Search NASA⌕ Search

SEARCH · Search NASA

Results for “exascale computing”

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 181 records · Page 10
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

ORBIT-2 Dataset for Scaling Exascale Vision Foundation Models for Weather and Climate Downscaling

This dataset release corresponds to the work conducted in ORBIT-2: Scaling Exascale Vision Foundation Models for Weather and Climate Downscaling, where large-scale AI methods were applied to improve climate and weather resolution. The collection integrates four widely used, publicly available datasets: ERA5, PRISM, DAYMET, and IMERG. To prepare the data for ORBIT-2 model training and evaluation, we applied a preprocessing pipeline that generates paired low-resolution and high-resolution samples, enabling supervised downscaling experiments. The transformation from coarse to fine scales was performed using bilinear regridding, consistent with the procedures described in WeatherBench2, a community benchmark for weather and climate AI models. This dataset supports the development and evaluation of foundation models designed for weather and climate downscaling at exascale. Additional details on methodology and applications can be found in Wang et al., ORBIT-2 (arXiv:2505.04802, 2025).

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2024

ESM data downscaling: a comparison of super-resolution deep learning models

Abstract Climate projections at fine spatial resolutions are required to conduct accurate risk assessment for critical infrastructure and design adaptation planning. Generating these projections using advanced Earth system models (ESM) requires significant computational resources. To address this issue, various statistical downscaling techniques have been introduced to generate fine-resolution data from coarse-resolution simulations. In this study, we evaluate and compare five deep learning-based downscaling techniques, namely, super-resolution convolutional neural networks, fast super-resolution convolutional neural network ESM, efficient sub-pixel convolutional neural network, enhanced deep residual network (EDRN), and super-resolution generative adversarial network (SRGAN). These techniques are applied to a dataset generated by the Energy Exascale Earth System Model (E3SM), focusing on key surface variables such as surface temperature, shortwave heat flux, and longwave heat flux. Models are trained and validated using paired fine-resolution (0.25 $$^{\circ }$$ ∘ ) and coarse-resolution (1 $$^{\circ }$$ ∘ ) monthly data obtained from a 9-year simulation. Next, blind testing is performed using monthly data obtained from two different years outside of the training and validation set. To evaluate the efficiency of each technique, different statistical metrics are used, including mean squared error (MSE), peak signal-to-noise ratio (PSNR), structural similarity index measure (SSIM), and learned perceptual image patch similarity (LPIPS). The results show that EDRN outperforms other algorithms in terms of PSNR, SSIM, and MSE, but struggles to capture fine-scale features in the data. In contrast, SRGAN, a generative model that uses perceptual loss, excels in capturing fine details at boundaries and internal structures, resulting in lower LPIPS than other methods.

Pawar, Nikhil M. (ORCID:0000000211613289)↗
DOE OSTI2024

EQSIM: Exascale Predictions of Earthquake Effects on Critical Infrastructure

The great “San Francisco” earthquake of 1906 is one of the most recognized, and sobering, demonstrations of the havoc that can be caused by the sudden and violent movement of Earth’s tectonic plates. The estimated 7.9-magnitude quake and subsequent fires decimated the major metropolis and surrounding areas: buildings turned to ruins, hundreds of thousands of people left homeless, and a death toll exceeding 3,000. Today, as evidenced by the catastrophic 7.8-magnitude earthquake that struck Turkey in February 2023, these events still present a significant danger to life and economic security. To mitigate the potential devastation of future earthquakes and better prepare for these inevitable events, researchers are turning to high-performance computers to simulate the underlying geophysical processes and accurately quantify associated risks to critical infrastructure.

42 ENGINEERING↗
DOE OSTI2026

A roadmap toward scaling, reasoning and self-evolving foundation models for nuclear and particle physics

Foundation models have revolutionized artificial intelligence, with Large Language Models demonstrating unprecedented capabilities in multimodal understanding, reasoning and tool use. Nuclear and particle physics stands at a critical juncture where similar transformative potential awaits realization. The field generates exabytes of experimental data, exascale simulations, and decades of theoretical insights — yet these remain largely disconnected from modern Artifical Intelligence (AI) capabilities, with most physics AI applications confined to narrow, task-specific models that suffer from domain shifting when applied to real experimental data. We present a roadmap for FM4NPP (Foundation Model for Nuclear and Particle Physics), systematically scaling from current proof-of-concept models to trillion-parameter architectures capable of autonomous discovery. Our approach advances three critical frontiers: unified data infrastructure integrating detector data, scientific knowledge and computational tools across global facilities; multi-facility foundation models enabling cross-experiment knowledge transfer and accelerated discovery; and agentic AI capabilities for reasoning and autonomous tool use. The resulting self-evolving FM4NPP will transform physics research by converting time-intensive data analysis, theory derivation and computational bottlenecks into rapid AI–human collaborative discovery. This paradigm shift promises to fundamentally accelerate scientific progress in nuclear and particle physics, enabling researchers to focus on high-level insights while AI handles routine analysis and explores vast parameter spaces beyond human capacity.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2025

Experiences with SYCL on AMD GPUs with Kokkos

With the recent diversification of the hardware landscape in the high-performance computing (HPC) community, performance-portability solutions are becoming more and more important. One of the most popular choices is Kokkos, which recently became a Linux Foundation project. Most of its development is supported by the US Department of Energy and the French Alternative Energies and Atomic Energy Commission. Kokkos is implemented as a C++ library with multiple backends to support CPUs as well as various GPU architectures. These backends include OpenMP, CUDA, HIP, and also SCYL. This approach enables users to leverage the preferred vendor toolchain for the respective platform (e.g. CUDA, ROCm, OneAPI). The SYCL backend is used to target Intel GPUs, in particular to support the Aurora exascale supercomputer. However, SYCL itself also offers a large degree of portability, and in fact Kokkos’ CI for SYCL has been running on NVIDIA hardware due to a lack of access to Intel GPUs. In this report, we describe our experience with using Kokkos SYCL backend on AMD GPUs targeting the Frontier supercomputer at Oak Ridge National Laboratory. The two major SYCL implementations are DPC++ and AdaptiveCpp. While the Kokkos SYCL backend has been implemented using the former, the latter was the first implementation to target AMD GPUs. We will discuss the experience with both of these SYCL implementations in terms of functionality and performance. Using Kokkos to evaluate SYCL toolchains has a number of benefits. Kokkos’ use of SYCL is fairly complex, exercising features such as graphs, relocatable device functions, atomics – including for non-arithmetic types, as well as pinned and page migratable memory allocations. Kokkos also needs to implement capabilities such as Kokkos’ hierarchical parallelism that are not a straight-forward mapping to SYCL capabilities. Furthermore, a large number of libraries and applications that represent diverse use cases are implemented in Kokkos, providing readily available test cases for a toolchain evaluation. Preliminary results show that support for AMD GPUs in DPC++ is much less mature than for NVIDIA GPUs or Intel GPUs. While the situation has improved significantly over the last year, we still encounter many runtime failures, dispatching problems, and code generation issues. With AdaptiveCpp the challenges arise even earlier in the evaluation process. Since Kokkos’ SYCL implementation is largely focused on supporting Intel GPUs, we opted to leverage SYCL extensions which are available in DPC++ but not in AdaptiveCpp. Furthermore, AdaptiveCpp appears to be less conformant with the SYCL2020 standard which Kokkos relies on. In some cases, we are able to work around the lack of feature support, in other cases we have to disable certain Kokkos capabilities to evaluate the toolchain. Our evaluation will leverage Kokkos’ unit tests to establish basic functionality and feature completeness. We then use simple benchmarks for components of a CG implementation as a measure of usability and performance of the SYCL toolchains.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Enhanced climate reproducibility testing with false discovery rate correction

Simulating the Earth's climate is an important and complex problem, thus climate models are similarly complex, comprised of millions of lines of code. In order to appropriately utilize the latest computational and software infrastructure advancements in Earth system models running on modern hybrid computing architectures to improve their performance, precision, accuracy, or all three; it is important to ensure that model simulations are repeatable and robust. This introduces the need for establishing statistical or non-bit-for-bit reproducibility, since bit-for-bit reproducibility may not always be achievable. Here, we propose a short-simulation ensemble-based test for an atmosphere model to evaluate the null hypothesis that modified model results are statistically equivalent to that of the original model. We implement this test in version 2 of the US Department of Energy's Energy Exascale Earth System Model (E3SM). The test evaluates a standard set of output variables across the two simulation ensembles and uses a false discovery rate correction to account for multiple testing. The false positive rates of the test are examined using re-sampling techniques on large simulation ensembles and are found to be lower than the currently implemented bootstrapping-based testing approach in E3SM. We also evaluate the statistical power of the test using perturbed simulation ensemble suites, each with a progressively larger magnitude of change to a tuning parameter. The new test is generally found to exhibit more statistical power than the current approach, being able to detect smaller changes in parameter values with higher confidence.

Kelleher, Michael E. [Oak Ridge National Laborator↗
DOE OSTI2024

Uncertainty Quantification and Sensitivity Analysis of Low-Dimensional Manifold via Co-Kurtosis PCA in Combustion Modeling

For multi-scale multi-physics applications e.g., the turbulent combustion code Pele, robust and accurate dimensionality reduction is crucial to solving problems at exascale and beyond. A recently developed technique, Co-Kurtosis based Principal Component Analysis (CoK-PCA) which leverages principal vectors of co-kurtosis, is a promising alternative to traditional PCA for complex chemical systems. To improve the effectiveness of this approach, we employ Artificial Neural Networks for reconstructing thermo-chemical scalars, species production rates, and overall heat release rates corresponding to the full state space. Our focus is on bolstering confidence in this deep learning based non-linear reconstruction through Uncertainty Quantification (UQ) and Sensitivity Analysis (SA). UQ involves quantifying uncertainties in inputs and outputs, while SA identifies influential inputs. One of the noteworthy challenges is the computational expense inherent in both endeavors. To address this, we employ the Monte Carlo methods to effectively quantify and propagate uncertainties in our reduced spaces while managing computational demands. Our research carries profound implications not only for the realm of combustion modeling but also for a broader audience in UQ. By showcasing the reliability and robustness of CoK-PCA in dimensionality reduction and deep learning predictions, we empower researchers and decision-makers to navigate complex combustion systems with greater confidence.

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

Development of a River Dynamical Core for E3SM to simulate compound flooding on Exascale-class heterogeneous supercomputers

Flooding events pose significant risk to human life, property, and infrastructure. Physically-consistent quantification of altered flood risks in global models requires hyper-resolution (~1 km) or fine flood simulations using two-dimensional (2D) physics schemes, both of which are unavailable in the current generation Earth System Models. Here, in this work, we have developed the River Dynamical Core (RDycore), which is an open-source, 2D shallow water equation (SWE) library for the U.S. Department of Energy's Energy Exascale Earth System Model (E3SM). RDycore uses PETSc and libCEED libraries that allows it to run efficiently on CPUs and GPUs, as well as select a time-integration algorithm at runtime without requiring any code modifications. RDycore achieves spatial error convergence rates for problems with analytical and manufactured solutions similar to those reported previously in the literature, or consistent with the implemented first-order spatial discretization scheme. RDycore's accuracy in predicting flooding for a well-studied dam break problem is comparable to existing SWE models. For a problem with 471 million grid cells, RDycore achieves a speedup of 6.6x and 7.6x on GPUs compared to CPUs when using 320 compute nodes on DOE's Perlmutter and Frontier supercomputers, respectively. The one-way coupling of the RDycore library within E3SM is demonstrated by performing multiple 5-day flooding simulations during Hurricane Harvey driven by five precipitation datasets. The E3SM--RDycore simulations at 30 m spatial resolution accurately simulate maximum water height during the hurricane when benchmarked against a previously published study and achieve a speedup of 15x (Perlmutter) and 21x (Frontier) on GPUs relative to CPUs. The work presented here is the foundational step in providing hardware and algorithmic portability framework for simulating kilometer-scale river dynamics within E3SM.

Flood Simulation↗
DOE OSTI2024

Pretraining Billion-Scale Geospatial Foundational Models on Frontier

As AI workloads increase in scope, generalization capability becomes challenging for small task-specific models and their demand for large amounts of labeled training samples increases. On the contrary, Foundation Models (FMs) are trained with internet-scale unlabeled data via self-supervised learning and have been shown to adapt to various tasks with minimal fine-tuning. Although large FMs have demonstrated significant impact in natural language processing and computer vision, efforts toward FMs for geospatial applications have been restricted to smaller size models, as pretraining larger models requires very large computing resources equipped with state-of-the-art hardware accelerators. Current satellite constellations collect 100+TBs of data a day, resulting in images that are billions of pixels and multimodal in nature. Such geospatial data poses unique challenges opening up new opportunities to develop FMs. We investigate billion scale FMs and HPC training profiles for geospatial applications by pretraining on publicly available data. We studied from end-to-end the performance and impact in the solution by scaling the model size. Our larger 3B parameter size model achieves up to 30% improvement in top1 scene classification accuracy when comparing a 100M parameter model. Moreover, we detail performance experiments on the Frontier supercomputer, America's first exascale system, where we study different model and data parallel approaches using PyTorch's Fully Sharded Data Parallel library. Specifically, we study variants of the Vision Transformer architecture (ViT), conducting performance analysis for ViT models with size up to 15B parameters. By discussing throughput and performance bottlenecks under different parallelism configurations, we offer insights on how to leverage such leadership-class HPC resources when developing large models for geospatial imagery applications.

Tsaris, Aristeidis (aris)↗
DOE OSTI2024

Enhancing Photosynthesis Simulation Performance in ESMs with Machine Learning-Assisted Solvers

When simulating vegetation dynamics, photosynthesis accounts for a large fraction of the computational cost in most Earth System Models (ESMs). This is largely since photosynthesis is represented as a system of nonlinear equations, and the solution requires the use of an initial guess followed by many iterations of the numerical solver to obtain a solution. We use machine learning (ML) to replicate the response surface of the model’s numerical solver to improve the choice of initial guess, therefore requiring fewer iterations to obtain a final solution. We implemented this test on the leaf-level calculations as well as at the canopy scale, and for both we observed fewer iterations of the photosynthesis solver when a ML-based initial guess was implemented. The model tested here is the Energy Exascale Earth System Model - Land Model (ELM). The ML-based algorithms used here are trained on simulations from the model itself and used only to improve the initial guess for the solver; therefore, the model maintains its own set of physics to obtain the final solution. This work shows novel ways to utilize ML-based methods to improve the performance of numerical solvers in ESMs.

Massoud, Elias [ORNL] (ORCID:0000000217725361)↗
DOE OSTI2024

Program Generators for Exascale and Beyond (Final Technical Report)

The major goals of this project were to devise programming models and supporting tools that simplify the construction of specialized high‐performance generators, leading to improvements over the state of the art along the various dimensions of portability, performance, and productivity. In particular, the project achieved these goals by focusing on domain‐specific languages (DSLs) as intermediate layers in a translation and optimization framework, on composability of program generators, on re‐targeting legacy code, and on fundamental methodologies such as stage polymorphism or "generic programming in time".

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

A graphics processing unit accelerated sparse direct solver and preconditioner with block low rank compression

We present the GPU implementation efforts and challenges of the sparse solver package STRUMPACK. The code is made publicly available on github with a permissive BSD license. STRUMPACK implements an approximate multifrontal solver, a sparse LU factorization which makes use of compression methods to accelerate time to solution and reduce memory usage. Multiple compression schemes based on rank-structured and hierarchical matrix approximations are supported, including hierarchically semi-separable, hierarchically off-diagonal butterfly, and block low rank. Here, in this paper, we present the GPU implementation of the block low rank (BLR) compression method within a multifrontal solver. Our GPU implementation relies on highly optimized vendor libraries such as cuBLAS and cuSOLVER for NVIDIA GPUs, rocBLAS and rocSOLVER for AMD GPUs and the Intel oneAPI Math Kernel Library (oneMKL) for Intel GPUs. Additionally, we rely on external open source libraries such as SLATE (Software for Linear Algebra Targeting Exascale), MAGMA (Matrix Algebra on GPU and Multi-core Architectures), and KBLAS (KAUST BLAS). SLATE is used as a GPU-capable ScaLAPACK replacement. From MAGMA we use variable sized batched dense linear algebra operations such as GEMM, TRSM and LU with partial pivoting. KBLAS provides efficient (batched) low rank matrix compression for NVIDIA GPUs using an adaptive randomized sampling scheme. The resulting sparse solver and preconditioner runs on NVIDIA, AMD and Intel GPUs. Interfaces are available from PETSc, Trilinos and MFEM, or the solver can be used directly in user code. We report results for a range of benchmark applications, using the Perlmutter system from NERSC, Frontier from ORNL, and Aurora from ALCF. For a high frequency wave equation on a regular mesh, using 32 Perlmutter compute nodes, the factorization phase of the exact GPU solver is about 6.5× faster compared to the CPU-only solver. The BLR-enabled GPU solver is about 13.8× faster than the CPU exact solver. For a collection of SuiteSparse matrices, the STRUMPACK exact factorization on a single GPU is on average 1.9× faster than NVIDIA’s cuDSS solver.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Towards Exascale Astrophysics of Mergers and Supernovae (TEAMS)

The TEAMS project brought together cutting-edge simulations, theoretical insights, and collaborative efforts to deepen our understanding of some of the universe’s most extreme phenomena—supernovae, neutron star mergers, and the powerful signals they emit. Using one of the largest suites of 3D supernova simulations ever conducted, researchers uncovered new insights into how massive stars explode, how those explosions vary by stellar mass, and what conditions lead to the birth of neutron stars or black holes. They also studied the radiation and gravitational wave signals emitted during these events, revealing how future observations can be used to uncover what happens deep inside collapsing stars. The team developed improved tools for modeling how light and neutrinos behave in such explosive environments, enabling more accurate predictions of what astronomers might observe. Work also explored how the chemical composition and geometry of kilonovae—the visible explosions that follow neutron star mergers—influence their signals and can reveal the origins of heavy elements like gold. These efforts not only advanced scientific knowledge, but also trained a new generation of researchers at the intersection of astrophysics, computational science, and nuclear theory.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2026

ExaChem/exachem

Open Source Exascale Quantum Chemistry Software

Panyala, Ajay [Pacific Northwest National Laborato↗
DOE OSTI2025

Performance-portable Binary Neutron Star Mergers with AthenaK

We introduce an extension to the AthenaK code for general-relativistic magnetohydrodynamics (GRMHD) in dynamical spacetimes using a 3+1 conservative Eulerian formulation. Like the fixed-spacetime GRMHD solver, we use standard finite-volume methods to evolve the fluid and a constrained-transport scheme to preserve the divergence-free constraint for the magnetic field. We also utilize a first-order flux correction (FOFC) scheme to reduce the need for an artificial atmosphere and optionally enforce a maximum principle to improve robustness. We demonstrate the accuracy of AthenaK using a set of standard tests in flat and curved spacetimes. Using a SANE accretion disk around a Kerr black hole, we compare the new solver to the existing solver for stationary spacetimes using the so-called "HARM-like" formulation. We find that both formulations converge to similar results. We also include the first published binary neutron star (BNS) mergers performed on graphical processing units (GPUs). Thanks to the FOFC scheme, our BNS mergers maintain a relative error of $\mathcal{O}$(10 –11 ) or better in baryon mass conservation up to collapse. Finally, we perform scaling tests of AthenaK on OLCF Frontier, where we show excellent weak scaling of ≥80% efficiency up to 32,768 GPUs and 74% up to 65,536 GPUs for a GRMHD problem in dynamical spacetimes with six levels of mesh refinement. AthenaK achieves an order-of-magnitude speedup using GPUs compared to CPUs, demonstrating that it is suitable for performing numerical relativity problems on modern exascale resources.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2024

exadigitUE5

This project provides the AR/VR interface to ORNL's exascale digital twin. The main functionality is implemented using Unreal Engine 5.1 for Desktop or Microsoft Hololens2 based visualization and interation with the system. The digital twin provides data ingestion from telemetry, as well as triggering and interacting with simulations developed for the wider ExaDigiT project at ORNL, as well as for the LUMI system at CSC and other CrayEX Supercomputers. For the overarching project, see ExaDigiT at https://exadigit.github.io, with the code repositories at https://code.ornl.gov/exadigit.

Maiterth, Matthias [Oak Ridge National Laboratory ↗
DOE OSTI2025

Hydraulic Conductivity Measurements, Utqiagvik (Barrow), Alaska, 2014

Six individual ice cores were collected from the Barrow Environmental Observatory in Barrow, Alaska, in May of 2013 as part of the Next Generation Ecosystem Experiment (NGEE). Each core was drilled at a different location to varying depths. After drilling, the cores were stored in coolers packed with dry ice and flown to Lawrence Berkeley National Laboratory (LBNL) in Berkeley, CA. 3-dimensional images of the cores were constructed using medical X-ray computed tomography (CT) scanner at 120kV. Hydraulic conductivity samples were extracted from these cores at LBNL Richmond Field Station in Richmond, CA, in February 2014 by cutting 5 to 8 inch segments using a chop saw. Samples were packed individually and stored at -20C freezing temperatures to minimize any changes in structure or loss of ice content prior to analysis. Hydraulic conductivity was determined through falling head tests using a permeameter [ELE International, Model #: K-770B] (Appendix A). Samples were placed in a latex membrane via a membrane stretcher while frozen. Use of a membrane stretcher made the membranes easier to secure and minimized contact with the sample. A clear polycarbonate sleeve, fabricated with a stainless steel ring at the bottom to keep the sleeve from floating, was placed around the sample inside the permeameter to minimize deformation during analysis. The permeameter was filled with water and 1.0 PSI of air was applied for confining pressure during sample defrost. Outflow valves were left open to allow for incremental thawing and samples were left to thaw for approximately 12 hours. After approximately 12 hours of thaw, initial falling head tests were performed. When the flow was significantly too fast or too slow, the analysis was stopped and the burette size was adjusted accordingly (i.e. a larger diameter burette was used for flows that were faster than desired or a smaller diameter burette was used for flows that were slower than desired). Two to four measurements were collected on each sample and collection stopped when the applied head load exceeded 25% change from the original load. Analyses were performed between 2 to 3 times for each sample. The final hydraulic conductivity calculations were computed using methodology of Das et al., 1985.The Next-Generation Ecosystem Experiments: Arctic (NGEE Arctic), was a 15-year research effort (2012-2027) to reduce uncertainty in Earth System Models by developing a predictive understanding of carbon-rich Arctic ecosystems and feedbacks to climate. NGEE Arctic was supported by the Department of Energy's Office of Biological and Environmental Research. The NGEE Arctic project had two field research sites: 1) located within the Arctic polygonal tundra coastal region on the Barrow Environmental Observatory (BEO) and the North Slope near Utqiagvik (Barrow), Alaska and 2) multiple areas on the discontinuous permafrost region of the Seward Peninsula north of Nome, Alaska. Through observations, experiments, and synthesis with existing datasets, NGEE Arctic provided an enhanced knowledge base for multi-scale modeling and contributed to improved process representation at global pan-Arctic scales within the Department of Energy's Earth system Model (the Energy Exascale Earth System Model, or E3SM), and specifically within the E3SM Land Model component (ELM).

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2026

Numerical Water Tracers in the Atmospheric Component of the Energy Exascale Earth System Model: Implementation and Changes in Moisture Origin

Numerical water tracers are implemented in the Energy Exascale Earth System Model version 2. Simulations performed with the water‐tag‐enabled model for both pre‐industrial and future greenhouse gas concentrations reveal a marked increase in the role of mid‐latitude and southern subtropical regions as exporters of atmospheric moisture—to the extratropical upper troposphere and the tropical free troposphere. For the latter, the northward shift of the Intertropical Convergence Zone increases cross‐hemispheric transport of subtropical water vapor to the Northern Hemisphere. In the polar regions, most of the lower tropospheric moistening instead arises from increases in local evaporation. These findings illustrate the utility of the water tags, underscore critical changes in global hydrologic cycle, and provide insight into atmospheric dynamics under future climate scenarios. For applications when a global grid is desired, we additionally propose a novel statistical reconstruction, based on copula modeling, of the joint distribution of origin of water vapor, which reduces the number of tracers from order $\mathcal{O}\left({n}^{2}\right)$to order $\mathcal{O}(n)$, substantially ameliorating the considerable computational cost of water tracers. This statistical reconstruction is particularly beneficial to the interpretation of the relationship between latitude and longitude of origin of moisture over the tropical oceans and in the lower troposphere over land.

copula modeling↗