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DOE OSTI2024

Determining reference standard strength for neutron-irradiated reduced activation ferritic/martensitic steel F82H by Bayesian method

The deterministic approach widely adopted in the design of structural components relies on systematically defined design limits using empirically determined safety factors. However, this approach is not always appropriate because structures are subjected to a variety of loads in the practical environment, which may result in excessively conservative design limits. In recent years, a more rigorous probabilistic approach that incorporates material strength distributions has become an important solution. In the probabilistic approach, the probability density functions of material strength properties underpin the design criteria. Here, the objective of this study is to identify the density distribution functions that best describe tensile properties of irradiated F82H to define a reference strength for DEMO design. Due to the limited number of existing data, this study specifically employs a Bayesian prediction method based on Monte Carlo simulations to determine a material reference value with statistical reliability and to investigate its effectiveness. For example, the dependence of tensile properties of 300 °C irradiated materials on irradiation damage and the range predicted by 95% Bayesian estimation was evaluated. As a statistical model for the dose dependence of statistical parameters, the normal distribution exhibited a better fit for 0.2% proof strength and tensile strength, whereas the distribution of total elongation data gave comparable reference values for both the normal and Weibull distribution models. Both models gave comparable criteria for the distribution of total elongation data. The Weibull model also gave better results for uniform elongation. The function best describing the model was a logarithmic law for both 0.2% proof strength and tensile strength, while a power law for both total and uniform elongation, which allowed for more comprehensive data prediction of irradiation data with statistical accuracy for DEMO reactor design.

36 MATERIALS SCIENCE↗

DOE OSTI2022

A machine learning inversion scheme for determining interaction from scattering

Small angle scattering techniques have now been routinely used to quantitatively determine the potential of mean force in colloidal suspensions. However the numerical accuracy of data interpretation is often compounded by the approximations adopted by liquid state analytical theories. To circumvent this long standing issue, here we outline a machine learning strategy for determining the effective interaction in the condensed phases of matter using scattering. Via a case study of colloidal suspensions, we show that the effective potential can be probabilistically inferred from the scattering spectra without any restriction imposed by model assumptions. Comparisons to existing parametric approaches demonstrate the superior performance of this method in accuracy, efficiency, and applicability. This method can effectively enable quantification of interaction in highly correlated systems using scattering and diffraction experiments.

74 ATOMIC AND MOLECULAR PHYSICS↗

DOE OSTI2021

Validation Metrics for Fixed Effects and Mixed-Effects Calibration

The modern scientific process often involves the development of a predictive computational model. To improve its accuracy, a computational model can be calibrated to a set of experimental data. A variety of validation metrics can be used to quantify this process. Some of these metrics have direct physical interpretations and a history of use, while others, especially those for probabilistic data, are more difficult to interpret. In this work, a variety of validation metrics are used to quantify the accuracy of different calibration methods. Frequentist and Bayesian perspectives are used with both fixed effects and mixed-effects statistical models. Through a quantitative comparison of the resulting distributions, the most accurate calibration method can be selected. Two examples are included which compare the results of various validation metrics for different calibration methods. It is quantitatively shown that, in the presence of significant laboratory biases, a fixed effects calibration is significantly less accurate than a mixed-effects calibration. This is because the mixed-effects statistical model better characterizes the underlying parameter distributions than the fixed effects model. The results suggest that validation metrics can be used to select the most accurate calibration model for a particular empirical model with corresponding experimental data.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS2006

Hard and Soft Constraints in Reliability-Based Design Optimization

This paper proposes a framework for the analysis and design optimization of models subject to parametric uncertainty where design requirements in the form of inequality constraints are present. Emphasis is given to uncertainty models prescribed by norm bounded perturbations from a nominal parameter value and by sets of componentwise bounded uncertain variables. These models, which often arise in engineering problems, allow for a sharp mathematical manipulation. Constraints can be implemented in the hard sense, i.e., constraints must be satisfied for all parameter realizations in the uncertainty model, and in the soft sense, i.e., constraints can be violated by some realizations of the uncertain parameter. In regard to hard constraints, this methodology allows (i) to determine if a hard constraint can be satisfied for a given uncertainty model and constraint structure, (ii) to generate conclusive, formally verifiable reliability assessments that allow for unprejudiced comparisons of competing design alternatives and (iii) to identify the critical combination of uncertain parameters leading to constraint violations. In regard to soft constraints, the methodology allows the designer (i) to use probabilistic uncertainty models, (ii) to calculate upper bounds to the probability of constraint violation, and (iii) to efficiently estimate failure probabilities via a hybrid method. This method integrates the upper bounds, for which closed form expressions are derived, along with conditional sampling. In addition, an l(sub infinity) formulation for the efficient manipulation of hyper-rectangular sets is also proposed.

Crespo, L.uis G.↗

DOE OSTI2020

Analysis of Hydrology and Interim Measure Performance in the Chromium Plume at Los Alamos National Laboratory - 20498

Hexavalent chromium, Cr(VI), is present in the regional aquifer beneath Los Alamos National Laboratory (LANL) at concentrations greater than the New Mexico groundwater standard (50 ppb). The source of the Cr is blowdown from a power plant where potassium dichromate was used as a corrosion inhibitor from 1956 to 1972. Blowdown comingled with effluent from another source and migrated approximately 3 miles down a canyon, primarily as surface flow, until reaching an alluvial infiltration zone. The infiltrating water moved vertically through a thick (approximately 900 ft), complex stratigraphic sequence of unsaturated tuffs, basalt, and unconsolidated alluvial deposits. Horizontal transport also took place within spatially limited perched groundwater horizons in the vadose zone (VZ). Breakthrough at the regional aquifer water table is thought to have occurred in multiple hydraulic windows or 'drip points' located approximately 3 miles from the initial release site at the power plant. The regional aquifer is highly heterogeneous and anisotropic, and displays unconfined behavior in an upper zone and confined behavior at depth. The Cr(VI) contamination has so far remained confined to the upper approximately 25 m of the aquifer. The ambient hydraulic gradient in the regional aquifer is toward the east/southeast, but the water table is extremely flat in the Cr plume area compared to surrounding areas. There is some local variability in flow direction, potentially related to effects of seasonal pumping from nearby Los Alamos county water-supply wells, which draw from the regional aquifer several hundred feet below the water table. An interim measure (IM) is currently operating at the site to achieve hydraulic control of plume migration using a series of extraction wells, ion-exchange treatment, and injection along the downgradient portion of the plume. The migration of Cr contamination in the aquifer has been previously evaluated using a calibrated probabilistic numerical model, which has also been used to design the spatial configuration of extraction and injection wells for the IM, and to identify locations for monitoring wells. The model is built using the Finite Element Heat and Mass (FEHM) transfer code, and the Model Analysis and Decision Support (MADS) software is used for analyses. To date, the portion of the IM system that has operated in a continuous manner has been very effective at hydraulic control. Modifications to the flow field associated with the influence of IM pumping and injection are evaluated here, with modeling constraints to be provided by recent observations of tracers injected into both monitoring wells and IM injection wells, as well as opportunistic observations of geochemical signatures of treated ion-exchange effluent injected into IM injection wells. The tracer and geochemical data provide insights into modified flow directions within the Cr plume area that cannot be otherwise obtained. Results of the modeling work are being used to inform adaptive management of IM operations and will be a valuable tool for development and refinement of long-term remediation strategies. (authors)

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗

DOE OSTI2021

Seawater sea-sand engineered/strain-hardening cementitious composites (ECC/SHCC): Assessment and modeling of crack characteristics

Highlights: • A probabilistic model was proposed to model the crack width evolution of SS-ECC at different strain levels. • The Weibull distribution fit the crack width distribution better than the log-normal distribution. • A 5-D representation was proposed to assess the SS-ECC by considering both cracking and mechanical performance. • Larger sea-sand size and lower fiber dosage led to larger crack widths in SS-ECC under the same tensile strain. • 18-mm PE fiber led to larger crack widths in SS-ECC at strain >2%, due to a large fraction of fiber rupture. Seawater sea-sand Engineered Cementitious Composites (SS-ECC) is a new version of ECC for marine constructions facing the scarcity of freshwater and river/manufactured sand. This study aims to assess and model the crack characteristics of SS-ECC, which are critical for its applications with non-corrosive reinforcements. The influence of sea-sand size, fiber length and fiber dosage on the crack characteristics of SS-ECC was explored. A five-dimensional representation was proposed to assess the overall performance of SS-ECC, by comprehensively considering both the crack characteristics (i.e., crack width and its variation) and the mechanical properties (i.e., compressive and tensile properties). A probabilistic model was also proposed to describe the stochastic nature and evolution of crack width, and it can be used to estimate the critical tensile strain on SS-ECC for a given crack-width limit and cumulative probability. The findings and proposed methods can facilitate the design of SS-ECC in marine and coastal structures.

36 MATERIALS SCIENCE↗

DOE OSTI2022

Earthquake Phase Association Using a Bayesian Gaussian Mixture Model

Earthquake phase association algorithms aggregate picked seismic phases from a network of seismometers into individual seismic events and play an important role in earthquake monitoring and research. Dense seismic networks and improved phase picking methods produce massive seismic phase datasets, particularly for earthquake swarms and aftershocks occurring closely in time and space, making phase association a challenging problem. Here, we present a new association method, the Gaussian Mixture Model Association (GaMMA), that combines the Gaussian mixture model with earthquake location, origin time, and magnitude estimation. We treat earthquake phase association as an unsupervised clustering problem in a probabilistic framework, where each earthquake corresponds to a cluster of P and S phases with a hyperbolic moveout of arrival times and a decay of amplitude with distance. We use the multivariate Gaussian distribution to model the collection of phase picks of an event; and the mean of the multivariate Gaussian distribution is given by the predicted arrival time and amplitude from the causative event. We carry out the pick assignment to each earthquake and determine earthquake source parameters (i.e., earthquake location, origin time, and magnitude) under the maximum likelihood criterion using the Expectation-Maximization algorithm. The GaMMA method does not require typical association steps of other algorithms, such as grid-search or supervised training. The results for both synthetic tests and for the 2019 Ridgecrest earthquake sequence show that GaMMA effectively associates phases from a temporally and spatially dense earthquake sequence while producing useful estimates of earthquake location and magnitude.

58 GEOSCIENCES↗

NASA NTRS1990

Probabilistic fatigue methodology for six nines reliability

Fleet readiness and flight safety strongly depend on the degree of reliability that can be designed into rotorcraft flight critical components. The current U.S. Army fatigue life specification for new rotorcraft is the so-called six nines reliability, or a probability of failure of one in a million. The progress of a round robin which was established by the American Helicopter Society (AHS) Subcommittee for Fatigue and Damage Tolerance is reviewed to investigate reliability-based fatigue methodology. The participants in this cooperative effort are in the U.S. Army Aviation Systems Command (AVSCOM) and the rotorcraft industry. One phase of the joint activity examined fatigue reliability under uniquely defined conditions for which only one answer was correct. The other phases were set up to learn how the different industry methods in defining fatigue strength affected the mean fatigue life and reliability calculations. Hence, constant amplitude and spectrum fatigue test data were provided so that each participant could perform their standard fatigue life analysis. As a result of this round robin, the probabilistic logic which includes both fatigue strength and spectrum loading variability in developing a consistant reliability analysis was established. In this first study, the reliability analysis was limited to the linear cumulative damage approach. However, it is expected that superior fatigue life prediction methods will ultimately be developed through this open AHS forum. To that end, these preliminary results were useful in identifying some topics for additional study.

Everett, R. A., Jr.↗

DOE OSTI2022

Realistic galaxy image simulation via score-based generative models

ABSTRACT We show that a denoising diffusion probabilistic model (DDPM), a class of score-based generative model, can be used to produce realistic mock images that mimic observations of galaxies. Our method is tested with Dark Energy Spectroscopic Instrument (DESI) grz imaging of galaxies from the Photometry and Rotation curve OBservations from Extragalactic Surveys (PROBES) sample and galaxies selected from the Sloan Digital Sky Survey. Subjectively, the generated galaxies are highly realistic when compared with samples from the real data set. We quantify the similarity by borrowing from the deep generative learning literature, using the ‘Fréchet inception distance’ to test for subjective and morphological similarity. We also introduce the ‘synthetic galaxy distance’ metric to compare the emergent physical properties (such as total magnitude, colour, and half-light radius) of a ground truth parent and synthesized child data set. We argue that the DDPM approach produces sharper and more realistic images than other generative methods such as adversarial networks (with the downside of more costly inference), and could be used to produce large samples of synthetic observations tailored to a specific imaging survey. We demonstrate two potential uses of the DDPM: (1) accurate inpainting of occluded data, such as satellite trails, and (2) domain transfer, where new input images can be processed to mimic the properties of the DDPM training set. Here we ‘DESI-fy’ cartoon images as a proof of concept for domain transfer. Finally, we suggest potential applications for score-based approaches that could motivate further research on this topic within the astronomical community.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2020

Enhanced Component Performance Study: Emergency Diesel Generators 1998–2018

This report presents an enhanced performance evaluation of emergency diesel generators (EDGs) at U.S. commercial nuclear power plants. This report evaluates component performance over time using (1) Institute of Nuclear Power Operations (INPO) Consolidated Events Database (ICES) data from 1998 through 2018 and (2) maintenance unavailability (UA) performance data from Mitigating Systems Performance Index (MSPI) Basis Document data from 2002 through 2018. The objective is to show estimates of current failure probabilities and rates related to EDGs, trend these data on an annual basis, determine if the current data are consistent with the probability distributions currently recommended for use in NRC probabilistic risk assessments, show how the reliability data differ for different EDG manufacturers and for EDGs with different ratings; and summarize the subcomponents, causes, detection methods, and recovery associated with each EDG failure mode. Engineering analyses were performed with respect to time period and failure mode without regard to the actual number of EDGs at each plant. The factors analyzed are: sub-component, failure cause, detection method, recovery, manufacturer, and EDG rating. A statistically significant increasing trend was identified in the frequency of FTLR demands for emergency power system (EPS) and high pressure core spray (HPCS) EDGs and a statistically significant decreasing trend was identified in the frequency of run > 1H hours for EPS and HPCS EDGs.

99 GENERAL AND MISCELLANEOUS↗

DOE OSTI2020

Neuromorphic scaling advantages for energy-efficient random walk computations

Computing stands to be radically improved by neuromorphic computing (NMC) approaches inspired by the brain's incredible efficiency and capabilities. Most NMC research, which aims to replicate the brain's computational structure and architecture in man-made hardware, has focused on artificial intelligence; however, less explored is whether this brain-inspired hardware can provide value beyond cognitive tasks. We demonstrate that high-degree parallelism and configurability of spiking neuromorphic architectures makes them well-suited to implement random walks via discrete time Markov chains. Such random walks are useful in Monte Carlo methods, which represent a fundamental computational tool for solving a wide range of numerical computing tasks. Additionally, we show how the mathematical basis for a probabilistic solution involving a class of stochastic differential equations can leverage those simulations to provide solutions for a range of broadly applicable computational tasks. Despite being in an early development stage, we find that NMC platforms, at a sufficient scale, can drastically reduce the energy demands of high-performance computing platforms.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2022

Enhanced Component Performance Study: Emergency Diesel Generators 1998-2020

This report presents an enhanced performance evaluation of emergency power system (EPS) and high-pressure core spray (HPCS) emergency diesel generators (EDGs) at U.S. commercial nuclear power plants. This report evaluates component performance over time using (1) Institute of Nuclear Power Operations (INPO) Industry Reporting and Information System (IRIS) data from 1998 through 2020 and (2) maintenance unavailability performance data from Mitigating Systems Performance Index (MSPI) Basis Document data from 2002 through 2020. The objective is to show estimates of current failure probabilities and rates related to EDGs, trend these data on an annual basis, determine if the current data are consistent with the probability distributions currently recommended for use in NRC probabilistic risk assessments, show how the reliability data differ for different EDG manufacturers and for EDGs with different ratings; and summarize the subcomponents, causes, detection methods, and recovery associated with each EDG failure mode. The EDG failure modes considered are fail to start (FTS), fail to load and run (FTLR), and fail to run after one hour of operation (FTR>1H). Engineering analyses were performed with respect to time period and failure mode without regard to the actual number of EDGs at each plant. The factors analyzed are: subcomponent, failure cause, detection method, recovery, manufacturer, and EDG rating. The following increasing trends were identified for EDGs for the most recent 10-year period: • HPCS EDG FTS failure probability • EPS and HPCS EDG frequency of start demands (demands per reactor year) • EPS and HPCS EDG frequency of FTLR demands. The following decreasing trends were identified for EDGs for the most recent 10-year period: • EPS EDG FTS failure probability • EPS EDG FTLR failure probability • EPS EDG unavailability • EPS and HPCS EDG frequency of FTLR events (failures per reactor year).

99 GENERAL AND MISCELLANEOUS↗

DOE OSTI2023

Enhanced Component Performance Study: Emergency Diesel Generators 1998–2022

This report presents an enhanced performance evaluation of the emergency power system (EPS) and high-pressure core spray (HPCS) emergency diesel generators (EDGs) at U.S. commercial nuclear power plants. This report evaluates component performance over time using (1) Institute of Nuclear Power Operations (INPO) Industry Reporting and Information System (IRIS) data from 1998 through 2022 and (2) maintenance unavailability performance data from Mitigating Systems Performance Index (MSPI) Basis Document data from 2002 through 2022. The objective is to show estimates of current failure probabilities and rates related to EDGs, trend these data on an annual basis, determine if the current data are consistent with the probability distributions currently recommended for use in Nuclear Regulatory Commission (NRC) probabilistic risk assessments, show how the reliability data differ for different EDG manufacturers and for EDGs with different ratings; and summarize the subcomponents, causes, detection methods, and recovery associated with each EDG failure mode. The EDG failure modes considered are fail to start (FTS), fail to load and run (FTLR), and fail to run after one hour of operation (FTR>1H). Engineering analyses were performed with respect to time-period and failure mode without regard to the actual number of EDGs at each plant. The factors analyzed include subcomponent, failure cause, detection method, recovery, manufacturer, and EDG rating. The following increasing trends were identified for EDGs for the most recent 10-year period: (1) HPCS EDG FTS failure probability, (2) EPS and HPCS EDG frequency of start demands (demands per reactor year), and (3) EPS and HPCS EDG frequency of FTLR demands. The following decreasing trends were identified for EDGs for the most recent 10-year period: (1) EPS EDG FTS failure probability, (2) EPS EDG FTLR failure probability, (3) EPS EDG unavailability, and (4) EPS and HPCS EDG frequency of FTLR events (failures per reactor year).

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

DOE OSTI2025

Enhanced Component Performance Study: Emergency Diesel Generators 1998-2024

This report presents an enhanced performance evaluation of the emergency power system (EPS) and high-pressure core spray (HPCS) emergency diesel generators (EDGs) at U.S. commercial nuclear power plants. This report evaluates component performance over time using (1) Institute of Nuclear Power Operations (INPO) Industry Reporting and Information System (IRIS) data from 1998 through 2024 and (2) maintenance unavailability performance data from Mitigating Systems Performance Index (MSPI) Basis Document data from 2002 through 2024. The objective is to show estimates of current failure probabilities and rates related to EDGs, trend these data on an annual basis, determine if the current data are consistent with the probability distributions currently recommended for use in Nuclear Regulatory Commission (NRC) probabilistic risk assessments, show how the reliability data differ for different EDG manufacturers and for EDGs with different ratings; and summarize the subcomponents, causes, detection methods, and recovery associated with each EDG failure mode. The EDG failure modes considered are fail to start (FTS), fail to load and run (FTLR), and fail to run after one hour of operation (FTR>1H). Engineering analyses were performed with respect to time-period and failure mode without regard to the actual number of EDGs at each plant. The factors analyzed include subcomponent, failure cause, detection method, recovery, manufacturer, and EDG rating. The following increasing trends were identified for EDGs for the most recent 10-year period: • EPS and HPCS EDG frequency of start demands (demands per reactor year) • EPS and HPCS EDG frequency of FTLR demands • EPS and HPCS EDG frequency of run>1H hours. The following decreasing trends were identified for EDGs for the most recent 10-year period: • EPS EDG FTR>1H failure rate • EPS EDG unreliability • EPS and HPCS EDG frequency of FTR>1H events (failures per reactor year).

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

NASA NTRS2025

Assessment of Model Outcomes Between the Integrated Medical Model (IMM) and the Medical Extensible Dynamic Probabilistic Risk Assessment Tool (MEDPRAT)

The Medical Extensible Dynamic Probabilistic Risk Assessment Tool (MEDPRAT) is a computational model that provides human health and medical risk predictions for crewed spaceflight missions. MEDPRAT utilizes discrete event modeling and dynamic probabilistic simulation to predict critical mission outcomes (total medical events, crew health index, quality time lost, loss of crew life, removal to definitive care), condition occurrences, and resource consumption. Input parameters for MEDPRAT include crew attributes (e.g., sex), types of mission activities (e.g., whether and where crew members perform an extravehicular activity (EVA)), available resources, treatment information, and probability distributions for medical conditions. As an evolution of the Integrated Medical Model (IMM), MEDPRAT provides enhanced capabilities and higher fidelity, and incorporates more appropriate assumptions for long-duration spaceflight. IMM is the currently accepted standard for quantifying spaceflight mission medical risk in NASA operations that uses a probabilistic risk assessment (PRA) approach. MEDPRAT builds on the same logical foundation as IMM but implements the model architecture through highly optimized Monte Carlo sampling methods. An analysis is performed comparing the outputs from IMM with those from MEDPRAT V1.0 and V2.0 for the same reference missions in order to quantify similarities and differences in the model outcomes. The juxtaposition between IMM and MEDPRAT V1.0 and 2.0 shown in this report demonstrates that these two models generate very similar results; where differences in outcomes are shown, these are in accordance with the underlying assumptions and differences in the model architectures. This validation effort further establishes the credibility and reliability of the MEDPRAT software.

Matthew T Prelich↗

DOE OSTI2024

Short-Term Load Forecasting Considering EV Charging Loads with Prediction Interval Evaluation

Short-term load forecasting plays a critical role in power system planning and operation. Along with the electrification of various loads, electricity demands are becoming increasingly hard to predict. Notably, the recent rise in electric vehicles (EVs) has further contributed to this unpredictability. To address this issue, this paper proposes a probabilistic load forecasting strategy utilizing Gaussian process regression, structured in a day-ahead manner. While many works focus on deterministic prediction, probabilistic forecasting offers additional insights into variability and uncertainty, enabling more flexible and reliable operation for power systems. To enhance the accuracy of the load forecasting model, the inputs include features related to EV charging habits as well as commonly used weather information. The load forecasting results are evaluated using various metrics, including conventional ones that assess the accuracy of point forecasts, as well as additional metrics that test the reliability of prediction intervals. The proposed load forecasting method is finally tested on real residential power consumption data and EV charging data sampled from real-world sources. The results prove that the new features can greatly improve the performance of the load forecasting method.

electrical vehicle↗

NASA NTRS2003

A Probabilistic Approach to Technology Identification, Evaluation, and Selection for Affordable Transport Concepts

Most grant goals were accomplished over the course of the grant, albeit only partial success was achieved in applying ASDL s existing advanced design methods to HAVOC. The limited success in applying ASDL's advanced design methods occurred due to: fundamental 20 of 197 limitations in the RSM metamodeling process used; the highly computationally intensive nature of HAVOC as compared to many other advanced system synthesis and sizing codes; and the highly curved nature of low I(sub sp) rocket system spaces. These challenges encountered during the grant led to ongoing efforts in more efficient and robust sequential metamodeling/surrogate optimization based advanced design methods which are expected to overcome all of the challenges encountered with RSM during the grant. Other achievements over the course of the grant included the documentation of the existing HAVOC cost module and formulation of new cost estimation relationships in support of APS analysis of 3rd Gen RLV. Further achievements beyond the scope of the grant as originally defined were also achieved such as HAVOC code and portability improvements, historical propulsion performance and reliability regression relationships, and new visualization routines to vacilitate visualization of HAVOC DOE case file results.

Source record↗

DOE OSTI2026

Discovery of Probabilistic Dirichlet-to-Neumann Maps on Graphs

Dirichlet-to-Neumann maps enable the coupling of multiphysics simulations across computational subdomains by ensuring continuity of state variables and fluxes at artificial interfaces. We present a novel method for learning Dirichlet-to-Neumann maps on graphs using Gaussian processes, specifically for problems where the data obey a conservation law arising from an underlying partial differential equation. Our approach combines discrete exterior calculus and nonlinear optimal recovery to infer relationships between vertex and edge values. This framework yields data-driven predictions with uncertainty quantification across the entire graph, even when observations are limited to a subset of vertices and edges. By minimizing the reproducing kernel Hilbert space norm while penalizing kernel complexity through maximum likelihood estimation, our method ensures that the resulting surrogate strictly enforces conservation laws without overfitting. We demonstrate our method on two representative applications: subsurface flow in fracture networks and arterial blood flow. Finally, the results demonstrate that the method maintains high accuracy and well-calibrated uncertainty estimates even under severe data scarcity, highlighting its potential for scientific applications where limited data and reliable uncertainty quantification are critical.

Dirichlet-to-Neumann map↗