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Regularization by denoising diffusion models for solving inverse PDE problems with application to full waveform inversion

Partial differential equation (PDE)-governed inverse problems are fundamental across various scientific and engineering applications; yet they face significant challenges due to nonlinearity, ill-posedness, and sensitivity to noise. Here, we introduce a computational framework, regularization by denoising using diffusion models for partial differential equations (RED-DiffEq), by integrating physics-driven inversion and data-driven learning. RED-DiffEq leverages pretrained diffusion models as a regularization mechanism for PDE-governed inverse problems. We apply RED-DiffEq to solve the full waveform inversion problem in geophysics, a challenging seismic imaging technique that seeks to reconstruct high-resolution subsurface velocity models from seismic measurement data. Our method shows enhanced accuracy and robustness compared to benchmark methods. Additionally, it exhibits strong generalization and domain decomposition capacity, enabling the inversion of more complex velocity models with larger domains than those used in training the diffusion model. Our framework can also be directly applied to diverse PDE-governed inverse problems.

Shan, Siming [Yale University, New Haven, CT (Unit

Basin & Range Investigation for Developing Geothermal Energy

Hidden geothermal systems represent a potentially prolific energy resource that could support critical U.S. public and government energy priorities. Basin and Range Investigations for Developing Geothermal Energy (BRIDGE) addressed some the challenges associated with hidden system exploration by prioritizing cost-effective exploration early on through strategic workflow and informed decision-making that mitigates early risk and shifts resources to later exploration stages (e.g., drilling). Sandia National Laboratories partnered with U.S. Navy Geothermal Office, Geologic Geothermal Group, and independent consultants, with additional collaboration with U.S. Geological Survey and private industry. The primary tool of the BRIDGE project was to deploy a regional-scale airborne electromagnetic method to investigate the shallow resistivity structure in areas with high prospectivity. This was followed up at several prospects by a multidisciplinary exploration approach, including additional geologic, geophysical and geochemical studies. A central tenet to the BRIDGE methodology is that zones of low resistivity frequently occur over geothermal systems in the Basin and Range, and when paired with other data constraints, imaging these zones can enable discovery of these systems. In addition to exploring greenfield areas (i.e., Grover Point), the BRIDGE project also flew HTEM resistivity surveys over known geothermal systems including those with established power plants (Don A. Campbell and Salt Wells) and prospects that are known to the literature but remain undeveloped, at least in part, due to a lack of understanding on the location of their producible reservoirs. BRIDGE produced a comprehensive set of data from prospects identified in the Nevada Play Fairway Analysis along with conceptual models for top ranking prospects, wherein all of the observations are used to inform an interpreted model of the system. These models present a range of possible system parameters such as temperature and size, and they are further informed by system analogues in the Basin and Range province and elsewhere. The results of this work leave space for further exploration that may now occur at prospects ‘down the list’ rather than distribution exploration resources evenly across all prospects.

15 GEOTHERMAL ENERGY

"Hidden" hydrothermal technical potential & technoeconomics: Revealing permeability & fluids with more data

Historical hydrothermal estimates have largely relied on temperature or heat flow estimates ignoring the need for natural flowing fluids. More accurate hydrothermal estimates require some indication of permeability and fluids that naturally exist in the subsurface. This paper describes a novel approach that includes proxies of permeability and fluids in hydrothermal estimates by leveraging the relatively data-rich Great Basin. Specifically, nameplate capacities (megawatts) of operating geothermal plants, negative (0 megawatt) locations and 48 geophysical and geologic features are used to used in eXtreme Gradient Boosting (XGBoost) regression to make hydrothermal capacity predictions. Additionally, this work inputs the XGBoost-based hydrothermal predictions into the Renewable Energy Potential (reV) model to quantify technical capacity, its uncertainty and techno-economics. Compared to historical hydrothermal estimates, these predictions adhere to the 37 operating geothermal plants and negative locations. We present a method for subsampling the negative sites to bring the labels into balance that uses the geologic domain knowledge to proportionally represent negatives. Overall, the distributions of the hydrothermal technical capacity and the site levelized cost of energy are respectively much tighter, lower and more accurate than the previous estimates for the Great Basin, as they include geological and geophysical surrogates for permeability and fluids. Percentile (50th and 90th, median and high estimate, respectively) models provide bookends for these metrics.

13 HYDRO ENERGY

The GFDL‐CM4X Climate Model Hierarchy, Part II: Case Studies

This paper is Part II of a two‐part paper that documents the Climate Model version 4X (CM4X) hierarchy of coupled climate models developed at the Geophysical Fluid Dynamics Laboratory. Part I of this paper is presented in Griffies et al. (2025a, https://doi.org/10.1029/2024MS004861 ). Here we present a suite of case studies that examine ocean and sea ice features that are targeted for further research, which include sea level, eastern boundary upwelling, Arctic and Southern Ocean sea ice, Southern Ocean circulation, and North Atlantic circulation. The case studies are based on experiments that follow the protocol of version 6 from the Coupled Model Intercomparison Project. The analysis reveals a systematic improvement in the simulation fidelity of CM4X relative to its CM4.0 predecessor, as well as an improvement when refining the ocean/sea ice horizontal grid spacing from the 0.25° of CM4X‐p25 to the 0.125° of CM4X‐p125. Even so, there remain many outstanding biases, thus pointing to the need for further grid refinements, enhancements to numerical methods, and/or advances in parameterizations, each of which target long‐standing model biases and limitations.

54 ENVIRONMENTAL SCIENCES

Infrasonic directivity of monopole, dipole and bipole ground-surface reflected sources

Infrasound (acoustic waves below 20 Hz) can be used to detect, locate and quantify activity in the atmosphere such as volcanic eruptions and anthropogenic explosions. Attempts to quantify volcanic eruption parameters such as exit velocity, plume height and mass flow rate using infrasound data depend strongly on assumptions of the acoustic source type. Infrasonic sources may produce omnidirectional or directional wavefields, while propagation effects, such as interaction with topography, can induce further wavefield directivity that is measured by field instrumentation. Limited sampling of these wavefields can hinder our ability to infer the underlying source, and thus our understanding of the eruption characteristics. Equivalent sources are often used to represent acoustic source mechanisms and resultant wavefields. In this study, we review equivalent acoustic sources as they pertain to infrasonic scale and wavelengths commonly encountered in very local (⁠<5 km range) geophysical field deployments. We highlight the equivalent infrasonic bipole source that can be induced by ground-reflection of an elevated monopole; we are not aware of any prior infrasound studies that use the bipole source concept. We use analytical and numerical methods to explore source directivity of monopole, dipole and bipole ground-reflected sources at infrasonic frequencies as well as the additional directivity complications introduced by interactions with topography. We illustrate that for typical volcano-infrasound wavelengths, increasing height above the ground as well as increasing source frequency leads to increased wavefield directivity. Numerical modelling using a simple omnidirectional monopole source embedded in topography further illustrates that both horizontal and vertical infrasound directionality can be induced by topography at the distance scales appropriate for local volcano infrasound monitoring. Information summarized in this analytical and numerical exploration of infrasound directivity may be used to help guide future volcano-infrasound field deployments intended to estimate source parameters or quantify wavefield directivity. Analytic solutions for simple whole-space or half-space atmospheres provide useful formulations for planning or initially analysing geophysical field-scale experimental data; however, especially at very local distances from the source (⁠<5 km), 3-D simulations are necessary to account for complex topography commonly encountered in volcano-infrasound applications.

Infrasound

Advances in geophysical forensic event monitoring

Forensic analysis of man-made, non-nuclear events (such as industrial accidents, explosion experiments and mine collapses) has become more frequent and detailed owing to advancements in geophysical monitoring. Here, in this Technical Review, we demonstrate how geophysical forensic monitoring using seismic, infrasound and hydroacoustic recordings provides insights on events in the solid earth, atmosphere and underwater. Advanced techniques, including machine-learning-based models, have been developed to detect, identify and investigate these events, providing information on location, subevents, sources and explosive yield. The increase in data availability, application of advanced methods and computation and the growth of multitechnology approaches have increased the accuracy of forensic event analysis and enabled more realistic characterization of uncertainties. For example, the 2020 Beirut explosion in Lebanon demonstrated that various seismic, acoustic and other methods could be used to estimate explosive yield (and yield uncertainties) of about 1 ktonne, providing confidence in the application of these methods to smaller events where data are available. However, forensic investigations remain largely limited to known events with identified sources. Increased access to data, sophisticated analysis methods and high-resolution earth models will improve forensic event analysis further, enabling civil and scientific applications, such as localization in the search for the lost ARA San Juan submarine.

geophysics

Ecosystems and Networks Integrated with Genes and Molecular Assemblies (ENIGMA): Molecular and Computational Technologies for Environmental Microbiology (Final Scientific/Technical Report)

The ENIGMA science focus area (SFA) is a multi-disciplinary, multi-institutional research effort focused on addressing foundational knowledge gaps in environmental microbial communities by studying groundwater and sediment microbiomes in the shallow subsurface at the contaminated Oak Ridge Reservation (ORR). We seek to discover and characterize the reciprocal interactions between the microbial communities and the geochemical and geophysical parameters of the shallow subsurface within the contamination plume. The primary goal of this subcontract was to develop experimental and computational tools to advance our understanding of microbial adaptation and community assembly in contaminated environments, with specific efforts in high-throughput genomic methods, microbial ecology tools, and studies of heavy metal contamination impacts.

54 ENVIRONMENTAL SCIENCES

Cloud Fusion of Big Data and Multi-Physics Models using Machine Learning for Discovery, Exploration, and Development of Hidden Geothermal Resources

The primary goals of this project are identifying hidden geothermal resources in the USA and designing profitable enhanced geothermal systems (EGS). Many non-obvious processes and parameters could characterize geothermal resources and could control the ultimate energy potential of geothermal fields. Diverse datasets (e.g., geology, geochemistry, geophysics, satellite, airborne geophysics) are available to help characterize geothermal resources, but this data is sparse and multi-scale. This has hindered attempts to leverage the datasets for geothermal exploration and profitable EGS design. Recent advancements in machine learning (ML) give promise to overcome these issues. Modern ML methods and tools can (1) analyze large datasets, (2) assimilate model ensembles that include a multitude of inputs and outputs, (3) process sparse datasets, (4) perform transfer learning between sites with different data quality, (5) extract hidden geothermal signatures from field and simulation data, (6) label geothermal resources and processes, (7) identify high-value data acquisition targets, and (8) guide geothermal exploration and production by selecting optimal exploration, production, and drilling strategies. In this work, we implement ML-based geothermal exploration and an enhanced geothermal systems (EGS) design tool to achieve the above goals. Our exploration tool is GeoThermalCloud (GTC) EGS design tool is GeoDT-ML. GTC (github.com/SmartTensors/GeoThermalCloud.jl) utilizes a LANL unsupervised ML platform called SmartTensors (https://tensors.lanl.gov/) to automate data analyses and interpretations by extracting hidden signatures to identify geothermal prospects. It enables the identification of critical measurements needed to identify geothermal resource signatures. GeoDT-ML (github.com/SmartTensors/GeoThermalCloud.jl/tree/master/) adds coupling to GeoDT (https://github.com/GeoDesignTool/GeoDT.git) for stochastic EGS design optimization and performance prediction. GeoDT-ML leverages recent advances in deep learning and high-performance computing. Contributors to this effort include LANL, PNNL, Google, Stanford, and Julia Computing.

15 GEOTHERMAL ENERGY

Basic Research Needs for Inverse Methods for Complex Systems under Uncertainty

Inverse problems, which aim to infer unknown properties of a system using experimental and observational data, are central to addressing many of the U.S. Department of Energy’s (DOE) most critical scientific and engineering challenges. Accurate, computationally efficient, and data-efficient solutions to inverse problems are essential for advancing DOE mission-critical science drivers, including analyzing data from large-scale experimental facilities, optimizing fusion reactor performance, accelerating materials discovery, enhancing geophysical imaging, improving wildfire predictions, and enabling autonomous systems and digital twins. However, these problems are becoming increasingly complex, often involving nonlinear, highdimensional, and interconnected systems and models that span multiple physics and scales, while relying on data with varying quantity, quality, and information content. Compounding these challenges is the uncertainty inherent in DOE-relevant systems, where errors in inputs, noise in data, incompleteness of data, and discrepancies between models and reality constrain the accuracy and precision of solutions. At the same time, the convergence of recent scientific computing trends—scientific machine learning, artificial intelligence, and computing advances such as exascale computing—is creating unprecedented opportunities for tackling these challenges. The cross-cutting nature of inverse problems, combined with their growing complexity and rapidly evolving data and algorithmic demands, strongly motivates the formulation of a prioritized research agenda to maximize their capabilities and impact. In response to this need, DOE’s Advanced Scientific Computing Research (ASCR) program in the Office of Science convened the Workshop on Basic Research Needs for Inverse Problems for Complex Systems Under Uncertainty in June 2025. This workshop brought together experts across disciplines to identify grand challenges and major opportunities in the field. Through collaborative discussions, the workshop defined transformative research directions aimed at addressing the mathematical, statistical, and computational challenges posed by inverse problems under uncertainty. As a result of these efforts, four priority research directions (PRDs) were identified to guide future research and development in this area. These PRDs, summarized below, represent a roadmap for advancing the foundational science and mathematics of inverse problems, enabling robust, scalable, and uncertainty-aware solutions that are critical for DOE applications.

97 MATHEMATICS AND COMPUTING

Rapid wavefield forecasting for earthquake early warning via deep sequence to sequence learning

We propose a deep learning model, WaveCastNet, to forecast high-dimensional wavefields. WaveCastNet integrates a convolutional long expressive memory architecture into a sequence-to-sequence forecasting framework, enabling it to model long-term dependencies and multiscale patterns in both space and time. By sharing weights across spatial and temporal dimensions, WaveCastNet requires significantly fewer parameters than more resource-intensive models such as transformers, resulting in faster inference times. Crucially, WaveCastNet also generalizes better than transformers to rare and critical seismic scenarios, such as high-magnitude earthquakes. Here, we show the ability of the model to predict the intensity and timing of destructive ground motions in real time, using simulated data from the San Francisco Bay Area. Furthermore, we demonstrate its zero-shot capabilities by evaluating WaveCastNet on real earthquake data. Our approach does not require estimating earthquake magnitudes and epicenters, steps that are prone to error in conventional methods, nor does it rely on empirical ground-motion models, which often fail to capture strongly heterogeneous wave propagation effects.

Geophysics

Quantitative kinetic rules for plastic strain-induced α - ω phase transformation in Zr under high pressure

Plastic strain-induced phase transformations (PTs) and chemical reactions under high pressure are broadly spread in modern technologies, friction and wear, geophysics, and astrogeology. However, because of very heterogeneous fields of plastic strain $E$ p and stress σ tensors and volume fraction c of phases in a sample compressed in a diamond anvil cell (DAC) and impossibility of measurements of σ and $E$ p , there are no strict kinetic equations for them. Here, we develop a kinetic model, finite element method (FEM) approach, and combined FEM-experimental approaches to determine all fields in strongly plastically predeformed Zr compressed in DAC, and specific kinetic equation for α-ω PT consistent with experimental data for the entire sample. Since all fields in the sample are very heterogeneous, data are obtained for numerous complex 7D paths in the space of 3 components of the plastic strain tensor and 4 components of the stress tensor. Kinetic equation depends on accumulated plastic strain (instead of time) and pressure and is independent of plastic strain and deviatoric stress tensors, i.e., it can be applied for various above processes. Our results initiate kinetic studies of strain-induced PTs and provide efforts toward more comprehensive understanding of material behavior in extreme conditions.

36 MATERIALS SCIENCE

Multi-frequency progressive refinement for learned inverse scattering

Interpreting scattered acoustic and electromagnetic wave patterns is a computational task that enables remote imaging in a number of important applications, including medical imaging, geophysical exploration, sonar and radar detection, and nondestructive testing of materials. However, accurately and stably recovering an inhomogeneous medium from far-field scattered wave measurements is a computationally difficult problem, due to the nonlinear and non-local nature of the forward scattering process. We design a neural network, called Multi-Frequency Inverse Scattering Network (MFISNet), and a training method to approximate the inverse map from far-field scattered wave measurements at multiple frequencies. We consider three variants of MFISNet, with the strongest performing variant inspired by the recursive linearization method — a commonly used technique for stably inverting scattered wavefield data — that progressively refines the estimate with higher frequency content. MFISNet outperforms past methods in regimes with high-contrast, heterogeneous large objects, and inhomogeneous unknown backgrounds.

97 MATHEMATICS AND COMPUTING

Sensitivity of Fine‐Resolution Urban Heat Island Simulations to Soil Moisture Parameterization

ABSTRACT Urban areas experience the impact of natural disasters, such as heatwaves and flash floods, disparately in different neighbourhoods across a city. The demand for precise urban hydrometeorological and hydroclimatological modelling to examine this disparity, and the interacting challenges posed by climate change and urbanisation, has thus surged. The Weather Research and Forecasting (WRF) model has served such operational and research purposes for decades. Recent advancements in WRF, including enhanced numerical schemes and sophisticated urban atmospheric‐hydrological parameterizations, have empowered the simulation of urban geophysical processes at high resolution (~1 km), but even this resolution misses significant urban microclimate variability. This study applies the large‐eddy simulations (LES) mode within WRF, coupled with single‐layer urban canopy models (SLUCM), to enable even finer‐scale modelling (150 m) of the Urban Heat Island (UHI) effect in the Baltimore metropolitan area. We run nine scenarios to evaluate various methods of initializing soil moisture and various spinup lead times, and to assess the impact of WRF's Mosaic approach in depicting subgrid‐scale processes. We evaluate the scenarios by comparing the WRF simulated land surface temperature (LST) against Landsat LST and the WRF simulated hourly 2‐m air temperatures (AT) with observations from eight weather stations across the domain. Results underscore the paramount influence of the lead spinup time on the spatiotemporal distribution of simulated soil moisture, consequently shaping WRF's efficacy in predicting the UHI. Furthermore, interpolating soil moisture‐related parameters from the parent for child domain initialization yields a notable reduction in mean and root‐mean‐squared errors. This improvement was particularly evident in simulations with the longest spinup time, affirming the importance of carefully designing the initialization of soil moisture for improved urban temperature predictions.

Talebpour, Mahdad

Nonlinear Ensemble Filtering with Diffusion Models: Application to the Surface Quasigeostrophic Dynamics

The intersection between classical data assimilation methods and novel machine learning techniques has attracted significant interest in recent years. Here, we explore another promising solution in which diffusion models are used to formulate a robust nonlinear ensemble filter for sequential data assimilation. Unlike standard machine learning methods, the proposed ensemble score filter (EnSF) is completely training free and can efficiently generate a set of analysis ensemble members. Here, in this study, we apply the EnSF to a surface quasigeostrophic model and compare its performance against the popular local ensemble transform Kalman filter (LETKF), which makes Gaussian assumptions in the analysis step. Numerical tests demonstrate that EnSF maintains stable performance in the absence of localization and for a variety of experimental settings. We find that while LETKF maintains optimal performance in the case of linear observations of the entire state and a perfect model, EnSF shows improvements over LETKF when nonlinear observations are assimilated and the system is subject to unexpected model errors. A spectral decomposition of the analysis results in this nonlinear observation regime shows that the largest improvements over LETKF occur at large scales (small wavenumbers), where LETKF lacks sufficient ensemble spread. Overall, this initial application of EnSF to a geophysical model of intermediate complexity motivates further development of the algorithm for more realistic problems.

Artificial intelligence

Complementary Subsurface Characterization Methods to Develop a Geologic Model for the EGS Collab Experiment, Sanford Underground Research Facility

The EGS (Enhanced Geothermal Systems) Collab project was performed within the Sanford Underground Research Facility (SURF) with a goal of understanding processes and evaluation of models related to hydraulic stimulation of rock at depth. The present work deals with the development of Testbed 2 where experiments were conducted at a depth of 1.25 km and were located within a well-characterized testbed in a metamorphic, amphibolite host rock. A total of eleven boreholes varying in length between 10.6 m and 81.2 m were continuously cored to develop the testbed. In addition to the continuous coring of the amphibolite host rock, geophysical instrumentation supporting electrical resistivity tomography (ERT), microearthquake (MEQ) detection, and optical fiber providing distributed temperature sensing (DTS) and distributed acoustic sensing (DAS) were installed in the monitoring boreholes, all of which produced a comprehensive complementary suite of characterization and monitoring technologies. Pre-stimulation characterization of the groundwater conditions identified only two hydraulically significant fractures, neither of which transects the central portion of the testbed. Flow and pressure monitoring indicated low preexisting pore pressure conditions likely affected by the mine openings.

15 GEOTHERMAL ENERGY

Complementary Subsurface Characterization Methods to Develop a Geologic Model for the EGS Collab Experiment, Sanford Underground Research Facility

The EGS (Enhanced Geothermal Systems) Collab project was performed within the Sanford Underground Research Facility (SURF) with a goal of understanding processes and evaluation of models related to hydraulic simulation of rock at depth. The present work deals with the development of Testbed 2 where experiments were conducted at a depth of 1.25 km and were located within a well-characterized testbed in a metamorphic, amphibolite host rock. A total of eleven boreholes varying in length between 10.6 m and 81.2 m were continuously cored to develop the testbed. In addition to the continuous coring of the amphibolite host rock, geophysical instrumentation supporing electrical resisivity tomography (ERT), microearthquake (MEQ) detection, and optical fiber providing distributed temperature sensing (DTS) and distributed acoustic sensing (DAS) were installed in the monitoring boreholes, all of which produced a comprehensive complementary suite of characterization and monitoring technologies.

15 GEOTHERMAL ENERGY

Unsupervised Clustering of Microseismic Events and Focal Mechanism Analysis at the CO 2 Injection Site in Decatur, Illinois

Characterization of induced microseismicity at a carbon dioxide (CO 2 ) storage site is critical for preserving reservoir integrity and mitigating seismic hazards. We apply a multilevel machine learning (ML) approach that combines the nonnegative matrix factorization and hidden Markov model to extract spectral representations of microseismic events and cluster them to identify seismic patterns at the Illinois Basin-Decatur Project. Unlike traditional waveform correlation methods, this approach leverages spectral characteristics of first arrivals to improve event classification and detect previously undetected planes of weakness. By integrating ML-based clustering with focal mechanism analysis, we resolve small-scale fault structures that are below the detection limits of conventional seismic imaging. Our findings reveal temporal bursts of microseismicity associated with brittle failure, providing insights into the spatio-temporal evolution of fault reactivation during CO 2 injection. This approach enhances seismic monitoring capabilities at CO 2 injection sites by improving fault characterization beyond the resolution of standard geophysical surveys.

Willis, Rachel Marie [Sandia National Laboratories

FY 2026 Midyear Report: Seismic Monitoring of Underground Vibration Sources Using Distributed Acoustic Sensing and Seismometers

Safeguards-relevant temporal changes in underground facilities can be observed using geophysical monitoring techniques. Seismic waves, in particular, provide valuable insights into subsurface activities and can serve as an important tool for detecting anomalous events that may indicate containment breaches at geological repositories. This midyear report summarizes ongoing efforts to automatically and rapidly detect and locate anomalous vibration signals that could be indicative of potential containment breaches. Previous work during FY25 focused on compiling continuous seismic datasets from two underground sites and developing a database of continuous waveforms and ground-truth event data derived from multiple sensing modalities. Building on this foundation, we are adapting anomaly detection and geolocation algorithms to explore methods for monitoring underground activities using two relatively low-maintenance sensing technologies: a dense surface geophone array deployed at the Pleasant Gap mine in Pennsylvania, and a three-dimensional fiber-optic cable array for distributed acoustic sensing (DAS) installed in the subsurface at the Sanford Underground Research Facility (SURF) in South Dakota. This report summarizes work conducted during the first two quarters of FY26, during which we refined a dynamic power spectral density (PSD)-based detector, applied it independently to each geophone station, and then combined the per‑station detections with density-based spatial clustering of applications with noise (DBSCAN) to cluster events and produce spatial maps over a nine‑day interval. In addition, we outline plans for a field trial at the Waste Isolation Pilot Plant (WIPP) in New Mexico to compare traditional seismic monitoring approaches with DAS techniques and to evaluate the benefits of combined data analysis. Activities during the past two quarters have included the preparation and submission of a Field Test Plan to WIPP for approval, as well as submission to headquarters for review and feedback.

58 GEOSCIENCES