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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 73 records · Page 4
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Applying deep learning methods to develop new models of molecular charge transfer, nonadiabatic dynamics, and nonlinear spectroscopy in the condensed phase

Photon- and field-induced charge transfer has central importance in the generation and storage of electricity, the novel properties of materials, photo-induced catalysis, and electro-optic activity (e.g., photovoltaic cells, fuel cells, and organic chromophores for use in optical fibers and light-emission diodes). These non-equilibrium electronic and chemical transformations are probed by ultrafast, nonlinear spectroscopies. Accurate simulations play a crucial role in our ability to understand, optimize, and control these transformations. This project applies modern deep learning and machine learning (ML) methods to dramatically improve models of electronic dynamics, electronic-nuclear dynamics, and spectroscopic measurements for improved simulations of chemistry in complex environments, far from equilibrium phenomena, and processes in extreme environments, such as materials exposed to strong or resonant fields. This project develops accurate neural net models that go beyond predictive capability to also provide new insight into the fundamental physics underlying electron and nuclear dynamics. To achieve its objectives, this project explores and develops customized versions of high-capacity deep learning algorithms/models. These techniques are developed with an emphasis on fundamental chemical insight, not just predictive accuracy, to assist the development of the next generation of quantum simulation methods.

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

DOE FAIR Surrogate Benchmarks Supporting AI and Simulation Research (SBI Surrogate Benchmark Initiative) (Final Report)

Computational Science is being revolutionized by integrating AI and simulation and, in particular, by deep learning surrogate models that can replace all or part of traditional large‐scale HPC computations. Such surrogates can achieve remarkable performance improvements, as much as several orders of magnitude, and save both compute time and energy. The Surrogate Benchmark Initiative (SBI) project creates a community repository and FAIR (Findable, Accessible, Interoperable, and Reusable) data ecosystem for HPC application surrogate benchmarks. The SBI team comes from Argonne National Laboratory (ANL), Indiana University (IU), Rutgers University, the University of Tennessee, Knoxville (UTK), and the University of Virginia(UVA). SBI repositories include data, code, and all relevant collateral artifacts, that the science and engineering community needs to use and reuse these data sets and surrogates. SBI repositories generate active research from both participants in SBI and the broader AI and domain science communities. This project develops surrogates that use several different neural nets to learn and quickly infer the results of simulations and data systems and capture them as surrogate benchmarks with a rich set of metadata, covering. Data; Model; Metrics specification; Machine specification; Science, Speed, Power Results, We research FAIR metadata for these benchmarks. We develop application surrogate examples as benchmarks across many fields (ANL, UTK, IU, UVA). We also study non Surrogate benchmarks that have many common features and similar issues regarding FAIRness. We work with MLCommons (UVA, UTK), which is a major machine learning benchmarking activity where we get metadata ontologies, software, and benchmarks, benchmarks have datasets, models, and metadata, and they need a technical framework developed by UTK and Rutgers and deployed by UVA. We study features of Surrogates, including performance, training set size, and uncertainty quantification (Rutgers, UVA and IU).

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

FAIR Surrogate Benchmarks Supporting AI and Simulation Research (Final Report)

Computational Science is being revolutionized by integrating AI and simulation and, in particular, by deep learning surrogate models that can replace all or part of traditional large‐scale HPC computations. Such surrogates can achieve remarkable performance improvements, as much as several orders of magnitude, and save both compute time and energy. The Surrogate Benchmark Initiative (SBI) project creates a community repository and FAIR (Findable, Accessible, Interoperable, and Reusable) data ecosystem for HPC application surrogate benchmarks. The SBI team comes from Argonne National Laboratory (ANL), Indiana University (IU), Rutgers University, the University of Tennessee, Knoxville (UTK), and the University of Virginia (UVA). SBI repositories include data, code, and all relevant collateral artifacts that the science and engineering community need to use and reuse these data sets and surrogates. SBI repositories generate active research from both the participants in SBI and the broad community of AI and domain scientists. This project develops surrogates that use several different neural nets to learn and quickly infer the results of simulations and data systems and captures them as surrogate benchmarks with a rich set of metadata covering: Data; Model; Metrics specification; Machine specification; and Science, Speed, and Power Results. We research FAIR metadata for these benchmarks. We develop application surrogate examples as benchmarks across many fields (ANL, UTK, IU, UVA). We also study non-Surrogate benchmarks that have many common features and similar issues as regards FAIRness. We work with MLCommons (UVA, UTK), which is a major machine learning benchmarking activity where we get metadata ontologies, software, and benchmarks, Benchmarks have datasets, models, and metadata and they need a technical framework developed by UTK and Rutgers and deployed by UVA. We study features of Surrogates including performance, training set size, and uncertainty quantification (Rutgers, UVA and IU).

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Active Optical Control with Machine Learning: A Proof of Concept for the Vera C. Rubin Observatory

Abstract The Active Optics System of the Vera C. Rubin Observatory (Rubin) uses information provided by four wave front sensors to determine deviations between the reconstructed wave front and the ideal wave front. The observed deviations are used to adjust the control parameters of the optical system to maintain image quality across the 3.°5 field of view. The baseline approach from the project is to obtain amplitudes of the Zernike polynomials describing the distorted wave front from out-of-focus images collected by the wave front sensors. These Zernike amplitudes are related via an “influence matrix” to the control parameters necessary to correct the wave front. In this paper, we use deep-learning methods to extract the control parameters directly from the images captured by the wave front sensors. Our neural net model uses anti-aliasing pooling to boost performance, and a domain-specific loss function to aid learning and generalization. The accuracy of the control parameters derived from our model exceeds Rubin requirements even in the presence of full-moon background levels and mis-centering of reference stars. Although the training process is time consuming, model evaluation requires only a few milliseconds. This low latency should allow for the correction of the optical configuration during the readout and slew interval between successive exposures.

Astronomy & Astrophysics↗
DOE OSTI2020

Big Sky Regional Carbon Sequestration Partnership (Phase III Final Scientific/Technical Report)

The Big Sky Carbon Sequestration Partnership (BSCSP) pursued a Phase III demonstration project at Kevin Dome in north central Montana. Kevin Dome covers approximately 700 square miles and is a naturally occurring CO 2 reservoir that is flanked by oil and gas fields. The carbon dioxide (CO 2 ) is in the upper Devonian Duperow (carbonate) formation and does not reach the spill point of the dome; therefore, the dome has potential as a CO 2 sequestration reservoir, a CO 2 supply, or as both if anthropogenic sources and enhanced oil recovery (EOR) operations are associated with the dome. Kevin Dome could potentially act as a buffer to continue accepting anthropogenic CO 2 when EOR flooding operations are interrupted or completed. The project objective was to produce one million tonnes of CO 2 from the gas cap of Kevin Dome, pipe it laterally, inject, and re-store it in the brine leg of the same formation to test the hub / buffer storage concept. This was to be accomplished by drilling up to five production wells, building a short pipeline and compression facilities, and drilling an injection well and several monitoring wells. BSCSP commenced outreach and site characterization activities including acquiring baseline data for near-surface insurance monitoring, acquiring 3-dimensional, 9-component surface seismic over the project area, drilling two test wells (one in the production area and one in the injection area), coring key intervals, and performing comprehensive logging. Well tests of those wells revealed two barriers to the project. The production (Danielson 33-17) well was perforated in multiple zones but failed to produce any significant CO 2 . This was despite being drilled in the near vicinity of a historic well that had produced 3700 MCF per day in a drill stem test. Modeling indicated that this was likely due to a phase change during production causing a temperature drop resulting in hydrate and/or water ice formation that clogged the formation. Tests of the injection zone (Wallewein 22-1) well indicated total dissolved solids (TDS) slightly below the EPA required 10,000 parts per million (ppm) for a Class VI underground injection control permit. While the project was initiated before Class VI rules were promulgated, and this was an experimental project (seemingly qualified for a Class V permit), the Environmental Protection Agency (EPA) indicated that injection would require a Class VI permit. The low salinity result was unexpected as contours plotted based on regional formation water quality data indicated an expected TDS above 20,000 ppm, and wells between the recharge zone and the Wallewein well tested above 10,000 ppm. Faced with the inability to obtain an injection permit, the demonstration project could not proceed. However, the project had generated valuable samples and data on a large natural analog including 32 sq. mi. of 3-D, 9-C seismic, 430 ft. of carbonate core covering seven different depositional environments taken from areas with, and without the presence of CO 2 , 30 ft. of core of two caprocks, a tight carbonate and an anhydrite, a full set of modern logs on both wells, and well tests. DOE decided to re-scope the project around completing studies utilizing this data. This report covers both the initial scope and the re-scope (Task / Section numbers preceded with an R). While the report covers a wide range of project activities, highlights of this work include: Development of a geostatic model using neural nets to match well logs to facies and using multi-waveform seismic to inform reservoir heterogeneity; Unique mechanical testing of permeability – stress relationship in two caprock materials; Development of full waveform inversion to generate a high resolution velocity model; Model development for dual permeability (fracture and matrix) systems to better account for matrix-matrix interactions; Joint seismic wave inversion (including the first quadr-joint inversion) exhibiting better imaging of a challenging reservoir zone in stiff rock; Core flow and core flood results on a reactive carbonate; and Innovative laboratory measurements of seismic response of fractured core as a function of fluid fill.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2024

Studies of e+e- Pair Photo-Production on Proton Target at 8 GeV in the GlueX Experiment

Lepton pair production has played an important role in both nuclear and particle physics, being among the earliest calculations utilizing QED, and seen most famously in the discovery of the J= at BNL. A technique is presented for measuring the linear polarization of GeV scale photon beams through the detection of e+e? pairs photo-produced in the target. This technique is applied to the analysis of GlueX data on proton target. Simulation predicts the analyzing power for pair production to be :5725 ? 0:0025 for the GlueX data, and the analysis of experimental data gives a linear polarization of approximately 35%, in good agreement with other measurements of beam polarization. The pair production technique is complementary to other electromagnetic and hadronic measurements of beam polarization, and is generally applicable in experiments that allow for forward angle electron and positron identi?fication and tracking. To facilitate this study, a neural net was trained for e=? separation to eliminate the pion background. Further, it is demonstrated that these e+e? pairs are sensitive to the proton charge form factor, which opens up the possibility for a new method to measure the proton RMS charge radius.

Schick, Andrew↗
DOE OSTI2023

Peri-Net-Pro: the neural processes with quantified uncertainty for crack patterns

Abstract This paper develops a deep learning tool based on neural processes (NPs) called the Peri-Net-Pro, to predict the crack patterns in a moving disk and classifies them according to the classification modes with quantified uncertainties. In particular, image classification and regression studies are conducted by means of convolutional neural networks (CNNs) and NPs. First, the amount and quality of the data are enhanced by using peridynamics to theoretically compensate for the problems of the finite element method (FEM) in generating crack pattern images. Second, case studies are conducted with the prototype microelastic brittle (PMB), linear peridynamic solid (LPS), and viscoelastic solid (VES) models obtained by using the peridynamic theory. The case studies are performed to classify the images by using CNNs and determine the suitability of the PMB, LBS, and VES models. Finally, a regression analysis is performed on the crack pattern images with NPs to predict the crack patterns. The regression analysis results confirm that the variance decreases when the number of epochs increases by using the NPs. The training results gradually improve, and the variance ranges decrease to less than 0.035. The main finding of this study is that the NPs enable accurate predictions, even with missing or insufficient training data. The results demonstrate that if the context points are set to the 10th, 100th, 300th, and 784th, the training information is deliberately omitted for the context points of the 10th, 100th, and 300th, and the predictions are different when the context points are significantly lower. However, the comparison of the results of the 100th and 784th context points shows that the predicted results are similar because of the Gaussian processes in the NPs. Therefore, if the NPs are employed for training, the missing information of the training data can be supplemented to predict the results.

Mathematics↗
DOE OSTI2023

Phonon-informed Neural Thermal Scattering (NeTS) Optimization for Crystalline Graphite and Beryllium Metal

Fast neutrons born from fission lose energy through scattering interactions in the process of slowing-down. As neutrons thermalize to the order of $k$ $b$ $T$ (where $k$ $b$ is the Boltzmann constant, and $T$ is the temperature of the medium), their de Broglie wavelength and energy approaches the order of inter-atomic spacing and quantized lattice vibrations, i.e., phonons. At thermal energies, the thermal scattering law (TSL), i.e., $S$($α, β$), captures crystal binding contributions to the total reaction rate, or cross section. This dimensionless material property describes the energy ($β$) and momentum ($α$) exchanges available in a medium. Currently, $S$($α, β$) is evaluated in the Full Law Analysis Scattering System Hub (FLASSH) code for discrete inputs and stored as ENDF/B File 7 for 0-phonon elastic (MT 2) and n-phonon inelastic (MT 4) processes. Further processing recasts $S$($α, β$) into cumulative distribution functions for sampling post-collision scattering kinematics. In practice, interpolation schemes are employed to access data between tabulated values. An improvement to this juncture of the nuclear data pipeline is supplying cross sections on-the-fly (OTF), as has been developed for the un-resolved resonance region to minimize non-physical interpolation errors. This capability may improve simulation accuracy for accident and transient analyses, where rapidly varying changes in temperature and pressure are difficult to predict beforehand. To do so, deep artificial neural networks (ANNs) can be employed which collapse non-linear, complex data into a lightweight dictionary of neural weights and biases. This has been successfully demonstrated for the hydrogen in light water $S$($α, β$) dataset in the form of a Neural Thermal Scattering (NeTS) module. In this work, the NeTS framework is extended to consider the impact of material-dependent dynamical features on optimal neural pre-processing and architecture design decisions, such as number of neurons per hidden layer, residual skip connections and neural depth. New NeTS modules for crystalline graphite and beryllium metal illuminate a novel correlation between dynamical nonlinearity and optimal neural parametrization when deploying $S$($α, β$) on-the-fly.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2022

Neural Thermal Scattering (NeTS) Modules for Graphite & Beryllium [Slides]

This presentation discusses the motivations behind the project, which include facilitating compact formulation for TSL data, extending AI knowledge, and providing advanced reactor simulation framework. It also discusses how the project accounts for beryllium and graphite complexity, including atomistic, dynamical, and neural. Additionally, new material-informed neural thermal scattering (NeTS) Modules are examined. The implications, conclusions, and plan future work for the project are also discussed.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2024

Embedding Neural Thermal Scattering (NeTS) Modules in SERPENT for Higher Fidelity Advanced Reactor Analysis

When a neutron born in fission thermalizes to the order of $k$ $B$ $T$, it’s de-Broglie wavelength and energy approach the order of inter-atomic spacing and elementary lattice oscillations, respectively. $S$($a,β,t$) or the scattering law, uuantify these temperature-dependent crystallographic contributions to total cross section (or reaction rate). In a Monte Carlo analysis, cumulative distribution functions (CDFs) of $S$($a,β,t$) are loaded to memory from “A Compact ENDF” (ACE) files for stochastically selecting thermal scattered neutron trajectories. In this work, novel neural thermal scattering (NeTS) modules for $S$($a,β,t$) CDFs are designed, trained, serialized and embedded within SERPENT using Python’s limited C-API for on-the-fly deployment of crystalline graphite $S$($a,β,t$) sampling. Torchscript tracing and Numba just-in-time (JIT) compilation streamline neural inference on NVIDIA GPUs with CUDA libraries. Demonstrations of bare sphere thermalization of fast and thermal sources show excellent agreement between embedded NeTS in SERPENT and MCNP. With an explicit model of the reactor, NeTS can predict on-the-fly changes in TREAT neutron spectra as a function of local temperature, which can serve to improve transient and accident predictions in a multiphysics analysis framework. This framework can be further extended to account on-the-fly for changes in local graphitic microstructure to scattering cross sections, and outlines a novel coupling of modern machine learning with state-of-the-art reactor physics methods.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Deep learning model to detect various synchrophasor data anomalies

High-density synchrophasors provide valuable information for power grid situational awareness, operation and control. Unfortunately, due to factors including communication instability and hardware failure, their data quality can be greatly deteriorated by anomalies. Since the anomalies can impact the performance of the synchrophasor applications, it is of paramount significance to propose a model to detect anomalies in synchrophasor. In this study, a convolutional neural network model is established to detect and classify the anomalies in the synchrophasor measurements. Additionally, four types of anomalies observed in actual synchrophasors including erroneous patterns, random spikes, missing points and high-frequency interferences are considered in this study. The proposed model is extensively evaluated via field-collected measurements from the synchrophasor network in Jiangsu grid, China. The superior performance of the proposed model indicates the great potential of using deep learning for the detection of abnormal synchrophasor measurements.

42 ENGINEERING↗
DOE OSTI2020

Boosting the Performance of Plug-and-Play Priors via Denoiser Scaling

Plug-and-play priors (PnP) is an image reconstruction framework that utilizes an image denoiser as an imaging prior. Unlike traditional regularized inversion, PnP does not require the prior to be expressible in the form of a regularization function. This flexibility enables PnP algorithms to exploit the most effective image denoisers, leading to their state-of-the-art performance in various imaging tasks. However, many powerful denoisers, such as the ones based on convolutional neural networks (CNNs), do not have tunable parameters that would allow controlling their influence within PnP. To address this issue, in this paper, we introduce a scaling parameter that adjusts the magnitude of the denoiser input and output. We theoretical justify the denoiser scaling from the perspectives of proximal optimization, statistical estimation, and consensus equilibrium. Finally, we provide numerical experiments demonstrating the ability of denoiser scaling to systematically improve the performance of PnP for denoising CNN priors that do not have explicitly tunable parameters.

Boosting↗
DOE OSTI2024

UFNet: Joint U-Net and Fully Connected Neural Network to Bias Correct Precipitation Predictions from

Paper information. Shuang Yu, Indrasis Chakraborty, Gemma J. Anderson, Donald D. Lucas, Yannic Lops, and Daniel Galea. UFNet: Joint U-Net and fully connected neural network to bias correct precipitation predictions from climate models. Artificial Intelligence for the Earth Systems, 2024. Overview. This work develops the UFNet methodology to correct E3SM historical precipitation projection bias. The UFNet deep learning framework consists of a two-part architecture: a U-Net convolutional network to capture the spatiotemporal distribution of precipitation and a fully connected network to capture the distribution of higher-order statistics. The joint network, termed UFNet, can simultaneously improve the spatial structure of the modeled precipitation and capture the distribution of extreme precipitation values. Below we provide guidance for applying UFNet to correct the Energy Exascale Earth System Model (E3SM; Golaz et al. 2019) daily precipitation projection over the contiguous United States (CONUS). Getting started 1. Obtain the historical climate simulation and observation data. The E3SM historical simulation data are available through https://aims2.llnl.gov/search/cmip6/. The CPC unified gauge-based analysis of daily precipitation can be found through https://psl.noaa.gov/data/gridded/data.cpc.globalprecip.html. The ECMWF atmospheric reanalysis of the 20th century (ERA-20C) data are available through https://www.ecmwf.int/en/forecasts/datasets/reanalysis-datasets/era-20c. The spatial resolution of E3SM and observed datasets are both regridded to a common 1° resolution grid using conservative interpolation. The regridded E3SM, CPC and ERA-20C with 1° resolution can be found throught ./data/. 2. Train the fully connected network (DNN) Python train_dnn.py 3. Train the UFNet Python train_ufnet.py 4. Evaluation and compared with the baseline Python evaluation.py

Lucas, DonaldD↗
DOE OSTI2026

VRN3P: Variational Recurrent Neural Network Based Net-Load Prediction under High Solar Penetration

This is the final technical report for the SETO-funded VRN3P project (PNNL# 76914). The goal of this project, led by Pacific Northwest National Laboratory (PNNL), in collaboration with Lawrence Livermore National Laboratory (LLNL) and Portland General Electric (PGE), was to develop and validate a deep variational recurrent neural network-based net-load prediction (VRN3P) framework for probabilistic time-series forecasting of day-ahead net-load under high solar penetration scenarios. The project team reports successful design of a novel probabilistic net-load forecasting architecture, comprising of a variational autoencoder and a recurrent neural network, which demonstrates 30% improvement in forecast performance, 60% improvement in training time, and consumes 44% less memory, when compared with conventional baseline models. The team tested the VRN3P model performance on GridLAB-D test-cases representing varying BTM solar penetration levels of 20%, 30%, and 50%, with integrated time-series net-load profiles provided by the utility partner (PGE). The VRN3P model demonstrate <2% hourly MAPE (averaged over the year) for day- ahead net-load forecast on the test scenario with 20% BTM solar. Transfer learning extension of the VRN3P model has demonstrated 8.33× speed-up in training, while still achieving acceptable forecast performance of 2.24% hourly MAPE on the 30% BTM solar penetration test-scenario. A preliminary version of the VRN3P GridAPPS-D™has been developed, along with a web-based interactive user-interface (named ‘Forte’) which has made available on GitHub for public use.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2020

AP-Net: An atomic-pairwise neural network for smooth and transferable interaction potentials

Intermolecular interactions are critical to many chemical phenomena, but their accurate computation using ab initio methods is often limited by computational cost. The recent emergence of machine learning (ML) potentials may be a promising alternative. Useful ML models should not only estimate accurate interaction energies but also predict smooth and asymptotically correct potential energy surfaces. However, existing ML models are not guaranteed to obey these constraints. Indeed, systemic deficiencies are apparent in the predictions of our previous hydrogen-bond model as well as the popular ANI-1X model, which we attribute to the use of an atomic energy partition. As a solution, we propose an alternative atomic-pairwise framework specifically for intermolecular ML potentials, and we introduce AP-Net—a neural network model for interaction energies. The AP-Net model is developed using this physically motivated atomic-pairwise paradigm and also exploits the interpretability of symmetry adapted perturbation theory (SAPT). We show that in contrast to other models, AP-Net produces smooth, physically meaningful intermolecular potentials exhibiting correct asymptotic behavior. Initially trained on only a limited number of mostly hydrogen-bonded dimers, AP-Net makes accurate predictions across the chemically diverse S66x8 dataset, demonstrating significant transferability. On a test set including experimental hydrogen-bonded dimers, AP-Net predicts total interaction energies with a mean absolute error of 0.37 kcal mol−1, reducing errors by a factor of 2–5 across SAPT components from previous neural network potentials. The pairwise interaction energies of the model are physically interpretable, and an investigation of predicted electrostatic energies suggests that the model “learns” the physics of hydrogen-bonded interactions.

Glick, Zachary L. (ORCID:0000000309002849)↗
DOE OSTI2020

NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations

In the last 50 years there has been a tremendous progress in solving numerically the Navier-Stokes equations using finite differences, finite elements, spectral, and even meshless methods. Yet, in many real cases, we still cannot incorporate seamlessly (multi-fidelity) data into existing algorithms, and for industrial-complexity applications the mesh generation is time consuming and still an art. Moreover, solving ill-posed problems (e.g., lacking boundary conditions) or inverse problems is often prohibitively expensive and requires different formulations and new computer codes. Here, we employ physics-informed neural networks (PINNs), encoding the governing equations directly into the deep neural network via automatic differentiation, to overcome some of the aforementioned limitations for simulating incompressible laminar and turbulent flows. We develop the Navier-Stokes flow nets (NSFnets) by considering two different mathematical formulations of the Navier-Stokes equations: the velocity-pressure (VP) formulation and the vorticity-velocity (VV) formulation. Since this is a new approach, we first select some standard benchmark problems to assess the accuracy, convergence rate, computational cost and flexibility of NSFnets; analytical solutions and direct numerical simulation (DNS) databases provide proper initial and boundary conditions for the NSFnet simulations. The spatial and temporal coordinates are the inputs of the NSFnets, while the instantaneous velocity and pressure fields are the outputs for the VP-NSFnet, and the instantaneous velocity and vorticity fields are the outputs for the VV-NSFnet. This is unsupervised learning and, hence, no labeled data are required beyond boundary and initial conditions and the fluid properties. The residuals of the VP or VV governing equations, together with the initial and boundary conditions, are embedded into the loss function of the NSFnets. No data is provided for the pressure to the VP-NSFnet, which is a hidden state and is obtained via the incompressibility constraint without extra computational cost. Unlike the traditional numerical methods, NSFnets inherit the properties of neural networks (NNs), hence the total error is composed of the approximation, the optimization, and the generalization errors. Here, we empirically attempt to quantify these errors by varying the sampling (“residual”) points, the iterative solvers, and the size of the NN architecture. For the laminar flow solutions, we show that both the VP and the VV formulations are comparable in accuracy but their best performance corresponds to different NN architectures. The initial convergence rate is fast but the error eventually saturates to a plateau due to the dominance of the optimization error. For the turbulent channel flow, we show that NSFnets can sustain turbulence at , but due to expensive training we only consider part of the channel domain and enforce velocity boundary conditions on the subdomain boundaries provided by the DNS data base. We also perform a systematic study on the weights used in the loss function for balancing the data and physics components, and investigate a new way of computing the weights dynamically to accelerate training and enhance accuracy. In the last part, we demonstrate how NSFnets should be used in practice, namely for ill-posed problems with incomplete or noisy boundary conditions as well as for inverse problems. We obtain reasonably accurate solutions for such cases as well without the need to change the NSFnets and at the same computational cost as in the forward well-posed problems. As a result, we also present a simple example of transfer learning that will aid in accelerating the training of NSFnets for different parameter settings.

97 MATHEMATICS AND COMPUTING↗