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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 271 records · Page 15
All sourcesNASA NTRSDOE OSTI
DOE OSTI2026

Federated learning for 2D synchrotron x-ray diffractometry: a cross-institutional approach for phase quantification of Ti–6Al–4V alloy

High-energy Two dimensional (2D) synchrotron x-ray diffractometry provides important insights into the atomistic structure and phase evolution of materials, yet traditional analysis methods remain complex, knowledge-intensive, and computationally demanding. Deep-learning models offer a powerful alternative for automating their analysis. Institutions that hold these datasets may be unwilling to share their data due to privacy and security policies, as well as the challenges associated with large-scale data transfer. As a result, models trained on local datasets often perform well only on their own data but exhibit bias and poor generalization across different instruments or facilities. To overcome these limitations, we explore federated learning (FL) for 2D synchrotron diffractograms, enabling collaborative model training without exchanging raw data. In this study, 2D synchrotron diffractograms of Ti–6Al–4V alloy collected from two independent facilities are used to train convolutional neural networks for predicting the β-phase volume fraction. Experimental results show that federated global models significantly outperform locally trained models in terms of generalization and achieve accuracy comparable to centralized trained models. These findings demonstrate the potential of FL to enable secure, cross-institutional collaboration and enhance the scalability of deep-learning-based materials characterization.

36 MATERIALS SCIENCE↗
DOE OSTI2026

Physicochemical evolution of uranium nitride kernel microstructure with varying carbon distribution for advanced TRISO fuel forms

Uranium nitride (UN) has emerged as a fuel candidate for advanced nuclear reactor concepts due to its superior uranium density, thermal conductivity, and high melting temperature. However, the fabrication route for converting UO 2 to UN is complex and difficult to standardize. Although the chemistry of this conversion process is well-studied, more insight into the physicochemical dynamics of this conversion using advanced characterization techniques can help further our understanding of this material system. This work leveraged thermogravimetric analysis (TGA), X-ray diffraction (XRD), and nondestructive 3D X-ray computed tomography (XCT) to characterize dynamic microstructural changes in the UO 2 → UCO → UN fabrication pathway for two kernels with a varying carbon distribution in the starting composition. TGA and XRD were used to quantify changes in the mass, density, and chemical composition of the two kernels, while three-dimensional image processing and segmentation of XCT data were used to quantify the volume, surface area, and spatial distribution of features within each kernel for multiple steps along the fabrication pathway. The analysis indicates distinct differences between the two kernels that are correlated to downstream conversion efficiency. In conclusion, this work is among the first to perform 3D quantification of physicochemical evolution during UN conversion, providing quantitative correlation between processing, properties, and expected fuel performance.

Nuclear fuel↗
DOE OSTI2026

Integration of Online Cross-Section Generation Capability with Depletion and Transient Solvers in Griffin

Griffin is a Multiphysics Object-Oriented Simulation Environment (MOOSE)-based reactor multiphysics analysis application jointly developed by Argonne and Idaho National Laboratories under the DOENE Nuclear Energy Advanced Modeling and Simulation (NEAMS) program. In FY25, an online crosssection generation capability based on the Self-Shielding Application Programming Interface (SSAPI) was demonstrated for TRISO-fueled reactor problems under steady-state conditions. This fiscal year, that capability was extended to support depletion and transient multiphysics calculations, enabling high-fidelity analyses that generate self-shielded cross sections on the fly from the actual evolving composition and temperature states rather than from pre-tabulated libraries. For depletion, a two-way coupling was established in which SSAPI computes compact-averaged self-shielded cross sections that the depletion solver then uses to advance the Bateman equations, with the updated compositions returned to SSAPI at each step; the depletion module was refactored to support both library-based and SSAPI-based cross sections, and additional logic was added to track daughter isotopes and to exclude minor isotopes for efficiency. For transient analysis, the SSAPI multigroup library was extended with the kinetics data required for time-dependent calculations, the Improved Quasi-Static (IQS) scheme was coupled with SSAPI, and several supporting capabilities were implemented, including a self-shielding treatment that lets control rods and drums move within a self-shielded model, which had previously been impossible and had ruled out rod- and drum-movement transients with on-the-fly cross sections altogether, a new mixing scheme for delayed-neutron precursor decay constants, a checkpoint-based restart workflow, and performance improvements such as pointwise cross-section interpolation and the bypassing of unnecessary Dancoff factor calculations. The implemented capabilities were verified against Serpent Monte Carlo solutions. For depletion, a prismatic pin-cell problem based on a Next Generation Nuclear Plant (NGNP) Very High Temperature Reactor benchmark showed excellent agreement, with eigenvalue differences within 200 pcm over the entire burnup range (up to 140 MWD/kgU) and fission-product and actinide inventories agreeing to within 0.8% and 2.5%, respectively; a heat-pipe microreactor assembly problem with a much higher fuel loading confirmed the same behavior and quantified the bias introduced when the multigroup equivalence effect is neglected. For transient analysis, a pin-cell problem with a step reactivity insertion and temperature feedback reproduced the analytically expected asymptotic power and showed close agreement between the direct and IQS solutions, and a two-dimensional microreactor core problem with control-drum rotation exercised the new moving-drum self-shielding treatment and demonstrated successful coupling of the online crosssection generation with both the direct and IQS transient methods. The capability was further exercised on a full-core pebble-bed problem, in which Griffin was coupled with the System Analysis Module (SAM) to simulate load-following operation of the gPBR with the Doppler feedback resolved at the TRISO fuel kernel temperature. These developments in Griffin provide a convenient, high-fidelity approach to cross-section generation for advanced thermal reactors with geometrically complex and highly heterogeneous configurations, including TRISO-fueled prismatic and pebble-bed systems, and support steady-state, depletion, and transient multiphysics calculations. They also enable self-shielded cross sections to be evaluated directly at the actual coupled state of the system, thereby establishing a foundation for high-fidelity, fully coupled multiphysics analysis of advanced reactors

Park, H.↗
DOE OSTI2024

Atomic-scale identification of active sites of oxygen reduction nanocatalysts

Heterogeneous nanocatalysts play a crucial role in both the chemical and energy industries. Despite substantial advancements in theoretical, computational and experimental studies, identifying their active sites remains a major challenge. Here we utilize atomic electron tomography to determine the three-dimensional atomic structure of PtNi and Mo-doped PtNi nanocatalysts for the electrochemical oxygen reduction reaction. We then employ the experimental atomic structures as input to first-principles-trained machine learning to identify the active sites of the nanocatalysts. Through the analysis of the structure–activity relationships, we formulate an equation termed the local environment descriptor, which balances the strain and ligand effects to provide physical and chemical insights into active sites in the oxygen reduction reaction. The ability to determine the three-dimensional atomic structure and chemical composition of realistic nanoparticles, combined with machine learning, could transform our fundamental understanding of the active sites of catalysts and guide the rational design of optimal nanocatalysts.

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

A Two-Dimensional Non-Linear Magnetic Equivalent Circuit Model to Facilitate the Preliminary Design of a Normal Conducting Quadrupole

Normal conducting quadrupoles have been used to focus charged particle beams in synchrotrons, beam transfer lines, medical linacs, etc. for a long time. Optimization techniques based on analytical expressions combined with the use of a numerical field analysis tool exist in the literature for the design of an optimal pole tip shape. However, the initial shape and dimensions of the remaining yoke (including the pole itself and yoke base that act as the return path) are usually less well defined. This article discusses a design methodology for a normal conducting quadrupole based on a two-dimensional magnetic equivalent circuit. Here, this approach considers the geometry of the entire magnet and the non-linear behavior of the yoke material, thereby eliminating the initial iterations. The design outcome of this exercise serves as a competent starting point and can then be used to refine the pole tip, pole taper, yoke size, and add other geometrical features to achieve the required field quality, gradient, coil considerations, etc. by employing a finite element analysis tool. An example design study to demonstrate the proposed methodology is presented.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2025

Direct Observations of Solute Dispersion in Rocks With Distinct Degree of Sub‐Micron Porosity

Abstract The transport of chemical species in rocks is affected by their structural heterogeneity to yield a wide spectrum of local solute concentrations. To quantify such imperfect mixing, advanced methodologies are needed that augment the traditional breakthrough curve analysis by probing solute concentration within the fluids locally. Here, we demonstrate the application of asynchronous, multimodality imaging by X‐ray computed tomography (XCT) and positron emission tomography (PET) to the study of passive tracer experiments in laboratory rock cores. The four‐dimensional concentration maps measured by PET reveal specific signatures of the transport process, which we have quantified using fundamental measures of mixing and spreading. We observe that the extent of solute spreading correlate strongly with the strength of subcore‐scale porosity heterogeneity measured by XCT, while dilution is enhanced in rocks containing substantial sub‐micron porosity. We observe that the analysis of different metrics is necessary, as they can differ in their sensitivity to the strength and forms of heterogeneity. The multimodality imaging approach is uniquely suited to probe the fundamental difference between spreading and mixing in heterogeneous media. We propose that when multi‐dimensional data is available, mixing and spreading can be independently quantified using the same metric. We also demonstrate that one‐dimensional transport models have limited predictive ability toward the internal evolution of the solute concentration, when the model is solely calibrated against the effluent breakthrough curves. The data set generated in this study can be used to build realistic digital rock models and to benchmark transport simulations that account deterministically for rock property heterogeneity.

Kurotori, Takeshi [Department of Chemical Engineer↗
DOE OSTI2026

Feasibility Study on Implementing a Staggered-Grid Finite Volume Method for System Analysis Code Development Under the MOOSE Framework

Here, this work summarizes a feasibility study on testing numerical algorithms that are suitable and efficient for advanced system analysis code development under the mutli-physics framework, MOOSE. The key to the test bed is the implementation of high-order one-dimensional staggered-grid finite volume method (SG-FVM), and its direct interaction with the linear/nonlinear solver, PETSc. The test bed utilized a more flexible code structure to enable the finite volume method implementation and direct interacting with the solver package, instead of using the natively supported finite element method by the framework. Using a suite of selected test problems with different problem sizes and levels of complexity, the implemented SG-FVM demonstrated superior performance improvement against a direct finite element method implementation through MOOSE. On two computer systems, the speedup was observed to be significant, with at least one order of magnitude of solving time reduction. For a complex reactor model, transient simulation was performed using the newly developed finite volume method code, the results of which agree very well with the reference results from the finite element method code. Overall, this study demonstrates a successful feasibility study on the proposed numerical algorithms and software structure to support advanced system analysis tool development.

MOOSE↗
DOE OSTI2024

Understanding Formation of Irradiation-Induced Defects through 4D-STEM, Electron Tomography, and WBDF-STEM

A major challenge in advancing nuclear materials for next-generation fission and proposed fusion reactors is to comprehensively understand the formation of irradiation-induced defects. Here it is essential to correlate the evolution of irradiation-induced defects and the degradation of mechanical properties, as they collectively dictate the material's lifespan and ensure nuclear safety. Scanning transmission electron microscopy (STEM) based techniques have emerged as indispensable tools for irradiation-induced defect characterization, offering high spatial resolution imaging and chemical analysis, such as electron energy loss spectroscopy (EELS) and energy dispersive X-ray spectroscopy (EDXS). These techniques have been effectively used to obtain an atomic-scale view of the defect structure. Recent advances in electron microscopy, particularly in 4D-STEM, offer detailed insight into microstructural evolution by capturing full 2D diffraction patterns at every pixel position. Using high-speed direct electron detectors, this technology generates a four-dimensional dataset, overcoming the limitations of traditional STEM imaging.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Designing Remote Monitoring for Smart Manufacturing Facilities: Hazard Identification and Classification

This study investigates the process of hazard identification in complex manufacturing environments during the design phase, emphasizing the significance of the design process in developing designs that effectively mitigate hazards in contexts with numerous variables, such as a variety of machines, sensors, actuators, and agents. Through a mixed-methods approach, the objective of this work is to understand how the evolution of design outcomes across various stages might influence a designer’s ability to recognize both standard and novel hazards. To achieve this understanding, an experimental design task was conducted with six designers from a national lab specializing in manufacturing technologies. This approach combined qualitative and quantitative data analysis from a one-hour virtual session with participants. Findings suggest that the complexity of identifying hazards in a high-dimensional design space is challenging within a limited time frame and that the identification of hazards is significantly influenced by the stage of the design task and the initial design decisions, indicating the need for extended time and strategic initial planning in the design process to enhance hazard identification.

Ballestas, Caseysimone↗
DOE OSTI2026

radkit base v1.6

The radkit (base) software suite (python) consists of three primary libraries: stark, trajan, and curie. The trajan library provides the tools to analyze and manipulate data from lidar and inertial measurement unit (IMU) devices, cameras, as well as trajectories from algorithms such as simultaneous localization and mapping (SLAM). These components allow reading and writing standard data formats, performing rigid affine transformations, discretizing three-dimensional space, and visualizing data products. The curie library comprises a standard set of object-oriented tools for radiation data and analysis in the following modules: (1) listmode and binmode data classes with methods for manipulation, plotting, slicing and file IO; (2) radiological/nuclear source detection/identification analysis results; (3) source encounters of correlated analyses and (4) energy-dependent angular detector response functions. The stark package provides low-level tools that are leveraged by both curie and trajan. The tools are flexible for offline analysis as well as performant for real-time integrations.

Salathe, Marco [Lawrence Berkeley National Laborat↗
DOE OSTI2025

Toward Drilling the Perfect Geothermal Well: An International Research Coordination Network for Geothermal Drilling Optimization Supported by Deep Machine Learning and Cloud Based Data Aggregation

The EDGE project, supported by the U.S. Department of Energy Geothermal Technologies Office under award DE-EE0008793, established a data-driven framework for improving the efficiency, cost-effectiveness, and reliability of geothermal well drilling. The project focused on developing scalable data infrastructure, advanced machine learning and probabilistic models, and integrated analytics tools to support continuous drilling optimization. A central objective was to reduce geothermal drilling costs by up to seventy percent while minimizing the risk of well failure through predictive diagnostics and adaptive planning. Over the project period, a comprehensive data repository was designed and deployed, incorporating records from over one hundred geothermal wells across varied geological settings. This repository supported both structured and unstructured data and adhered to FAIR data principles, enabling provenance tracking, quality control, and standardized metadata. The project introduced automated ingestion pipelines and a cloud-hosted platform that facilitated access to raw, processed, and derived datasets. This infrastructure served as the foundation for model development and analysis. Machine learning workflows were developed to predict key drilling metrics including rate of penetration, non-productive time, and total drilling costs. Self-organizing maps and dimensionality reduction methods were used to uncover operational patterns and outliers, while supervised learning algorithms such as random forests and deep neural networks were applied to forecast performance outcomes. The models were validated on heterogeneous datasets from both U.S. and Icelandic fields, demonstrating variable but significant predictive accuracy. The results indicated that finer temporal resolution, inclusion of lithological data, and consistency in operational annotations could substantially improve model performance. The project also implemented process mining techniques to reconstruct state-transition models from drilling event logs. These models enabled the identification of deviations from optimal workflows and provided insights into recurring failure modes. Analysis of non-productive time highlighted the impact of equipment failures, geological challenges, and human factors, offering opportunities for targeted mitigation strategies. The EDGE Dashboard was developed as a web-based expert system integrating data visualization, model outputs, and user-driven queries. It provided an accessible interface for operators to explore historical data, evaluate predicted outcomes, and compare drilling scenarios. Initial feedback from project partners suggested that the dashboard could serve as a foundation for more advanced advisory and optimization tools. Overall, the EDGE project demonstrated the feasibility and value of applying modern data science techniques to geothermal drilling. It delivered a set of interoperable tools and models that can support more efficient, lower-risk well development. The findings point toward a viable path for transitioning from advisory analytics to semi-autonomous drilling systems, contingent on continued collaboration, expanded datasets, and field validation. The project results have immediate relevance for drilling operations, data management practices, and future geothermal R&D efforts aimed at achieving reliable, cost-competitive geothermal energy at scale.

15 GEOTHERMAL ENERGY↗
DOE OSTI2024

Spectral Analysis of Regular Material Point Method and its Application to Study High Pressure Reverse Osmosis Membrane Compaction and Embossing

Material Point Method (MPM) is gaining widespread interest in applied continuum mechanics. The fact that all the continuum properties are stored on the particles (or material points) and the governing equations are solved on these material points makes MPM extremely suited to problems involving severe material deformations, such as crack propagation, soil movement, and fluid flows. Despite its popularity, only a few studies have focused on the numerical properties of MPM. This presentation introduces a global spectral analysis of the regular material point method. Contrary to previous studies, the analysis focuses on the numerical properties of the method in the spectral space. The amplification factor is derived as a function of the non- dimensional wave numbers. It provides insights into the stability and dissipative properties of the method for various CFL and Fourier numbers. The effect of the grid shape functions, number of particles per cell and their locations inside the grid cell are also analyzed. The EXAGOOP MPM solver (https://github.com/NREL/Exagoop.git) is developed at the National Renewable Energy Laboratory as a part of the NAWI UHPRO project and is based on the AMReX framework. A single-level, uniform cartesian grid is used as the background mesh, while the particle class in AMReX is used to manage the material point operations. Linear hat and B-splines are used as grid shape functions, while the time integration is performed using explicit Euler time integration. EXAGOOP is both CPU and GPU compatible and has been demonstrated to work well on multiple compute architectures. The performance of EXAGOOP on various computing architectures is presented along with its application to study compaction and embossing of high-pressure reverse osmosis membranes. The MPM solution accurately reproduces the membrane deformation. The deformed pore size and structure simulated using MPM also agree well with experimental SEM images.

material point method↗
DOE OSTI2026

Mitigating Algorithmic Bias in Cancer Site Classification Models

Purpose Integrating artificial intelligence in cancer diagnostics has improved tumor classification beyond rule-based systems. Despite these advancements, these models may still encode demographic biases. We conducted a large-scale, applied bias-probing study of a deep learning–based cancer site classifier to quantify race information encoded in document embeddings. We then evaluated how performance changes when race-correlated embedding dimensions are removed in a post-training sensitivity analysis. Methods The cancer site classifier was trained using 3.5 million electronic cancer pathology reports from six of the National Cancer Institute's SEER registries. We trained a hierarchical self-attention network to generate 400-dimensional document embeddings. These embeddings were used to train two downstream, gradient-boosted decision tree classifiers: one to classify the cancer sites and another to predict racial categories. We identified overlapping features by intersecting the top 50 feature-importance rankings from the site and race models and computed their cumulative feature importance in each model. As a post hoc sensitivity analysis, we progressively pruned these overlapping dimensions, retrained the site model, and compared overall macro-F1 and accuracy, race-stratified macro-F1, and group fairness metrics on the basis of demographic parity and equalized odds before and after pruning. Results The analysis revealed minimal feature overlap between the cancer site and race prediction models, and the cumulative importance scores indicated a negligible influence of racial information on clinical predictions. Post-training pruning of overlapping features did not compromise the models' diagnostic accuracy, with a 0.07% loss in accuracy. Conclusion Our findings demonstrate that HiSAN-generated embeddings from SEER data can be used effectively in cancer site classification without significant demographic bias influencing the outcomes. Post-training pruning therefore functions as a practical audit and sensitivity check.

Shivanna, Abhishek [ORNL] (ORCID:0009000665228593)↗
DOE OSTI2025

Chemistry imaging and distribution analysis of rare earth elements in coal using LIBS and LA-ICP-MS instruments

Currently, demand for rare earth elements (REEs) increased significantly. Coal is actively evaluated as potential economic sources for extraction of REEs. Here, in this work, laser-induced breakdown spectroscopy (LIBS) was evaluated for rapid estimation of REEs content and their distribution in the natural coal samples. The results were compared with similar laser ablation–inductively coupled plasma–mass spectrometry (LA-ICP-MS) measurements. Thirteen coal samples (nine standard samples and five natural samples) were used in this study. Powder samples were pressed into pellets while coal chunks were directly ablated for data recording. Pellets of the powder standard samples were used to optimize the data acquisition system and then data recorded with this optimized system was used to identify the proper data acquisition and analysis models. After establishing the proper data acquisition system and analysis model using the standard samples, natural coal samples in powder form and their chunks were utilized to record LIBS and LA-ICP-MS spectra. Multivariate calibration models were developed using four of the natural samples, which were evaluated by predicting the REE content in the fifth sample. Principal component analysis was performed on the LIBS data obtained from the natural samples and it classified all the samples with high accuracy. Two-dimensional (2D) elemental mapping on coal chunk samples was also performed using both LIBS and LA-ICP-MS to study the distribution of REEs in the samples. The resulting elemental images and their correlations can be used to infer mineral distributions.

01 COAL, LIGNITE, AND PEAT↗
DOE OSTI2024

Tracking the topology of neural manifolds across populations

Neural manifolds summarize the intrinsic structure of the information encoded by a population of neurons. Advances in experimental techniques have made simultaneous recordings from multiple brain regions increasingly commonplace, raising the possibility of studying how these manifolds relate across populations. However, when the manifolds are nonlinear and possibly code for multiple unknown variables, it is challenging to extract robust and falsifiable information about their relationships. We introduce a framework, called the method of analogous cycles, for matching topological features of neural manifolds using only observed dissimilarity matrices within and between neural populations. We demonstrate via analysis of simulations and in vivo experimental data that this method can be used to correctly identify multiple shared circular coordinate systems across both stimuli and inferred neural manifolds. Conversely, the method rejects matching features that are not intrinsic to one of the systems. Further, as this method is deterministic and does not rely on dimensionality reduction or optimization methods, it is amenable to direct mathematical investigation and interpretation in terms of the underlying neural activity. We thus propose the method of analogous cycles as a suitable foundation for a theory of cross-population analysis via neural manifolds.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Coupled Multiphysics Modeling of Lithium-Ion Batteries for Automotive Crashworthiness Applications

Considerable advances have been made in battery safety models, but achieving predictive accuracy across a wide range of conditions continues to be challenging. Interactions between dynamically evolving mechanical, electrical, and thermal state variables make model prediction difficult during mechanical abuse scenarios. In this study, we develop a physics-based modeling approach that allows for choosing between different mechanical and electrochemical models depending on the required level of analysis. We demonstrate the use of this approach to connect cell-level abuse response to electrode-level and particle-level transport phenomena. A pseudo-two-dimensional model and simplified single-particle models are calibrated to electrical-thermal cycling data and applied to mechanically induced short-circuit scenarios to understand how the choice of electrochemical model affects the model prediction under abuse scenarios. These models are implemented using user-defined subroutines on ls-dyna finite element software and can be coupled with existing automotive crash safety models.

analysis and design of components↗
DOE OSTI2024

A Practical Solver for Scalar Data Topological Simplification

This paper presents a practical approach for the optimization of topological simplification, a central pre-processing step for the analysis and visualization of scalar data. Given an input scalar field f and a set of “signal” persistence pairs to maintain, our approaches produces an output field g that is close to f and which optimizes (i) the cancellation of “non-signal” pairs, while (ii) preserving the “signal” pairs. In contrast to pre-existing simplification algorithms, our approach is not restricted to persistence pairs involving extrema and can thus address a larger class of topological features, in particular saddle pairs in three-dimensional scalar data. Our approach leverages recent generic persistence optimization frameworks and extends them with tailored accelerations specific to the problem of topological simplification. Extensive experiments report substantial accelerations over these frameworks, thereby making topological simplification optimization practical for real-life datasets. Our approach enables a direct visualization and analysis of the topologically simplified data, e.g., via isosurfaces of simplified topology (fewer components and handles). We apply our approach to the extraction of prominent filament structures in three-dimensional data. Specifically, we show that our pre-simplification of the data leads to practical improvements over standard topological techniques for removing filament loops. Here, we also show how our approach can be used to repair genus defects in surface processing. Finally, we provide a C++ implementation for reproducibility purposes.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Bias-Variance Trade-Off in Physics-Informed Neural Networks with Randomized Smoothing for High-Dimensional PDEs

Physics-Informed Neural Networks (PINNs) have triggered a paradigm shift in scientific computing, leveraging mesh-free properties and robust approximation capabilities. While proving effective for low-dimensional partial differential equations (PDEs), the computational cost of PINNs remains a hurdle in high-dimensional scenarios. This is particularly pronounced when computing high-order and high-dimensional derivatives in the physics-informed loss. Randomized Smoothing PINN (RS-PINN) introduces Gaussian noise for stochastic smoothing of the original neural net model, enabling the use of Monte Carlo methods for derivative approximation, which eliminates the need for costly automatic differentiation. Despite its computational efficiency, especially in the approximation of high-dimensional derivatives, RS-PINN introduces biases in both loss and gradients, negatively impacting convergence, especially when coupled with stochastic gradient descent (SGD) algorithms. We present a comprehensive analysis of biases in RS-PINN, attributing them to the nonlinearity of the Mean Squared Error (MSE) loss as well as the intrinsic nonlinearity of the PDE itself. We propose tailored bias correction techniques, delineating their application based on the order of PDE nonlinearity. The derivation of an unbiased RS-PINN allows for a detailed examination of its advantages and disadvantages compared to the biased version. Specifically, the biased version has a lower variance and runs faster than the unbiased version, but it is less accurate due to the bias. To optimize the bias-variance trade-off, we combine the two approaches in a hybrid method that balances the rapid convergence of the biased version with the high accuracy of the unbiased version. In addition to methodological contributions, we present an enhanced implementation of RS-PINN. Extensive experiments on diverse high-dimensional PDEs, including Fokker-Planck, Hamilton-Jacobi-Bellman (HJB), viscous Burgers’, Allen-Cahn, and Sine-Gordon equations, illustrate the bias-variance trade-off and highlight the effectiveness of the hybrid RS-PINN. Empirical guidelines are provided for selecting biased, unbiased, or hybrid versions, depending on the dimensionality and nonlinearity of the specific PDE problem.

97 MATHEMATICS AND COMPUTING↗