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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.

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At least 199 records · Page 11

pnnl/HADREC (33077-E)

This HADREC GUI is designed to visualize power system emergency control time-series data and facilitate the comparison of the newly developed meta strategy optimization algorithm-based emergency control with existing out-of-step-based emergency control approach, and a no-action case for reference.

Wang, Heng [Pacific Northwest National Laboratory ↗

Variability in Wind Energy Generation across the Contiguous United States

ERA5 provides high-resolution, high-quality hourly wind speeds at 100 m and is a unique resource for quantifying temporal variability in likely wind-derived power production across the United States. Gross capacity factors (CF) in seven independent system operators (ISOs) are estimated using the location and rated power of each wind turbine, a simplified power curve, and ERA5 output from 1979 to 2018. Excluding the California ISO, the marginal probability of a calm (zero power production) is less than 0.1 in any ERA5 grid cell. When a calm occurs, the mean co-occurrence across wind-turbine-containing grid cells ranges from 0.38 to 0.39 for ISOs in the Midwest and central plains [Midcontinent (or Midwest) ISO (MISO), Southwest Power Pool (SPP), and the Electric Reliability Council of Texas (ERCOT) region], increasing to 0.54–0.58 for ISOs in the eastern United States [Pennsylvania–New Jersey–Maryland interconnection (PJM), New York ISO (NYISO), and New England ISO (NEISO)]. Periods with low gross CF have a median duration of ≤6 h, except in California, and are most likely during summer. Additionally, gross CF exhibit highest variance at periods of 1 day in ERCOT and SPP; on synoptic scales in MISO, NEISO, and NYISO; and on interannual time scales in PJM. This implies differences in optimal strategies for ensuring resilience of supply. Theoretical scenarios show adding wind energy capacity near existing wind farms is advantageous even in areas with high existing installed capacity (IC), while expanding into areas with lower IC is more beneficial to reducing ramps and the probability of gross CF falling below 20%. These results emphasize the benefits of large balancing areas and aggregation in reducing wind power variability and the likelihood of wind droughts.

17 WIND ENERGY↗

Techno-economic assessment for the production of algal fuels and value-added products: opportunities for high-protein microalgae conversion

Abstract Background Microalgae possess numerous advantages for use as a feedstock in producing renewable fuels and products, with techno-economic analysis (TEA) frequently used to highlight the economic potential and technical challenges of utilizing this biomass in a biorefinery context. However, many historical TEA studies have focused on the conversion of biomass with elevated levels of carbohydrates and lipids and lower levels of protein, incurring substantial burdens on the ability to achieve high cultivation productivity rates relative to nutrient-replete, high-protein biomass. Given a strong dependence of algal biomass production costs on cultivation productivity, further TEA assessment is needed to understand the economic potential for utilizing potentially lower-cost but lower-quality, high-protein microalgae for biorefinery conversion. Results In this work, we conduct rigorous TEA modeling to assess the economic viability of two conceptual technology pathways for processing proteinaceous algae into a suite of fuels and products. One approach, termed mild oxidative treatment and upgrading (MOTU), makes use of a series of thermo-catalytic operations to upgrade solubilized proteins and carbohydrates to hydrocarbon fuels, while another alternative focuses on the biological conversion of those substrates to oxygenated fuels in the form of mixed alcohols (MA). Both pathways rely on the production of polyurethanes from unsaturated fatty acids and valorization of unconverted solids for use as a material for synthesizing bioplastics. The assessment found similar, albeit slightly higher fuel yields and lower costs for the MA pathway, translating to a residual solids selling price of $899/ton for MA versus $1033/ton for MOTU as would be required to support a $2.50/gallon gasoline equivalent (GGE) fuel selling price. A variation of the MA pathway including subsequent upgrading of the mixed alcohols to hydrocarbon fuels (MAU) reflected a required solids selling price of $975/ton. Conclusion The slight advantages observed for the MA pathway are partially attributed to a boundary that stops at oxygenated fuels versus fungible drop-in hydrocarbon fuels through a more complex MOTU configuration, with more comparable results obtained for the MAU scenario. In either case, it was shown that an integrated algal biorefinery can be economical through optimal strategies to utilize and valorize all fractions of the biomass.

09 BIOMASS FUELS↗

Measurement of chromatic aberrations using phase retrieval

We explore the feasibility of measuring chromatic aberrations using a technique based on phase retrieval, primarily for the purpose of diagnostics on the OMEGA EP laser. A computational model and optimization strategy is described, issues of numerical efficiency are addressed, and the potential limitations of the method are studied using mathematical analysis and Monte Carlo simulations. As a result, the algorithm performs well in simulation and yields encouraging results in a small-scale laboratory experiment.

97 MATHEMATICS AND COMPUTING↗

Building epsilon near zero materials from layered uniaxial metamaterials

Recently, there has been an explosion of activity in the fields of optics and photonics with the advent of fabrication techniques which enable the design of metamaterials which possess properties not encountered in the natural world. In this work, we are concerned with zero permittivity materials and a new scheme to design metamaterials for which all components of the dielectric tensor are approximately zero. Our approach involves the alternate layering of many, very thin, slices of two constituent metamaterials, a uniaxial layered medium and a uniaxial nanowire array. With a simple optimization strategy we demonstrate a candidate configuration which very nearly satisfies our design goal of zero permittivity.

Gray, Stephen K. [Argonne National Laboratory (ANL↗

Dipolar molecule emulator of lattice gauge theories (Final Report)

The quantum many-body problem is a great unsolved, cross-cutting challenge in physics that is of fundamental importance. Our understanding of phenomena related to dense quark matter, in particular, is challenged by this practical intractability of classical simulations. Because of the sheer cost and challenge of performing experiments that probe the length and energy scales relevant to such physics, there are practical motivations for finding theoretical methods to address the many-body problem. One promising approach is based on the use of a programmable and controllable analog quantum systems to emulate the physics of many-body problems of interest. While there are currently broad efforts to develop mid- to large-scale quantum computers, we are still likely many years away from such devices outperforming classical supercomputers for useful calculations. Even though digital quantum computers are still at a stage too premature for such tasks, approaches based on analog quantum simulators have advanced rapidly over the past two decades and can now treat many-body problems of interest on small- to medium-scale systems. Tackling the challenging many-body problems of relevance to high energy physics represents one of the great new opportunities and challenges for the field of analog quantum simulation. To address this challenge, this award brought together an interdisciplinary team of physicists with expertise in high energy theory (El-Khadra and Draper), computational and condensed matter physics (Clark), and experimental quantum and atomic, molecular, optical (AMO) science (DeMarco and Gadway), with the goal of developing realistic and optimized strategies for the analog quantum simulation of lattice gauge theories using cold atoms and molecules. This track 1 project included both theoretical and experimental goals, stated as follows. The theory goals were to "develop and verify a novel approach to quantum emulation of (1+1)d quantum link models, based on arrays of trapped quantum spins with long-range interactions." This final project report details the progress made under this award, as well as new research directions developed under this award.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Modeling of Reactor Design and Optimization for Scale-Up of the Catalyxx Process for Ethanol Conversion to Higher Alcohol Biofuels

This report summarizes the results of a collaborative efforts between Oak Ridge National Laboratory (ORNL) and Catalyxx Inc. to investigate scale-up of Catalyxx’s Ethanol upgrading to higher alcohols process. The study was funded by the U.S. Department of Energy (DOE) Bioenergy Technologies Office (BETO) under CRADA (Cooperative Research and Development Agreement) No: NFE-20-08396. The project is part of the Direct Funding Opportunity (DFO) for Computational Science to Enable Bioenergy program which utilized computational toolsets developed by the Consortium for Computational Physics and Chemistry, a multi-laboratory consortium in BETO. The report here summarizes a packed-bed reactor modeling effort spanning the range from lab to industrial scale (from 4 gram to 5-ton catalyst beds), and examining reactor design, process optimization strategies, and suggested design and operating conditions for Catalyxx’s ethanol upgrading plants. The results in this report have been shared in monthly steering meetings and presentations are available in the shared data house owned by Catalyxx. The modeling effort helped to define optimum operation conditions for maximum alcohol selectivity and yield: i.e., temperature control scenarios ranging from adiabatic to isothermal, feed rate, pressure, and inlet H 2 /Ethanol ratio. The modeling results were verified at lab-(4 gram) and pre-pilot (4 kg) scales and has been used to evaluate a 5-ton packed-bed reactor and identify operating conditions to maximize the butanol yield. Special focus was given to understanding mass-transfer effects in the pre-pilot and pilot-scale reactors, over the domain of flow rate, pressure, feed composition, pellet size, shape, porosity, bed voidage, and reactor dimensions (i.e., length/diameter). Modeling was also used to evaluate innovative reactor design concepts such as water removal to improve alcohol selectivity and yield, and a reactor with an additional side inlet to facilitate quenching. These concepts were thoroughly explored, and potential benefits were disclosed. The results in this report are summarized and described qualitatively to protect the IP rights of Catalyxx. The details have been shared with the Catalyxx team in the regular steering meetings. At the end of the project, Catalyxx Inc. announced a successful demonstration of pilot scale operation in Seville, Spain.

02 PETROLEUM↗

Data-Enabled Fusion Technology (Final Scientific/Technical Report)

Advancing Scientific Understanding in Fusion Energy and Machine Learning This research represented a significant step forward in machine learning (ML) applications for fusion energy experiments. The project integrated advanced data-driven modeling, optimization techniques, and artificial intelligence to enhance the predictive capabilities and operational efficiency of plasma-based fusion systems. Specifically, tasks focused on ML-enhanced diagnostics, operator guidance tools, and predictive modeling helped improve the ability to interpret complex fusion experiments. Key areas of advancement included: 1) data-driven plasma control, i.e., using ML algorithms to optimize experimental conditions and classify plasma behaviors based on historical data; 2) spectroscopy and diagnostics, i.e., applying AI models to extract previously inaccessible insights from experimental spectroscopy data; and 3) configuration mapping and operator guidance, i.e., developing a predictive framework to assist scientists in identifying the most effective experimental parameters, reducing reliance on manual adjustments. By refining these ML-driven techniques, the project contributed to the broader scientific community’s understanding of plasma dynamics and fusion energy viability. Technical Effectiveness and Economic Feasibility The methods investigated demonstrated high technical effectiveness, as reflected in milestones assessing the predictive accuracy, performance, and optimization of fusion configurations. The development of an Operator Guidance Tool (OGT), for example, led to more precise control of plasma conditions by learning from experimental data and offering real-time adjustments. From an economic standpoint, DeFT provided: 1) the ability to reduce trial-and-error experimentation, which lowered operational costs; 2) improved data interpretation methods, which enabled more efficient resource allocation in large-scale fusion research projects; and 3) the automation of key diagnostic tasks, which reduced manual labor and human error, increasing overall efficiency. 13 The final assessments of predictive models and optimization strategies demonstrated that these approaches were scalable and could be implemented across multiple fusion energy research programs. Public Benefit and Societal Impact This project contributed directly to the broader goal of achieving sustainable and commercially viable fusion energy, which had profound implications for clean energy production and climate change mitigation. The integration of AI-driven solutions into fusion research: 1) sped up scientific discovery, accelerating progress towards achieving energy breakthroughs; 2) reduced the cost of experimentation, making fusion research more accessible; and 3) provided a framework for future AI applications in high-energy physics, benefiting adjacent fields like space exploration, material science, and renewable energy. Additionally, by fostering collaborations between AI researchers and plasma physicists, this project promoted interdisciplinary innovation that could lead to broader applications beyond fusion research.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

Rapid SACR Observations of Convection at Bankhead National Forest (RAPID) Field Campaign Report

Improving our representation of convective cell processes requires better quantification of convective clouds throughout their entire life cycle. This includes gaining a clearer understanding of the controls on key convective cloud properties, such as updraft intensity, particle size distributions, rainfall rates, and hydrometeor species. Our inability to improve convective cloud process modeling stems, in part, from a limited understanding of convective cell properties, particularly given how rapidly these storms evolve. This lack of detailed observations in the most intense and organized convective storms is especially significant, as large errors remain in representing these clouds, which are critical for severe weather prediction and Earth system model performance. Cloud and precipitation radars are essential tools for studying cloud microphysics and dynamics, particularly in deeper convective clouds. The recent U.S. Department of Energy Atmospheric Radiation Measurement (ARM) User Facility’s third Mobile Facility (AMF3) Bankhead National Forest (BNF) deployment provides a unique opportunity to investigate important land–atmosphere interactions, as well as the environmental controls on deep convective cloud processes, in a location favorable for frequent convection. We operate the X/Ka-band Scanning ARM Cloud Radar (X/Ka SACR) using a scan strategy optimized to capture these rapidly evolving clouds and their properties, thereby improving studies of deep convective cloud processes. This effort is strengthened by a complementary and coordinated partnership with ongoing university and multi-agency radar activities collocated in north Alabama—a unique opportunity to examine clouds and precipitation from a lifetime-centric perspective.

54 ENVIRONMENTAL SCIENCES↗

PET Imaging Agents (FES, FFNP, and FDHT) for Estrogen, Androgen, and Progesterone Receptors to Improve Management of Breast and Prostate Cancers by Functional Imaging

Many breast and prostate cancers are driven by the action of steroid hormones on their cognate receptors in primary tumors and in metastases, and endocrine therapies that inhibit hormone production or block the action of these receptors provide clinical benefit to many but not all of these cancer patients. Because it is difficult to predict which individuals will be helped by endocrine therapies and which will not, positron emission tomography (PET) imaging of estrogen receptor (ER) and progesterone receptor (PgR) in breast cancer, and androgen receptor (AR) in prostate cancer can provide useful, often functional, information on the likelihood of endocrine therapy response in individual patients. This review covers our development of three PET imaging agents, 16α-[ 18 F]fluoroestradiol (FES) for ER, 21-[ 18 F]fluoro-furanyl-nor-progesterone (FFNP) for PgR, and 16β-[ 18 F]fluoro-5α-dihydrotestosterone (FDHT) for AR, and the evolution of their clinical use. For these agents, the pathway from concept through development tracks with an emerging understanding of critical performance criteria that is needed for successful PET imaging of these low-abundance receptor targets. Progress in the ongoing evaluation of what they can add to the clinical management of breast and prostate cancers reflects our increased understanding of these diseases and of optimal strategies for predicting the success of clinical endocrine therapies.

60 APPLIED LIFE SCIENCES↗

Development and Evaluation of Occupancy-Aware HVAC Control for Residential Building Energy Efficiency and Occupant Comfort

Occupancy-aware heating, ventilation, and air conditioning (HVAC) control offers the opportunity to reduce energy use without sacrificing thermal comfort. Residential HVAC systems often use manually-adjusted or constant setpoint temperatures, which heat and cool the house regardless of whether it is needed. By incorporating occupancy-awareness into HVAC control, heating and cooling can be used for only those time periods it is needed. Yet, bringing this technology to fruition is dependent on accurately predicting occupancy. Non-probabilistic prediction models offer an opportunity to use collected occupancy data to predict future occupancy profiles. Smart devices, such as a connected thermostat, which already include occupancy sensors, can be used to provide a continually growing collection of data that can then be harnessed for short-term occupancy prediction by compiling and creating a binary occupancy prediction. Real occupancy data from six homes located in Colorado is analyzed and investigated using this occupancy prediction model. Results show that non-probabilistic occupancy models in combination with occupancy sensors can be combined to provide a hybrid HVAC control with savings on average of 5.0% and without degradation of thermal comfort. Model predictive control provides further opportunities, with the ability to adjust the relative importance between thermal comfort and energy savings to achieve savings between 1% and 13.3% depending on the relative weighting between thermal comfort and energy savings. In all cases, occupancy prediction allows the opportunity for a more intelligent and optimized strategy to residential HVAC control.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗

Analyzing the Effects of Atmospheric Turbulent Fluctuations on the Wake Structure of Wind Turbines and Their Blade Vibrational Dynamics

In recent trends, a rising demand for renewable energy has driven wind turbines to larger proportions, where lighter blade designs are often adopted to reduce the costs associated with logistics and production. This causes modern utility-scale wind turbine blades to be inherently more flexible, and their amplified aeroelastic sensitivity results in complex multi-physics reactions to variant atmospheric conditions, including dynamic patterns of aerodynamic loading at the rotor and vortex structure evolutions within the wake. In this paper, we analyze the influence of inflow variance for wind turbines with large, flexible rotors through simulations of the National Rotor Testbed (NRT) turbine, located at Sandia National Labs’ Scaled Wind Farm Technology (SWiFT) facility in Lubbock, Texas. The Common Ordinary Differential Equation Framework (CODEF) modeling suite is used to simulate wind turbine aeroelastic oscillatory behavior and wind farm vortex wake interactions for a range of flexible NRT blade variations, operating in differing conditions of variant atmospheric flow. CODEF solutions of turbine operation in Steady-In-The-Average (SITA) wind conditions are compared to SITA wind conditions featuring a controlled gust-like pulse overimposed, to isolate the effects of typical wind fluctuations. Finally, simulations of realistic time-varying wind conditions from SWiFT meteorological tower measurements are compared to the solutions of SITA wind conditions. These increasingly complex atmospheric inflow variations are tested to show the differing effects evoked by various patterns of spatiotemporal atmospheric flow fluctuations. An analysis is presented for solutions of wind turbine aeroelastic response and vortex wake evolution, to elucidate the consequences of variant inflow, which pertain to wind turbine dynamics at an individual and farm-collective scale. The comparisons of simulated farm flow for SITA and measured fluctuating wind conditions show that certain regions of the wake contain up to a 12% difference in normalized axial velocity, due to the introduction of wind fluctuations. The findings of this study prove valuable for practical applications in wind farm control and optimization strategies, with particular significance for modern utility-scale wind power plants operating in variant atmospheric conditions.

Farrell, Alayna (ORCID:000000023555720X)↗

Automatic Rural Road Centerline Detection and Extraction from Aerial Images for a Forest Fire Decision Support System

To effectively manage the terrestrial firefighting fleet in a forest fire scenario, namely, to optimize its displacement in the field, it is crucial to have a well-structured and accurate mapping of rural roads. The landscape’s complexity, mainly due to severe shadows cast by the wild vegetation and trees, makes it challenging to extract rural roads based on processing aerial or satellite images, leading to heterogeneous results. This article proposes a method to improve the automatic detection of rural roads and the extraction of their centerlines from aerial images. This method has two main stages: (i) the use of a deep learning model (DeepLabV3+) for predicting rural road segments; (ii) an optimization strategy to improve the connections between predicted rural road segments, followed by a morphological approach to extract the rural road centerlines using thinning algorithms, such as those proposed by Zhang–Suen and Guo–Hall. After completing these two stages, the proposed method automatically detected and extracted rural road centerlines from complex rural environments. This is useful for developing real-time mapping applications.

Lourenço, Miguel (ORCID:0000000157673394)↗

Young Stellar Objects, Accretion Disks, and Their Variability with Rubin Observatory LSST

Vera C. Rubin Observatory, through the Legacy Survey of Space and Time (LSST), will allow us to derive a panchromatic view of variability in young stellar objects (YSOs) across all relevant timescales. Indeed, both short-term variability (on timescales of hours to days) and long-term variability (months to years), predominantly driven by the dynamics of accretion processes in disk-hosting YSOs, can be explored by taking advantage of the multiband filters option available in Rubin LSST, in particular the u, g, r, i filters that enable us to discriminate between photospheric stellar properties and accretion signatures. The homogeneity and depth of sky coverage that will be achieved with LSST will provide us with a unique opportunity to characterize the time evolution of disk accretion as a function of age and varying environmental conditions (e.g., field crowdedness, massive neighbors, metallicity) by targeting different star-forming regions. In this contribution to the Rubin LSST Survey Strategy Optimization Focus Issue, we discuss how implementing a dense observing cadence to explore short-term variability in YSOs represents a key complementary effort to the Wide–Fast–Deep observing mode that will be used to survey the sky over the full duration of the main survey (≈10 yr). The combination of these two modes will be vital to investigate the connection between the inner-disk dynamics and longer-term eruptive variability behaviors, such as those observed on EX Lupi–type objects.

79 ASTRONOMY AND ASTROPHYSICS↗

Probabilistic Forecast of All-Sky Solar Radiation Using Enhanced WRF-Solar: Preprint

This study presents enhancements of the Weather Research and Forecasting model with solar extensions (WRF-Solar) to provide probabilistic forecasts of solar radiation. Our approach builds ensemble WRF-Solar runs by introducing stochastic perturbations of variables that produce the largest uncertainties in predicting surface irradiance and clouds. The key variables are identified using tangent linear sensitivity analysis of six physics packages responsible for all-sky irradiance variability. An optimal strategy to stochastically perturb the selected variables is developed and applied to WRF-Solar to generate ensemble members for day-ahead solar prediction. The National Solar Radiation Database (NSRDB) is used to validate the ensemble forecast at arbitrary locations on the model grid. Preliminary results indicate that the proposed technique can potentially produce WRF-Solar ensembles providing reliable information of solar prediction uncertainty. This study describes the implemented methodology and initial results as well as future research to improve ensemble-based probabilistic forecasts with WRF-Solar.

all-sky↗

Advancing Geophysical Techniques to Image a Stratigraphic Hydrothermal Resource

Sedimentary-hosted geothermal energy systems are permeable structural, structural-stratigraphic, and/or stratigraphic horizons with sufficient temperature for direct use and/or electricity generation. Sedimentary-hosted (i.e., stratigraphic) geothermal reservoirs may be present in multiple locations across the central and eastern Great Basin of the USA, thereby constituting a potentially large base of untapped, economically accessible energy resources. Sandia National Laboratories has partnered with a multi disciplinary group of collaborators to evaluate a stratigraphic system in Steptoe Valley, Nevada using both established and novel geophysical imaging techniques. The goal of this study is to inform an optimized strategy for subsequent exploration and development of this resource and analogous ones. Building from prior Nevada Play Fairway Analysis (PFA), this team is primarily 1) collecting additional geophysical data, 2) employing novel joint geophysical inversion/modeling techniques to update existing 3D geologic models, and 3) integrating the geophysical results to produce a working, geologically constrained thermo-hydrological reservoir model. Prior PFA work highlights Steptoe Valley as a favorable resource basin that likely has both sedimentary and hydrothermal characteristics. However, there remains significant uncertainty on the nature and architecture of the resource(s) at depth, which increases the risk in exploratory drilling. Newly acquired gravity, magnetic, magnetotelluric, and controlled-source electromagnetic data products, in conjunction with new and preexisting geoscientific measurements and observations, are being integrated and evaluated for efficacy in understanding stratigraphic geothermal resources and mitigating exploration risk. Furthermore, the influence of hydrothermal activity on sedimentary-hosted reservoirs in favorable structural settings, and whether fault-controlled systems may locally enhance temperature and permeability in some deep stratigraphic reservoirs, will also be evaluated.

Geothermal, Sedimentary Heat, Geophysics, Seismic,↗

Is Knowledge About Running Applications Helping Improve Runtime Prediction of HPC Jobs?

High-performance computing systems rely upon scheduling algorithms to achieve high utilization. These schedulers rely upon user estimates of job resource requirements, such as runtime, to determine optimal scheduling of incoming jobs. These user estimates, however, are prone to error. To mitigate this error, significant research has been directed at providing better estimates of job runtime, usually employing machine learning techniques. These techniques are dependent upon the input features selected. Among the possible features is the primary application used by the job. In a survey of more than 20 papers directed at improving runtime prediction, only four included primary application as an input feature. We focus this investigation specifically on the value of adding primary application as an input feature, and find that it does improve model performance, especially for jobs with longer runtimes, though this improvement varies based on the application used. We recommend further research to determine the cause of this variability as well as an optimal strategy for employing a mixture of models both including and not including primary application as a feature.

feature selection↗

Challenges in Training PINNs: A Loss Landscape Perspective

This paper explores challenges in training Physics Informed Neural Networks (PINNs), emphasizing the role of the loss landscape in the training process. We examine difficulties in minimizing the PINN loss function, particularly due to ill conditioning caused by differential operators in the residual term. We compare gradient-based optimizers Adam, L-BFGS, and their combination Adam+L-FGS, showing the superiority of Adam+L-BFGS, and introduce a novel secondorder optimizer, NysNewton-CG (NNCG), which significantly improves PINN performance. Theoretically, our work elucidates the connection between ill-conditioned differential operators and ill-conditioning in the PINN loss and shows the benefits of combining first- and second-order optimization methods. Our work presents valuable insights and more powerful optimization strategies for training PINNs, which could improve the utility of PINNs for solving difficult partial differential equations.

Rathore, Pratik↗