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All sourcesNASA NTRSDOE OSTI
NASA NTRS2018

Soil Moisture Active Passive (SMAP) Project Assessment Report for Version 4 of the L4_SM Data Product

This report provides an assessment of Version 4 of the SMAP Level 4 Surface and Root Zone Soil Moisture (L4_SM) product, released on 14 June 2018. The assessment includes comparisons of L4_SM soil moisture and temperature estimates with in situ measurements from core validation sites and sparse networks. The assessment further includes a global evaluation of the internal diagnostics from the ensemble-based data assimilation system that is used to generate the L4_SM product, including observation-minus-forecast (O-F) brightness temperature residuals and soil moisture analysis increments.Together, the core validation site comparisons and the statistics of the assimilation diagnostics areconsidered primary validation methodologies for the L4_SM product. Comparisons against in situ measurements from regional-scale sparse networks are considered a secondary validation methodology because such in situ measurements are subject to upscaling errors from the point-scale to the grid-cell scale of the data product.The Version 4 L4_SM product benefits from an improved land surface modeling system and from retrospective surface meteorological forcing data that are as consistent as possible with the present-day datain terms of their climatology. Specifically, the model changes include revised parameters and parameterizations for (i) the surface energy balance, (ii) recharge from below of the model's surface excess reservoir, and (iii) the snow depletion curve. Updated ancillary inputs include improved datasets for landcover, topography, and vegetation height. The Version 4 algorithm further includes a revised approach to precipitation corrections that improves the precipitation climatology in Africa and the high-latitudes. Moreover, for system calibration the model is forced retrospectively with MERRA-2 reanalysis data, which are more consistent with the near-real time GEOS forward processing (FP) data used during the SMAP period than the retrospective GEOS data that were available for previous L4_SM versions. An analysis of the time-average surface and root zone soil moisture shows that the global pattern ofarid and humid regions is captured by the Version 4 L4_SM estimates. Owing to the changes in the landsurface modeling system, surface soil moisture is typically drier by several volumetric percent in Version 4 compared to Version 3, whereas root zone soil moisture is wetter in Version 4 in some regions and drierin others. Because of these climatological differences, the Version 3 and Version 4 products should not be combined into a single dataset for use in applications.Results from the core validation site comparisons indicate that Version 4 of the L4_SM data product meets the self-imposed L4_SM accuracy requirement, which is formulated in terms of the RMSE after removal of the long-term mean difference (ubRMSE). The overall ubRMSE of the 3-hourly L4_SM dataat the 9 km scale is 0.039 m3 m-3 for surface soil moisture and 0.029 m3 m-3 for root zone soil moisture,below the 0.04 m3 m-3 requirement. The L4_SM estimates are an improvement over estimates from a model-only Nature Run version 7.2 (NRv7.2), which demonstrates the beneficial impact of the SMAP brightness temperature data. Overall, L4_SM surface and root zone soil moisture estimates are more skillful than NRv7.2 estimates, with statistically significant improvements at the 5% level for surface soil moisture R and anomaly R values. Results from comparisons of the L4_SM product to i

Reichle, Rolf H.↗
NASA NTRS2019

Diurnal Cycles in SST: Coupled Data Assimilation and Future Observational Requirements

Most operational centers are developing coupled (atmosphere-ocean) data assimilation systems as an alternative to uncoupled counterparts (atmosphere- or ocean-only). The Sea Surface Temperature (SST) is one of the key variables that tightly connects the atmosphere and ocean states and also air-sea fluxes. However, current prototype coupled data assimilation systems rely on external (L4) gridded SST or along-track (L3 or L2) SST retrievals as observed data or relaxation field. But in reality, SST are measurements are available from sparse in-situ network of ships, moorings, and buoys; all of them combined together are far less than those from satellites. However, satellites do not directly measure temperature, and inferring SST from satellite measured radiances requires a radiative transfer model, its calibration and also bias correction.The NASA Global Modeling and Assimilation Office (GMAO) is developing a coupled data assimilation system which assimilates SST directly from the raw observations, i.e., satellite radiances and in-situ observations. The methodology to directly assimilate radiances for SST became operational in Jan, 2017 in the GMAO's near-real time Weather Analysis and Prediction System. There were many modifications to the GMAO system in order to implement SST assimilation, most of which generally improved the predictability of the system. In order to maintain and further improve this system, we advocate for the availability of a microwave satellite radiometer in future beyond the currently operational GPM- GMI and AMSR-2 missions. For improved modeling of the near-surface temperature, salinity and mixing processes, we suggest adding more than one temperature sensor and salinity sensors to the drifting buoy network.

Akella, Santha↗
DOE OSTI2024

Predicting Missing Regions in Charged Particle Tracks Using a Sparse 3D Convolutional Neural Network

The 2x2 Demonstrator is a prototype of ND-LAr, the liquid argon time-projection chamber of the Deep Underground Neutrino Experiment’s Near Detector complex. Both the 2x2 Demonstrator and ND-LAr are modular detectors that will have pixelated charge readouts and inactive regions wherein there is no sensitivity to charge deposition and light signals that arise from charged particle interactions with liquid argon. In the 2x2, these inactive regions are located in between the active detector modules, which introduces the challenge of inferring what charge signals ought to look like in these regions. This study explores the use of a Sparse 3D Convolutional Neural Network (ConvNet) to infer missing regions in charged particle tracks. Hits corresponding to energy depositions are voxelized into a three-dimensional grid for each track. Voxels that fall into predefined inactive regions are removed to simulate the lack of detector output. The model is trained to infer the topology of the missing track voxels, with the ultimate goal of inferring the missing charge or energy values in these voxels as well. Results indicate that this approach shows promise in prediction of missing track regions with some accuracy.

Utaegbulam, Hilary↗
DOE OSTI2024

Predicting Missing Regions in Charged Particle Tracks Using a Sparse 3D Convolutional Neural Network

The 2x2 Demonstrator is a prototype detector for the Deep Underground Neutrino Experiment (DUNE)'s Near Detector. Both the 2x2 Demonstrator and the Near Detector itself will have inactive regions wherein there is no sensitivity to charge deposition and light signals that arise from charged particle interactions with liquid argon. In the 2x2, these inactive regions are positioned in-between the active detector modules, which introduces the challenge of inferring what charge signals ought to look like in these regions. This study explores the use of a Sparse 3D Convolutional Neural Network (ConvNet) to infer missing regions in charged particle tracks. Hits corresponding to energy depositions are voxelized into a three-dimensional (3D) grid for each track. Inactive regions within the tracks are replaced with a dense, rectangular 3D grid of voxels, ensuring consistent step sizes in X, Y, and Z directions. Voxels in these dense regions are initialized with an energy value of -1, indicating nonphysical energy or charge. The model is trained to predict which voxels should activate as part of the track and which should not, with the goal of eventually inferring the missing charge or energy values in these voxels. Results indicate that the model accurately predicts track voxels within ±1 unit in X, Y, or Z directions and effectively identifies non-track voxels, despite some overprediction. The approach shows promise in prediction of missing track regions with some accuracy.

43 PARTICLE ACCELERATORS↗
DOE OSTI2024

Predicting Missing Regions in Charged Particle Tracks Using a Sparse 3D Convolutional Neural Network

The 2x2 Demonstrator is a prototype detector for the Deep Underground Neutrino Experiment (DUNE)'s Near Detector. Both the 2x2 Demonstrator and the Near Detector itself will have inactive regions wherein there is no sensitivity to charge deposition and light signals that arise from charged particle interactions with liquid argon. In the 2x2, these inactive regions are positioned in-between the active detector modules, which introduces the challenge of inferring what charge signals ought to look like in these regions. This study explores the use of a Sparse 3D Convolutional Neural Network (ConvNet) to infer missing regions in charged particle tracks. Hits corresponding to energy depositions are voxelized into a three-dimensional (3D) grid for each track. Inactive regions within the tracks are replaced with a dense, rectangular 3D grid of voxels, ensuring consistent step sizes in X, Y, and Z directions. Voxels in these dense regions are initialized with an energy value of -1, indicating nonphysical energy or charge. The model is trained to predict which voxels should activate as part of the track and which should not, with the goal of eventually inferring the missing charge or energy values in these voxels. Results indicate that the model accurately predicts track voxels within 1 unit in X, Y, or Z directions and effectively identifies non-track voxels, despite some overprediction. The approach shows promise in prediction of missing track regions with some accuracy.

Utaegbulam, Hilary↗
NASA NTRS1984

Application of VISSR Atmospheric Sounder (VAS) data in weather analysis

A technique which analyzes irregularly spaced satellite data is described. An experiment with rawinsonde and VISSR Atmospheric Sounder (VAS) radiance measurements collected on March 6-7, 1982 is conducted to reveal the applicability of the technique. The rawinsonde data are analyzed on a 16 x 12 grid using the two pass analysis scheme of Barnes (1973). A scheme similar to the Barnes (1973) procedure is employed to produce gridded analysis of VAS data over a 200 x 15000 km region in central part of the U.S. The use of a correction pass on the initial gridded field is described; the technique is extremely effective on uniformly spaced observations. The incorporation of the limited fine mesh model to the scheme to analyze data in sparse and cloudy regions is examined. A comparison of rawinsonde data with VAS data is provided. The technique proves effective for studying cloudy and sparse areas with VAS data and produces a four-dimensional data set with significant mesoscale structure.

Jedlovec, G. J.↗
NASA NTRS1992

Mapping unstructured grid computations to massively parallel computers

Investigated here is this mapping problem: assign the tasks of a parallel program to the processors of a parallel computer such that the execution time is minimized. First, a taxonomy of objective functions and heuristics used to solve the mapping problem is presented. Next, we develop a highly parallel heuristic mapping algorithm, called Cyclic Pairwise Exchange (CPE), and discuss its place in the taxonomy. CPE uses local pairwise exchanges of processor assignments to iteratively improve an initial mapping. A variety of initial mapping schemes are tested and recursive spectral bipartitioning (RSB) followed by CPE is shown to result in the best mappings. For the test cases studied here, problems arising in computational fluid dynamics and structural mechanics on unstructured triangular and tetrahedral meshes, RSB and CPE outperform methods based on simulated annealing. Much less time is required to do the mapping and the results obtained are better. Compared with random and naive mappings, RSB and CPE reduce the communication time two fold for the test problems used. Finally, we use CPE in two applications on a CM-2. The first application is a data parallel mesh-vertex upwind finite volume scheme for solving the Euler equations on 2-D triangular unstructured meshes. CPE is used to map grid points to processors. The performance of this code is compared with a similar code on a Cray-YMP and an Intel iPSC/860. The second application is parallel sparse matrix-vector multiplication used in the iterative solution of large sparse linear systems of equations. We map rows of the matrix to processors and use an inner-product based matrix-vector multiplication. We demonstrate that this method is an order of magnitude faster than methods based on scan operations for our test cases.

Hammond, Steven Warren↗
NASA NTRSDate not supplied

MISR Level 2 TOA/Cloud Stereo parameters (MIL2TCST_V1)

The MISR Top-of-Atmosphere (TOA)/Cloud Stereo geophysical parameters include stereoscopically-derived cloud motion vectors (winds), cloud-top heights, and an accompanying cloud mask. The Stereo product geophysical parameters include a stereoscopically-derived cloud mask and cloud height on a 1.1 km grid. It also includes cloud motion vectors on a 70.4 km grid. The three types of stereo heights are: the BestWind heights are only calculated for those regions where the associated wind vectors passed the quality tests. Therefore, they have sparse coverage but since the wind correction is included, these contain our 'best guess' as to what the true heights are. The WithoutWind heights are calculated assuming a constant wind vector of zero. They have almost complete coverage and therefore form a nice 'pretty picture' of the relative cloud heights over small areas. The RawWind heights are a diagnostic product as they are calculated using all available wind vectors (even the bad ones). It is therefore recommended that one only use the Best and Without wind products. It is important to remember that the stereo matchers pick up the layer of maximum contrast, which is not necessarily the same as the highest cloud so all the stereo heights are keyed to this level of maximum contrast. Therefore, higher and thinner cirrus layers may not be detected by any of the height fields. [Location=GLOBAL] [Temporal_Coverage: Start_Date=2000-02-24; Stop_Date=] [Spatial_Coverage: Southernmost_Latitude=-90; Northernmost_Latitude=90; Westernmost_Longitude=-180; Easternmost_Longitude=180] [Data_Resolution: Latitude_Resolution=1.1 km; Longitude_Resolution=1.1 km; Horizontal_Resolution_Range=1 km - < 10 km or approximately .01 degree - < .09 degree; Temporal_Resolution=about 15 orbits/day; Temporal_Resolution_Range=Daily - < Weekly].

EOSDIS↗
NASA NTRSDate not supplied

MISR Level 2 TOA/Cloud Stereo parameters (MIL2TCST_V2)

The MISR Top-of-Atmosphere (TOA)/Cloud Stereo geophysical parameters include stereoscopically-derived cloud motion vectors (winds), cloud-top heights, and an accompanying cloud mask. The Stereo product geophysical parameters include a stereoscopically-derived cloud mask and cloud height on a 1.1 km grid. It also includes cloud motion vectors on a 70.4 km grid. The three types of stereo heights are: the BestWind heights are only calculated for those regions where the associated wind vectors passed the quality tests. Therefore, they have sparse coverage but since the wind correction is included, these contain our 'best guess' as to what the true heights are. The WithoutWind heights are calculated assuming a constant wind vector of zero. They have almost complete coverage and therefore form a nice 'pretty picture' of the relative cloud heights over small areas. The RawWind heights are a diagnostic product as they are calculated using all available wind vectors (even the bad ones). It is therefore recommended that one only use the Best and Without wind products. It is important to remember that the stereo matchers pick up the layer of maximum contrast, which is not necessarily the same as the highest cloud so all the stereo heights are keyed to this level of maximum contrast. Therefore, higher and thinner cirrus layers may not be detected by any of the height fields. [Temporal_Coverage: Start_Date=2000-02-24; Stop_Date=] [Spatial_Coverage: Southernmost_Latitude=-90; Northernmost_Latitude=90; Westernmost_Longitude=-180; Easternmost_Longitude=180] [Data_Resolution: Latitude_Resolution=1.1 km; Longitude_Resolution=1.1 km; Temporal_Resolution=about 15 orbits/day].

CLOUD MASK↗
DOE OSTI2026

A GPU-based compressible combustion solver for applications exhibiting disparate space and time scales

High-speed chemically active flows pose significant computational challenges due to their disparate space and time scales, with stiff chemistry often dominating simulation time. While modern scientific computing programs achieve exascale performance by leveraging graphics processing units (GPUs), existing GPU-based compressible combustion solvers face critical limitations in memory management, load balancing, and handling the highly localized nature of chemical reactions. To this end, we present a high-performance compressible reacting flow solver built on the AMReX framework and optimized for multi-GPU settings. Here, our approach addresses three GPU performance bottlenecks: memory access patterns through column-major storage optimization, computational workload variability via a bulk-sparse integration strategy for chemical kinetics, and multi-GPU load distribution for adaptive mesh refinement applications. The solver adapts existing matrix-based chemical kinetics formulations to multi-grid contexts. Using representative combustion applications, including 2D and 3D detonations and a 3D jet-in-crossflow configuration, we demonstrate 1.4–5× performance improvements over initial implementations on an in-house cluster of NVIDIA H100 GPUs, and near-ideal weak scaling on the Frontier supercomputer (Oak Ridge Leadership Computing Facility) with up to 1024 AMD Instinct MI250X GPUs. Roofline analysis reveals substantial improvements in arithmetic intensity for both convection (∼ 10 ×) and chemistry (∼ 4 ×) routines, confirming efficient utilization of GPU memory bandwidth and computational resources.

42 ENGINEERING↗
NASA NTRS1992

Search for evidence of low energy protons in solar flares

We searched for linear polarization in the H alpha line using the Stokes Polarimeter at Mees Solar Observatory and present observations of a flare from NOAA active region 6659 which began at 01:30 UT on 14 Jun. 1991. Our dataset also includes H alpha spectra from the Mees charge coupled device (MCCD) imaging spectrograph as well as hard x ray observations from the Burst and Transient Source Experiment (BATSE) instrument on board the Gamma Ray Observatory (GRO). The polarimeter scanned a 40 x 40 inch field of view using 16 raster points in a 4 x 4 grid. Each scan took about 30 seconds with 2 seconds at each raster point. The polarimeter stopped 8.5 inches between raster points and each point covered a 6 inch region. This sparse sampling increased the total field of view without reducing the temporal cadence. At each raster point, an H alpha spectrum with 20 mA spectral sampling is obtained covering 2.6 A centered on H alpha line center. The preliminary conclusions from the research are presented.

Metcalf, Thomas R.↗
NASA NTRS2000

Ordering Unstructured Meshes for Sparse Matrix Computations on Leading Parallel Systems

The ability of computers to solve hitherto intractable problems and simulate complex processes using mathematical models makes them an indispensable part of modern science and engineering. Computer simulations of large-scale realistic applications usually require solving a set of non-linear partial differential equations (PDES) over a finite region. For example, one thrust area in the DOE Grand Challenge projects is to design future accelerators such as the SpaHation Neutron Source (SNS). Our colleagues at SLAC need to model complex RFQ cavities with large aspect ratios. Unstructured grids are currently used to resolve the small features in a large computational domain; dynamic mesh adaptation will be added in the future for additional efficiency. The PDEs for electromagnetics are discretized by the FEM method, which leads to a generalized eigenvalue problem Kx = AMx, where K and M are the stiffness and mass matrices, and are very sparse. In a typical cavity model, the number of degrees of freedom is about one million. For such large eigenproblems, direct solution techniques quickly reach the memory limits. Instead, the most widely-used methods are Krylov subspace methods, such as Lanczos or Jacobi-Davidson. In all the Krylov-based algorithms, sparse matrix-vector multiplication (SPMV) must be performed repeatedly. Therefore, the efficiency of SPMV usually determines the eigensolver speed. SPMV is also one of the most heavily used kernels in large-scale numerical simulations.

Oliker, Leonid↗
DOE OSTI2025

Practical Implementation of GPU-based Computing at the Grid Edge for Resilience Scenarios

This paper presents a practical implementation of GPU-accelerated computing at the grid edge to enhance power system resilience through next-generation smart meters. Advanced Metering Infrastructure (AMI) systems rely predominantly on centralized processing architectures, which limit real-time response capabilities during grid disturbances. This work proposes the integration of GPU-enabled computational platforms directly within smart meter to enable local execution support for power system analytics, fault detection algorithms, and optimization routines. The proposed framework uses the Julia programming language to leverage highperformance parallel computing capabilities while maintaining code portability and development efficiency. We use two experimental scenarios to benchmark the computational feasibility of this approach: sparse linear system solutions representative of power flow analyses, and multi-stage production cost simulations incorporating unit commitment and economic dispatch operations. Results demonstrate that computationally intensive power system algorithms, such as those supporting resilience scenario calculations, can be effectively executed at the distribution edge using commercially available embedded GPU hardware. Keywords—GPU acceleration, edge computing, smart meters, grid resilience, AMI, resilience.

De Souza, Reubun [School of Electrical Engineering↗
NASA NTRS1994

Fractal characteristics of ozonometric network

The fractal (correlation) dimensions are calculated which characterize the distribution of stations in the ground-based total ozone measuring network and the distribution of nodes in a latitude-longitude grid. The dimension of the ground-based ozonometric network equals 1.67 +/- 0.1 with an appropriate scaling in the 60 to 400 km range. For the latitude-longitude grid two scaling regimes are revealed. One regime, with the dimension somewhat greater than one, is peculiar to smaller scales and limited from a larger scale by the latitudinal resolution of the grid. Another scaling regime, with the dimension equal 1.84, ranges up to 15,000 km scale. The fact that the dimension of a measuring network is less than two possesses problems in observing sparse phenomena. This has to have important consequences for ozone statistics.

Gruzdev, Alexander N.↗
DOE OSTI2022

Modeling Spatial Distribution of Snow Water Equivalent by Combining Meteorological and Satellite Data with Lidar Maps

Abstract An accurate characterization of the water content of snowpack, or snow water equivalent (SWE), is necessary to quantify water availability and constrain hydrologic and land surface models. Recently, airborne observations (e.g., lidar) have emerged as a promising method to accurately quantify SWE at high resolutions (scales of ∼100 m and finer). However, the frequency of these observations is very low, typically once or twice per season in the Rocky Mountains of Colorado. Here, we present a machine learning framework that is based on random forests to model temporally sparse lidar-derived SWE, enabling estimation of SWE at unmapped time points. We approximated the physical processes governing snow accumulation and melt as well as snow characteristics by obtaining 15 different variables from gridded estimates of precipitation, temperature, surface reflectance, elevation, and canopy. Results showed that, in the Rocky Mountains of Colorado, our framework is capable of modeling SWE with a higher accuracy when compared with estimates generated by the Snow Data Assimilation System (SNODAS). The mean value of the coefficient of determination R 2 using our approach was 0.57, and the root-mean-square error (RMSE) was 13 cm, which was a significant improvement over SNODAS (mean R 2 = 0.13; RMSE = 20 cm). We explored the relative importance of the input variables and observed that, at the spatial resolution of 800 m, meteorological variables are more important drivers of predictive accuracy than surface variables that characterize the properties of snow on the ground. This research provides a framework to expand the applicability of lidar-derived SWE to unmapped time points. Significance Statement Snowpack is the main source of freshwater for close to 2 billion people globally and needs to be estimated accurately. Mountainous snowpack is highly variable and is challenging to quantify. Recently, lidar technology has been employed to observe snow in great detail, but it is costly and can only be used sparingly. To counter that, we use machine learning to estimate snowpack when lidar data are not available. We approximate the processes that govern snowpack by incorporating meteorological and satellite data. We found that variables associated with precipitation and temperature have more predictive power than variables that characterize snowpack properties. Our work helps to improve snowpack estimation, which is critical for sustainable management of water resources.

54 ENVIRONMENTAL SCIENCES↗
NASA NTRS2019

Numerical Reanalyses as a Gateway to Arctic Synthesis

Reanalyses are regularly gridded, retrospective depictions of the physical earth system, which are produced through the correction of a short-term forecast to available observations. In the Arctic, reanalyses are particularly well suited to marshal the sparse observing network to provide a plausible, multivariate representation of conditions. Atmospheric reanalyses such as MERRA-2 (NASA Modern-Era Retrospective analysis for Research and Applications, version 2) and ocean reanalyses such as SODA3 (Univ. Maryland Simple Ocean Data Assimilation version 3) are widely used in Arctic research for diagnostic studies of circulation, model evaluation, and as boundary conditions for a variety of process models. Here, we provide examples that illustrate the utility of reanalyses for providing information on the spatial and temporal scales of recent, rapid changes in the Arctic. Recent trends in Arctic surface temperatures, surface melt over Greenland and Arctic glaciers, and evolving freshwater conditions in the Arctic Ocean are examples where reanalyses can provide information that cannot easily be obtained via other means. These examples provide information on the scale, magnitude, and the uncertainty of recent Arctic change and provide a context for future scenarios. We further quantify uncertainties in key reanalyses variables and approaches for addressing these issues.

Arctic↗
NASA NTRS2016

An Optimized Multicolor Point-Implicit Solver for Unstructured Grid Applications on Graphics Processing Units

In the field of computational fluid dynamics, the Navier-Stokes equations are often solved using an unstructuredgrid approach to accommodate geometric complexity. Implicit solution methodologies for such spatial discretizations generally require frequent solution of large tightly-coupled systems of block-sparse linear equations. The multicolor point-implicit solver used in the current work typically requires a significant fraction of the overall application run time. In this work, an efficient implementation of the solver for graphics processing units is proposed. Several factors present unique challenges to achieving an efficient implementation in this environment. These include the variable amount of parallelism available in different kernel calls, indirect memory access patterns, low arithmetic intensity, and the requirement to support variable block sizes. In this work, the solver is reformulated to use standard sparse and dense Basic Linear Algebra Subprograms (BLAS) functions. However, numerical experiments show that the performance of the BLAS functions available in existing CUDA libraries is suboptimal for matrices representative of those encountered in actual simulations. Instead, optimized versions of these functions are developed. Depending on block size, the new implementations show performance gains of up to 7x over the existing CUDA library functions.

Zubair, Mohammad↗
DOE OSTI2022

Learning Optimal Multigrid Smoothers via Neural Networks

Multigrid methods are one of the most efficient techniques for solving large sparse linear systems arising from partial differential equations (PDEs) and graph Laplacians from machine learning applications. One of the key components of multigrid is smoothing, which aims at reducing high-frequency errors on each grid level. However, finding optimal smoothing algorithms is problem-dependent and can impose challenges for many problems. In this paper, we propose an efficient adaptive framework for learning optimized smoothers from operator stencils in the form of convolutional neural networks (CNNs). Here, the CNNs are trained on small-scale problems from a given type of PDEs based on a supervised loss function derived from multigrid convergence theories and can be applied to large-scale problems of the same class of PDEs. Numerical results on anisotropic rotated Laplacian problems and variable coefficient diffusion problems demonstrate improved convergence rates and solution time compared with classical hand-crafted relaxation methods.

97 MATHEMATICS AND COMPUTING↗