for remote sensing of Cu accumulation in moss at μmol/cm2 levels. Though currently, the method would be limited to highly toxic sites, it illustrates the possibility and provides a framework for development of higher-sensitivity methods to detect nmol/cm2 that are viable for urban contamination level monitoring.

63 RADIATION, THERMAL, AND OTHER ENVIRON. POLLUTAN↗

Universal signal scaling in microwave impedance microscopy

Microwave impedance microscopy (MIM) is an emerging scanning probe technique that measures the local complex dielectric function using near-field microwave. Although it has made significant impacts in diverse fields, a systematic, quantitative understanding of the signal's dependence on various important design parameters is lacking. Here, we show that for a wide range of MIM implementations, given a complex tip-sample admittance change ΔΥ, the MIM signal—the amplified change in the reflected microwave amplitude—is –G · ΔΥ/2Υ 0 · η 2 · V in , where η is the ratio of the microwave voltage at the probe to the incident microwave amplitude, Y o is the system admittance, and G is the total voltage gain. For linear circuits, η is determined by the circuit design and does not depend on V in . We show that the maximum achievable signal for different designs scales with η 2 or η when limited by input power or sample perturbation, respectively. Furthermore, this universal scaling provides guidance on diverse design goals, including maximizing narrow-band signal for imaging and balancing bandwidth and signal strength for spectroscopy.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Data supporting the manuscript "Nanometer Scale Imaging to Develop Quantitative Descriptors of Bipolar Membrane Junction Structure"

This dataset contains atomic force microscopy images associated with the manuscript "Nanometer Scale Imaging to Develop Quantitative Descriptors of Bipolar Membrane Junction Structure" by Maria Kelly, Emily R. Dunn, Ellis A. Spickermann, Josephine N. Gruber, César A. Lasalde-Ramírez, P. N. Romero Zavala, Éowyn Lucas, Ankur Gupta, Harry A. Atwater, and Wilson A. Smith. The dataset contains both raw images as well as segmented images produced by the image processing workflow described in the manuscript. A readme file and meta data file are included to provide additional details regarding the sample identity, image acquisition parameters, and file naming scheme.

36 MATERIALS SCIENCE↗

Data-driven materials research enabled by natural language processing and information extraction

Given the emergence of data science and machine learning throughout all aspects of society, but particularly in the scientific domain, there is increased importance placed on obtaining data. Data in materials science are particularly heterogeneous, based on the significant range in materials classes that are explored and the variety of materials properties that are of interest. This leads to data that range many orders of magnitude, and these data may manifest as numerical text or image-based information, which requires quantitative interpretation. The ability to automatically consume and codify the scientific literature across domains - enabled by techniques adapted from the field of natural language processing - therefore has immense potential to unlock and generate the rich datasets necessary for data science and machine learning. This review focuses on the progress and practices of natural language processing and text mining of materials science literature and highlights opportunities for extracting additional information beyond text contained in figures and tables in articles. Here, we discuss and provide examples for several reasons for the pursuit of natural language processing for materials, including data compilation, hypothesis development, and understanding the trends within and across fields. Current and emerging natural language processing methods along with their applications to materials science are detailed. We, then, discuss natural language processing and data challenges within the materials science domain where future directions may prove valuable.

36 MATERIALS SCIENCE↗

A framework for quantitative analysis of spectral data in two channels

Simultaneous acquisition of emission intensity from two channels in dual-color fluorescence measurements can be used to determine the spectral properties of an emitter. We detail a method to extract quantitative information from dual-color measurements by considering two models: a single spectral component model for peak wavelength determination and a dual spectral component model for relative strength determination. The former model is useful for spectral characterization and establishing heterogeneity within an emitter species, while the latter model is useful for analyzing energy transfer efficiency. Both models are formalized to yield probability distributions that reflect uncertainties due to shot noise. We applied these analysis tools to single-particle measurements of CdSe/CdS quantum dots. Isolated emitters had narrow spectral-intensity distributions that reflected their static emission profiles and fluorescence stability. Small assemblies of mixed donor/acceptor particles exhibited dynamic behavior where emission configuration changes reveal different spectral characteristics due to energy transfer.

36 MATERIALS SCIENCE↗

Derivation of A Representative Elementary Volume (REV) for Upscaled Two-Phase Flow in Porous Media

Relative permeability plays an important role in the upscaling of multiphase flow in porous media from the pore scale to the Darcy scale. The entire concept of relative permeability is contingent on the existence of a representative elementary volume (REV). As we move to smaller samples to measure relative permeability, such as with digital core analysis, the concept of a classical REV has become increasingly unlikely when using the conventional approach to defining a representative volume. The “‘conventional”’ understanding of an REV is that a large enough volume must be considered such that spatial variability averages out. In digital rock methods, such as pore-scale simulations based on micro-computed tomography (CT) images, the domain size is typically 2 to 4 mm. This is approximately the length scale of a single-phase flow REV using the classic REV approach. However, the single-phase perspective does not consider the complex dynamics and fluctuations often observed in multiphase flow systems, even at centimeter-scale experiments and/or simulations. A fundamental question is, therefore, whether the domain size commonly used in digital rock simulations can provide a consistent energy budget such that the concept of relative permeability exists. Based on first principles, relative permeability accounts for the rate of energy dissipated in a stationary process. If the dynamics are fluctuating, the energy dissipated can vary but will average out over a long enough timescale. The key to determining the validity of the relative permeability is the timescale of the measurement, not the spatial scale. The conventional REV theory assumes that spatial, temporal, and ensemble averages are equivalent in an ergodic system, but it does not provide a way to test this assumption. We provide a formal way to identify the timescale where the relative permeability accurately captures energy dissipation as a way to validate relative permeability measurements and quantitatively assess their accuracy. This result will be tested for a practical SCAL test, determining how long a flow experiment needs to be run to accurately characterize the rate of energy dissipation by the flow. The outcome will be a best practice guide for the determination of relative permeability from core-scale experiments and/or digital core simulations that ensure the energy budget is fully accounted for in the relative permeability coefficient.

Mcclure, James [Virginia Tech, Blacksburg]↗

Deep nonparametric estimation of operators between infinite dimensional spaces

Learning operators between infinitely dimensional spaces is an important learning task arising in machine learning, imaging science, mathematical modeling and simulations, etc. This paper studies the nonparametric estimation of Lipschitz operators using deep neural networks. Non-asymptotic upper bounds are derived for the generalization error of the empirical risk minimizer over a properly chosen network class. Under the assumption that the target operator exhibits a low dimensional structure, our error bounds decay as the training sample size increases, with an attractive fast rate depending on the intrinsic dimension in our estimation. Our assumptions cover most scenarios in real applications and our results give rise to fast rates by exploiting low dimensional structures of data in operator estimation. We also investigate the influence of network structures (e.g., network width, depth, and sparsity) on the generalization error of the neural network estimator and propose a general suggestion on the choice of network structures to maximize the learning efficiency quantitatively.

97 MATHEMATICS AND COMPUTING↗