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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 235 records · Page 13
All sourcesNASA NTRSDOE OSTI
DOE OSTI2026

Ensemble Federated Machine Learning‐Based Cybersecurity Situational Awareness in Microgrid Network

Cyber-physical microgrids are vulnerable to stealthy cybersecurity threats that disguise their actions through the exploitation of system knowledge. Such actions can severely impacts microgrids deployed in defense bases, slowing the response time of military forces during national emergencies. Several machine-learning algorithms have been proposed to detect intrusions in the grid networks; however, these traditional machine-learning algorithms lack data privacy and are subject to several adversarial machine-learning threats. This paper proposes a novel federated machine learning (FML)-based three-model framework to detect and identify stealthy data-integrity attacks while ensuring data privacy in microgrid networks. The proposed architecture uses a variational mode decomposition technique to extract derived features from incoming measurement and control datasets. The extraction of these derived features allows FML models to learn minute variations in data patterns that allow them to perform significantly better than the models trained with generic datasets consisting of raw features. Our experimental results show the efficient performance of the proposed methodology against different types of data integrity attacks while considering primary and secondary controllers in microgrids. Further, the applied FML-integrated random forest ensemble algorithm outperforms the existing generic FML algorithms during noisy and noise-free datasets with prediction latencies of only 91–134 µs per sample within the 0.1 s sampling interval and requires communication bandwidth of around ∼8.25 KB/s at the control center and ∼2.7 KB/s per edge client for communication.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2020

miniGAN: A Generative Adversarial Network proxy application WBS 2.2.6.08 ECP-2.1.3 (Q3 FY2020 Milestone Report) (V.1.0)

In order to support the machine learning co-design needs of ECP applications in current and future DOE HPC hardware, we have developed a generative adversarial network (GAN) proxy application, miniGAN, that has been released through the ECP proxy application suite. The proxy application is representative of the needs of ExaLearn's target applications, specifically the Cosmoflow and ExaGAN cosmology applications and the ExaWind energy application. The proxy application also demonstrates the first use of performance portable kernels within widely-used machine learning frameworks: PyTorch (Facebook) and Horovod (Uber). We provide performance scaling results for similar workloads to ExaGAN and a profile of individual GAN training components.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Data efficiency assessment of generative adversarial networks in energy applications

This study investigates the data requirements of generative artificial intelligence (AI), particularly generative adversarial networks (GANs), for reliable data augmentation in energy applications. Generative AI, though seen as a solution to data limitations, requires substantial data to learn meaningful distributions—a challenge often overlooked. This study addresses the challenge through synthetic data generation for critical heat flux (CHF) and power grid demand, focusing on renewable and nuclear energy. Two variants of GAN employed are conditional GAN (cGAN) and Wasserstein GAN (wGAN). Our findings include the strong dependency of GAN on data size, with performance declining on smaller datasets and varying performance when generalizing to unseen experiments. Mass flux and heated length significantly influence CHF predictions. wGAN is more robust to feature exclusion, making it suitable for constrained synthetic data generation. In energy demand forecasting, wGAN performed well for solar, wind, and load predictions. Longer lookback hours and larger datasets improved predictions, especially for load power. Seasonal variations posed challenges, with wGAN achieving a relatively high error of Root Mean Squared Error (RMSE) of 0.32 for load power prediction, compared to RMSE of 0.07 under same-season conditions. Feature exclusions impacted cGAN the most, while wGAN showed greater robustness. This study concludes that, while generative AI is effective for data augmentation, it requires substantial data and careful training to generate realistic synthetic data and generalize to new experiments in engineering applications.

22 - GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2023

Improving multiwell petrophysical interpretation from well logs via machine learning and statistical models

Well-log interpretation estimates in situ rock properties along well trajectory, such as porosity, water saturation, and permeability, to support reserve-volume estimation, production forecasts, and decision making in reservoir development. However, due to measurement errors, variability of well logs caused by multiple measurement vendors, different borehole tools, and nonuniform drilling/borehole conditions, estimations of rock properties with original well logs without proper preprocessing may not be accurate, especially in the context of multiwell estimation. Well-log normalization techniques such as two-point scaling and mean-variance normalization are commonly used to improve the robustness of multiwell rock-property estimation. However, these techniques do not consider the correlation between well logs and require subjective knowledge for their effective implementation. To reduce uncertainties and processing time associated with multiwell rock-property estimation from well logs, we develop discriminative adversarial (DA) and linear constraint models for well-log normalization and rock-property estimation. The DA neural network model developed for well-log normalization and interpretation can perform linear and nonlinear well-log normalization while considering the joint distribution of each well log and rock properties. However, the linear constraint model uses an ensemble of predictions from linear models to constrain well-log normalization and rock-property estimation. We also develop a divergence-based type well identification method to select type (training) wells for a test well based on the statistical similarity of associated well-log distributions instead of the interwell distance. We apply the DA model to perform well-log normalization and prediction of permeability for the Seminole San Andres Unit carbonate reservoir. Compared with the permeability predicted with the classical machine learning model without well-log normalization and models with two-point scaling normalization, the DA model yields the most accurate permeability prediction by decreasing the mean-squared error of permeability prediction by 20%–50%.

Geochemistry & Geophysics↗
DOE OSTI2025

Advanced Reactor Safeguards & Security Program: Cybersecurity Scenarios

The use of digital control systems and automation in advanced nuclear power systems introduces different types of vulnerabilities compared to legacy (i.e. analog) control systems that cyber adversaries can exploit. These vulnerabilities pose a challenge to reactor operators and cyber operations staff due to the dynamic nature of the event in which a human response or a lack of response can potentially evolve into a worsening plant condition. Using the Department of Homeland Security Cyber and Infrastructure Security Agency’s (CISA) critical infrastructure exercise framework, this document presents several cyber security scenarios typical of digital control systems that could be used in advanced reactor designs. These scenarios can be used in tabletop exercises to evaluate cyber security posture or conduct training on different aspects of cyber security, including detection, threat hunting using indicators of compromise, evaluating incident response, risk mitigation, incident reporting, information sharing and recovery.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2026

Cyote-attack Chain Estimator

Attack Chain Estimator (ACE) Application Overview The Attack Chain Estimator (ACE) Application is a sophisticated tool designed for the ingestion, classification, sequencing, and enrichment of cybersecurity threat reports. This application leverages advanced machine learning models and extensive historical data to provide comprehensive insights into cyber threats, specifically targeting Industrial Control Systems (ICS). Purpose The primary functions of the ACE Application include: Ingestion of Cybersecurity Threat Reporting: Capable of ingesting text-based threat reports in markdown or text file format. Supports ingestion of structured data from other sources in STIX/JSON format. Classification of Report’s Text-Based Events: Utilizes a DeBERTa classifier, specifically trained on cybersecurity data, to map the events to MITRE ATT&CK for ICS Tactics and Techniques. Classification is performed using multiple Jupyter notebooks and machine learning workflows hosted as FastAPI microservices: regex_data deberta_base_35_train_hft_classifier_mlflow.ipynb hft_regex_classifier_mlflow.ipynb param_train_hft_classifier_mlflow.ipynb regex_tactic_tech.ipynb Ordering of Tactics, Techniques, and Observable Events: Sequences the identified tactics, techniques, and events to form a coherent attack chain. Enrichment with Historical Attack Chain Details: Enhances the attack chain with details from historical attacks using a Markov model developed from CyOTE Precursor Analysis Report data. The Markov model is available as a FastAPI endpoint for seamless integration. Enrichment with Adversary Emulation Capabilities Data: Integrates adversary emulation capabilities data using MITRE Caldera for OT adversary abilities UUIDs. Export of Output Files: Provides options to export the enriched attack chain in JSON or CSV formats. Routing of Output to Other Applications: Facilitates routing of output to various platforms and applications, including: Threat Intelligence Platforms COREII Scout for Threat Intelligence Analysis COREII Modeling and Simulation for Adversary Emulation Technical Description The ACE Application is an advanced cybersecurity tool designed to provide detailed threat analysis and sequence generation. It is built on a robust architecture that integrates natural language processing, machine learning, and historical data modeling. Key Components: Data Ingestion Module: Handles the input of threat reports and data from various formats, ensuring flexibility in data sources. Classification Engine: Employs DeBERTa-based classifiers hosted as FastAPI microservices to analyze and classify threat report events in accordance with the MITRE ATT&CK framework for ICS. Sequence Generator: Orders the classified events into a logical attack chain, providing clear insight into the sequence of tactics and techniques used in the threat. Enrichment Engine: Integrates historical data and adversary emulation capabilities to enhance the attack chain with valuable context and additional details. The historical data enrichment is powered by a Markov model, which is available as a FastAPI endpoint. Export and Routing Module: Facilitates the export of the enriched attack chain in multiple formats and routes the output to designated applications for further analysis or emulation.

Paul, Tony [Idaho National Laboratory (INL), Idaho↗
NASA NTRS1991

Real-time simulation of helicopter air-to-air combat

The AUTOMAN computer program develops automated maneuvering decisions for helicopters during air-to-air combat over hilly terrain. Recently, the capabilities of this program have been extended and enhanced significantly. The revised program was installed at the NASA Ames manned flight-simulation facility to drive a computer-generated image of an enemy helicopter, thereby providing an adversary for the human pilot. Maneuvers are selected by employing game theory. Enhancements include a guidance law for target acquisition when a firing opportunity arises; fire-control sequence logic; improved low-flying capabilities; line-of-sight computations for the cockpit field-of-view, terrain obstructions, and visual range limits; use of terrain for masking; air-to-air collision-avoidance maneuvers; decision on dispensing flares and chaff; and adjustable levels of pilot experience. The program was found to be extremely useful for both rotorcraft handling-quality evaluations and air-to-air combat training.

Austin, Fred↗
NASA NTRSDate not supplied

NEUROSPF: A Tool For the Symbolic Analysis of Neural Networks

This paper presents NEUROSPF, a tool for the symbolic analysis of neural networks. Given a trained neural network model, the tool extracts the architecture and model parameters and translates them into a Java representation that is amenable for analysis using the Symbolic PathFinder symbolic execution tool. Notably, NEUROSPF encodes specialized peer classes for parsing the model’s parameters, thereby enabling efficient analysis. With NEUROSPF the user has the flexibility to specify either the inputs or the network internal parameters as symbolic, promoting the application of program analysis and testing approaches from software engineering to the field of machine learning. For instance, NEUROSPF can be used for coverage-based testing and test generation, finding adversarial examples and also constraint-based repair of neural networks, thus improving the reliability of neural networks and of the applications that use them.

neural networks↗
DOE OSTI2019

Highly-scalable, physics-informed GANs for learning solutions of stochastic PDEs

Uncertainty quantification for forward and inverse problems is a central challenge across physical and biomedical disciplines. We address this challenge for the problem of modeling subsurface flow at the Hanford Site by combining stochastic computational models with observational data using physics-informed GAN models. The geographic extent, spatial heterogeneity, and multiple correlation length scales of the Hanford Site require training a computationally intensive GAN model to thousands of dimensions. We develop a highly optimized implementation that scales to {\herogpucount} NVIDIA Volta GPUs. We develop a hierarchical scheme based on a multi-player game-theoretic approach for exploiting domain parallelism, map discriminators and generators to multiple GPUs, and employ efficient communication schemes to ensure training stability and convergence. Our implementation scales to {\heronodecount} nodes on the Summit supercomputer with a \weakscalingeff\% scaling efficiency, achieving peak and sustained half-precision rates of {\heropeakperf} PF/s and {\herosustainedperf} PF/s.

physics informed ML, machine learning, generative ↗
DOE OSTI2021

Novel usage of deep learning and high-performance computing in long-baseline neutrino oscillation experiments

Mención Internacional en el título de doctorDeep-learning methods are playing a crucial role in numerous scientific and industrialapplications. Over the past two decades, these techniques have helped in the collection,reconstruction, and analysis of large data samples in particle physics experiments. Themain topic of this PhD research is the study of deep-learning techniques in long-baselineneutrino oscillation experiments. Neutrinos are mysterious light elementary particles,and their investigation is essential to shed light on some of the remaining open questionsin physics. The work presented here describes an algorithm based on a convolutionalneural network developed to provide highly accurate and efficient selections of electronneutrino and muon neutrino interactions in the Deep Underground Neutrino Experiment(DUNE). With this algorithm, the electron neutrino (antineutrino) selection efficiencypeaks at 90% (94%) and exceeds 85% (90%) for reconstructed neutrino energies between2-5 GeV. The selection efficiency for muon neutrino (antineutrino) interactions is foundto have a maximum of 96% (97%) and exceeds 90% (95%) efficiency for reconstructedneutrino energies above 2 GeV. When considering all electron neutrino and antineutrinointeractions as signal (both those appearing from oscillations and those intrinsic tothe beam), a selection purity of 90% is achieved. These event selections are criticalto maximise the sensitivity of the experiment to CP-violating effects, key to furtherunderstand the matter-antimatter asymmetry of the Universe.In high-energy physics experiments, deep learning has also been explored for producingfast simulations and physically-motivated manipulations of simulated images. Some ofthose simulations, such as the light production and detection, are very computationallyexpensive and require novel methods to produce the necessary samples while controllingthe varied underlying physics model parameters. To do so, we invented the model-assistedgenerative adversarial network (MAGAN), first validated on simple generic case studiesand then successfully applied to the DUNE photon-detector simulation.Moreover, we also developed graph neural networks for 3D-voxel classification ofambiguities and optical crosstalk for a different particle physics experiment, most preciselyfor the proposed SuperFGD. This novel 3D-granular plastic-scintillator neutrino detectorwill be used to upgrade the near detector of the T2K neutrino oscillation experiment, and our method reports efficiencies and purities of 94-96% per event in the classificationof particle track voxels.Due to the growth and complexity of deep neural networks, researchers have beeninvestigating techniques to train those networks in a more computationally-efficient way.Many efforts have been made by the community to optimise deep-learning models byparallelising or distributing their training computation across multiple devices. In thisthesis, we study an approach based on data locality for those neural networks that cannotbenefit from scaling their computation due to a significant bottleneck in the data I/O.The research also includes a detailed study on the performance of deep neural networkson hardware accelerator boards.Los métodos de aprendizaje profundo son cada vez más utilizados en numerosas aplicacionescientíficas e industriales hoy en día. Durante las dos últimas décadas, estastécnicas se han empleado en la recolección, reconstrucción y análisis de la gran cantidadde datos generados por experimentos de física de partículas. El tema principal de estatesis doctoral es el uso de estos modelos de aprendizaje profundo en experimentos defísica de neutrinos, en concreto en los experimentos de larga distancia DUNE y T2K. Losneutrinos, partículas fundamentales neutras, de las más ligeras del Universo, pueden serclave para explicar algunas de las cuestiones todavía sin resolver en física fundamental.Entre las diferentes contribuciones que esta tesis ha hecho a su estudio, cabe destacar eldesarrollo de un algoritmo basado en una red de neuronas convolucional para seleccionarcon gran eficiencia y precisión las interacciones de neutrinos electrónicos y muónicos enel Deep Underground Neutrino Experiment (DUNE). La eficiencia de selección obtenidapara neutrinos (antineutrinos) electrónicos alcanza un máximo del 90% (94%) y supera el85% (90%) para neutrinos con energías reconstruidas en el rango 2-5 GeV. La selección deneutrinos (antineutrinos) muónicos tiene una eficiencia máxima del 96% (97%) y excedeel 90% (95%) para neutrinos con energías reconstruidas de más de 2 GeV. Considerandocomo señal todas las interacciones de neutrinos y antineutrinos electrónicos (procedentestanto de oscilaciones como intrínsecos en el haz inicial), se logra una pureza en la seleccióndel 90%. Dichas selecciones de eventos son fundamentales para maximizar la sensibilidaddel experimento a los efectos de violació...

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2020

TensorFlow Quantum: A Software Framework for Quantum Machine Learning

We introduce TensorFlow Quantum (TFQ), an open source library for the rapid prototyping of hybrid quantum-classical models for classical or quantum data. This framework offers high-level abstractions for the design and training of both discriminative and generative quantum models under TensorFlow and supports high-performance quantum circuit simulators. We provide an overview of the software architecture and building blocks through several examples and review the theory of hybrid quantum-classical neural networks. We illustrate TFQ functionalities via several basic applications including supervised learning for quantum classification, quantum control, simulating noisy quantum circuits, and quantum approximate optimization. Moreover, we demonstrate how one can apply TFQ to tackle advanced quantum learning tasks including meta-learning, layerwise learning, Hamiltonian learning, sampling thermal states, variational quantum eigensolvers, classification of quantum phase transitions, generative adversarial networks, and reinforcement learning. We hope this framework provides the necessary tools for the quantum computing and machine learning research communities to explore models of both natural and artificial quantum systems, and ultimately discover new quantum algorithms which could potentially yield a quantum advantage.

Broughton, Michael↗
DOE OSTI2024

Resilient Control of Networked Microgrids Using Vertical Federated Reinforcement Learning: Designs and Real-Time Test-Bed Validations

Improving system-level resiliency of networked microgrids against adversarial cyber-attacks is an important aspect in the current regime of increased inverter-based resources (IBRs). To achieve that, this paper contributes in designing a hierarchical control layer, in conjunction with the existing control layers, resilient to adversarial attack signals. Considering model complexities, unknown dynamical behaviors of IBRs, and privacy issues regarding data sharing in multi-party-owned microgrids, designing such a control layer is non-trivial. Here, to tackle these issues, a novel federated reinforcement learning (Fed-RL) method is proposed. To grasp the interconnected dynamics of networked microgrids, the paper develops Federated Soft Actor-Critic (FedSAC) algorithm following the vertical structure of implementing Fed-RL. Next, utilizing the OpenAI Gym interface, we built a custom set-up in GridLAB-D/HELICS co-simulation platform, named Resilient RL Co-simulation (ResRLCoSIM), to train the RL agents with IEEE 123-bus benchmark comprising 3 interconnected microgrids. Finally, the learned policies in the simulation are transferred to the real-time hardware-in-the-loop (HIL) test-bed developed using the high-fidelity Hypersim platform. Finally, experiments show that the simulator-trained RL controllers achieve desirable performance with the test-bed platform, validating the minimization of the sim-to-real gap.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2022

Deep reinforcement learning with online data augmentation to improve sample efficiency for intelligent HVAC control

Deep Reinforcement Learning (DRL) has started showing success in real-world applications such as building energy optimization. Much of the research in this space utilized simulated environments to train RL-agent in an offline mode. Very few research have used DRL-based control in real-world systems due to two main reasons: 1) sample efficiency challenge---DRL approaches need to perform a lot of interactions with the environment to collect sufficient experiences to learn from, which is difficult in real systems, and 2) comfort or safety related constraints---user's comfort must never or at least rarely be violated. In this work, we propose a novel deep Reinforcement Learning framework with online Data Augmentation (RLDA) to address the sample efficiency challenge of real-world RL. We used a time series Generative Adversarial Network (TimeGAN) architecture as a data generator. We further evaluated the proposed RLDA framework using a case study of an intelligent HVAC control. With a ≈28% improvement in the sample efficiency, RLDA framework lays the way towards increased adoption of DRL-based intelligent control in real-world building energy management systems.

Kurte, Kuldeep↗
DOE OSTI2023

RX-ADS: Interpretable Anomaly Detection Using Adversarial ML for Electric Vehicle CAN Data

Recent year has brought considerable advancements in Electric Vehicles (EVs) and associated infrastructures/ communications. Intrusion Detection Systems (IDS) are widely deployed for anomaly detection in such critical infrastructures. This paper presents an Interpretable Anomaly Detection System (RX-ADS) for intrusion detection in CAN protocol communication in EVs. Contributions include: 1) window based feature extraction method; 2) deep Autoencoder based anomaly detection method; and 3) adversarial machine learning based explanation generation methodology. The presented approach was tested on two benchmark CAN datasets: OTIDS and Car Hacking. The anomaly detection performance of RX-ADS was compared against the state-of-the-art approaches on these datasets: HIDS and GIDS. The RX-ADS approach presented performance comparable to the HIDS approach (OTIDS dataset) and has outperformed HIDS and GIDS approaches (Car Hacking dataset). Further, the proposed approach was able to generate explanations for detected abnormal behaviors arising from various intrusions. Furthermore, these explanations were later validated by information used by domain experts to detect anomalies. Other advantages of RX-ADS include: 1) the method can be trained on unlabeled data; 2) explanations help experts in understanding anomalies and root course analysis, and also help with AI model debugging and diagnostics, ultimately improving user trust in AI systems.

42 ENGINEERING↗
DOE OSTI2022

Entangling Quantum Generative Adversarial Networks

Generative adversarial networks (GANs) are one of the most widely adopted machine learning methods for data generation. In this work, we propose a new type of architecture for quantum generative adversarial networks (an entangling quantum GAN, EQ-GAN) that overcomes limitations of previously proposed quantum GANs. Leveraging the entangling power of quantum circuits, the EQ-GAN converges to the Nash equilibrium by performing entangling operations between both the generator output and true quantum data. In the first multiqubit experimental demonstration of a fully quantum GAN with a provably optimal Nash equilibrium, we use the EQ-GAN on a Google Sycamore superconducting quantum processor to mitigate uncharacterized errors, and we numerically confirm successful error mitigation with simulations up to 18 qubits. Finally, we present an application of the EQ-GAN to prepare an approximate quantum random access memory and for the training of quantum neural networks via variational datasets.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

On the effectiveness of neural operators at zero-shot weather downscaling

Machine-learning (ML) methods have shown great potential for weather downscaling. These data-driven approaches provide a more efficient alternative for producing high-resolution weather datasets and forecasts compared to physics-based numerical simulations. Neural operators, which learn solution operators for a family of partial differential equations, have shown great success in scientific ML applications involving physics-driven datasets. Neural operators are grid-resolution-invariant and are often evaluated on higher grid resolutions than they are trained on, i.e., zero-shot super-resolution. Given their promising zero-shot super-resolution performance on dynamical systems emulation, we present a critical investigation of their zero-shot weather downscaling capabilities, which is when models are tasked with producing high-resolution outputs using higher upsampling factors than are seen during training. To this end, we create two realistic downscaling experiments with challenging upsampling factors (e.g., 8x and 15x) across data from different simulations: the European Centre for Medium-Range Weather Forecasts Reanalysis version 5 (ERA5) and the Wind Integration National Dataset Toolkit. While neural operator-based downscaling models perform better than interpolation and a simple convolutional baseline, we show the surprising performance of an approach that combines a powerful transformer-based model with parameter-free interpolation at zero-shot weather downscaling. We find that this Swin-Transformer-based approach mostly outperforms models with neural operator layers in terms of average error metrics, whereas an Enhanced Super-Resolution Generative Adversarial Network-based approach is better than most models in terms of capturing the physics of the ground truth data. We suggest their use in future work as strong baselines.

17 WIND ENERGY↗
NASA NTRSDate not supplied

Combining Data with Physical Knowledge for Uncertainty Quantification in Certification and Reliability Analysis

Unifying empirical data with predictive models can enable engineering cost-savings through certification by analysis and reliability-based design. Both concepts require rigorous uncertainty quantification (UQ) and robust understanding and treatment of relevant physics. Combining sampling-based UQ algorithms with high-fidelity simulations creates a computational bottleneck that is often alleviated through the use of machine learning (ML). ML can be used to create computationally efficient surrogates for simulations of complex or high-dimensional physical interactions (e.g., multi-phase interactions associated with melt pools in laser powder bed fusion or spatially-dependent material properties in functionally graded materials). However, negative side effects of ML may include a lack of interpretability and negative correlation between event rarity and simulation accuracy due to a lack of training data. As such, it is important to infuse ML algorithms with physics-based guardrails to provide confidence in their predictions. This talk will provide a brief review of recent NASA research at this intersection of physics-based simulation, ML, and UQ with a focus on certification and reliability analysis.

uncertainty quantification↗
DOE OSTI2020

Conditional Generative Adversarial Networks for Solving Heat Transfer Problems

Generative Adversarial Networks (GANs) have been used as a deep learning approach to solving physics and engineering problems. Using deep learning for these problems is attractive in that reasonably accurate models can be inferred from only raw data, eliminating the need to define the exact physical equations governing a problem. We expand on previous work using GANs to generate steady-state solutions to the two-dimensional heat equation. Using a basic conditional GAN (cGAN), we generate accurate solutions for rectangular domains conditioned on four edge boundary conditions (MAE < 0.5%). For finding steady-state solutions over arbitrary two-dimensional domains (not constrained to rectangles), we use a cGAN designed for image-to-image translation. We train this GAN on various types of geometric domains (circles, squares, triangles, shapes with one circular or rectangular hole), achieving accurate results on test data made up of geometries similar to those in training (MAE < 1%). For both of these GANs, we experiment with different loss function terms, showing that a term using the gradients of solution images significantly improves the basic cGAN but not the image-to-image GAN. Lastly, we show that the image-to-image GAN performs poorly when applied to two-dimensional geometries that vary in structure from training data (MAE < 8% for shapes with multiple holes or different shaped holes). This demonstrates the cGAN's lack of generalizability. While the cGAN is an accurate and computationally efficient method when trained and tested on similarly structured data, it is a much less reliable method when applied to data that is slightly different in structure from the training data.

97 MATHEMATICS AND COMPUTING↗