MATEY: multiscale adaptive foundation models for spatiotemporal physical systems
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Zhang, Pei [ORNL] (ORCID:0000000283510529)↗
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Modeling Communication Infrastructures of Cyber Physical Systems
This effort explores modeling cyber infrastructure, including communication devices like switches, routers, sensors, and controllers, and physical media attributes like propagation of radio signals, in coordination with power distribution system model layouts. To do this, the project studies real-world configurations to define axioms of how different communication media is deployed with control equipment. These axioms will be used to develop tools to generate realistic cyber infrastructure models from starting power system models. This effort leverages and builds on the co-simulation platform developed by the other RD2C projects. The axioms and tools will be validated and demonstrated with the NS3 simulation tool as part of a co-simulation to show the behavior/impacts of cyber infrastructure on control operations.
24 POWER TRANSMISSION AND DISTRIBUTION↗
Uncertainty Quantification for Data-Driven Machine Learning Models in Nuclear Engineering Applications: Where We Are and What Do We Need?
Machine learning (ML) has been leveraged to tackle a diverse range of tasks in almost all branches of nuclear engineering. Many of the successes in ML applications can be attributed to the recent performance breakthroughs in deep learning, the growing availability of computational power, data, and easy-to-use ML libraries. However, these empirical successes have often outpaced our formal understanding of the ML algorithms. An important but under-rated area is uncertainty quantification (UQ) of ML. ML-based models are subject to approximation uncertainty when they are used to make predictions, due to sources including but not limited to, data noise, data coverage, extrapolation, imperfect model architecture and the stochastic training process. The goal of this paper is to clearly explain and illustrate the importance of UQ of ML. We will elucidate the differences in the basic concepts of UQ of physics-based models and data-driven ML models. Various sources of uncertainties in physical modeling and data-driven modeling will be discussed, demonstrated, and compared. We will also present and demonstrate a few techniques to quantify the ML prediction uncertainties, including Monte Carlo dropout, deep ensemble, Bayesian neural networks, Gaussian Processes and conformal prediction. Lastly, we will discuss the need for building a verification, validation and UQ framework to establish ML credibility.
22 GENERAL STUDIES OF NUCLEAR REACTORS↗
Reduced‐Order Probabilistic Emulation of Physics‐Based Ring Current Models: Application to RAM‐SCB Particle Flux
Abstract In this work, we address the computational challenge of large‐scale physics‐based simulation models for the ring current. Reduced computational cost allows for significantly faster than real‐time forecasting, enhancing our ability to predict and respond to dynamic changes in the ring current, valuable for space weather monitoring and mitigation efforts. Additionally, it can also be used for a comprehensive investigation of the system. Thus, we aim to create an emulator for the Ring current‐Atmosphere interactions Model with Self‐Consistent magnetic field (RAM‐SCB) particle flux that not only improves efficiency but also facilitates forecasting with reliable estimates of prediction uncertainties. The probabilistic emulator is built upon the methodology developed by Licata and Mehta (2023), https://doi.org/10.1029/2022sw003345 . A novel discrete sampling is used to identify 30 simulation periods over 20 years of solar and geomagnetic activity. Focusing on a subset of particle flux, we use Principal Component Analysis for dimensionality reduction and Long Short‐Term Memory (LSTM) neural networks to perform dynamic modeling. Hyperparameter space was explored extensively resulting in about 5% median symmetric accuracy across all data sets for one‐step dynamic prediction. Using a hierarchical ensemble of LSTMs, we have developed a reduced‐order probabilistic emulator (ROPE) tailored for time‐series forecasting of particle flux in the ring current. This ROPE offers accurate predictions of omnidirectional flux at a single energy with no pitch angle information, providing robust predictions on the test set with an error score below 11% and calibration scores under 8% with bias under 2% providing a significant speed up as compared to the full RAM‐SCB run.
79 ASTRONOMY AND ASTROPHYSICS↗
A Roadmap for a Lightning Modeling Grand Challenge
This document is a roadmap for building an interconnected model of the physical processes that produce a lightning discharge, and its observable optical and radio signals. We call this a Lightning Modeling Grand Challenge, recognizing that significant effort and coordination of human and financial resources is required to realize the capability. The roadmap serves to outline the coordination of resources necessary to enable stitching together existing knowledge and model components to make a lightning prediction, and to test these predictions with observations. Such a capability does not currently exist. The roadmap is motivated not only by a spirit of scientific inquiry, but by practical challenges faced by US Federal and societal stakeholders. Advancements in lightning observations have outpaced our tests of integrated understanding, leaving many stakeholders unsure how to design their missions to properly detect and discriminate lightning, and unsure how to apply the sometimes-disagreeing lightning signals from diverse instruments. The time is right to connect existing theories and models to support stakeholders in understanding the signals they observe, for needs as diverse as climate monitoring, national security, weather forecasting, public safety, and protection of natural and built environments. The roadmap’s two main technical sections describe the components of a linked physical model, followed by a description of models of lightning signals and sensors that are driven by outputs from the physical model. The goal is to predict the time-varying physical properties of lightning that are self-consistent with the thunderstorm’s structure and dynamics. These lightning signals then propagate through the storm, with realistic dispersion and attenuation, to receivers on the ground or in space. At a high level, the model begins with weather (cloud) model output, including explicit prediction of the electrification of cloud particles. The cloud’s electrical structure drives a model of lightning physics, from initiation, through channel development, and discharges along those channels. Key lightning parameters, such as the temperature and currents in the channel, and their space and time distribution, are then used to produce optical and electromagnetic signal sources that propagate to modeled receivers. This architecture therefore generates a dataset suitable for comparison to existing and envisioned observing systems. The need for additional measurements and field campaigns to support model development is described. In each model sub-component, inputs, outputs, uncertainties, evaluation methods, and next steps are summarized, interleaved with references to the scientific literature. Identifying boundaries between the model sub-components aids in segmenting an integrated, complex model into practical work packages and system sub-components, allowing a diverse team to contribute and maintain the system. We estimate that at least five years of effort and a $\$$10M initial investment is necessary to make a significant step forward. Mechanisms to facilitate community coordination, including annual workshops and open-source code repositories, are described.
54 ENVIRONMENTAL SCIENCES↗
Generalizing the compressible pairwise interaction extended point-particle model
Ejecta physics plays an important role in material interfaces that are impacted by a strong shock wave. When a shock impacts a rough surface of solid material and melts it, the Richtmyer–Meshkov instability grows perturbations on the surface, which can eject particles. After release, the ejecta travel through the post-shock compressible flow. To accurately simulate a large number of ejecta particles, an Euler–Lagrange approach is preferred, which requires modeling the subgrid-scale physics involved with fluid–particle interactions. We generalize the previous work from Hsiao et al. (2023) to consider systems of moving particles subject to any loading shock. The following improvements were made: (1) Particles are allowed to move relative to each other (2) Non-planar shocks are accounted for along with allowing for variable shock speeds. As a result, the generalized algorithm was tested with particle-resolved simulations for canonical test cases. The results of these tests are discussed and analyzed.
97 MATHEMATICS AND COMPUTING↗
Improving the National Solar Radiation Database (NSRDB) Using a Physics-Based Direct Normal Irradiance (DNI) Model
The National Solar Radiation Database (NSRDB) is a widely used resource providing satellite-derived solar data across the United States and globally. While the NSRDB employs a physical model for computing global horizontal irradiance (GHI), its current method for estimating cloudy-sky direct normal irradiance (DNI) relies on surface observations and empirical models. Recently, a novel physics-based approach, the Fast All-Sky Radiation Model for Solar applications with DNI (FARMS-DNI), was developed to enhance the DNI forecasting. FARMS-DNI incorporates both direct and scattered solar radiation within the circumsolar region, resulting in improved day-ahead DNI predictions when integrated into the Weather Research and Forecasting model with Solar extensions (WRF-Solar). This study integrates FARMS-DNI into the NSRDB algorithm to generate high-resolution DNI data from satellite resources. Our findings reveal that FARMS-DNI effectively mitigates the substantial DNI overestimation present in the conventional NSRDB across surface sites, particularly in conditions categorized as cloudy overcast. Consequently, this innovative model substantially enhances the overall accuracy of the NSRDB.
Xie, Yu↗
Improving the National Solar Radiation Database (NSRDB) Using a Physics-Based Direct Normal Irradiance (DNI) Model: Preprint
The National Solar Radiation Database (NSRDB) is a widely used resource providing satellite-derived solar data across the United States and globally. While the NSRDB employs a physical model for computing global horizontal irradiance (GHI), its current method for estimating cloudy-sky direct normal irradiance (DNI) relies on surface observations and empirical models. Recently, a novel physics-based approach, the Fast All-Sky Radiation Model for Solar applications with DNI (FARMS-DNI), was developed to enhance the DNI forecasting. FARMS-DNI incorporates both direct and scattered solar radiation within the circumsolar region, resulting in improved day-ahead DNI predictions when integrated into the Weather Research and Forecasting model with Solar extensions (WRF-Solar). This study integrates FARMS-DNI into the NSRDB algorithm to generate high-resolution DNI data from satellite resources. Our findings reveal that FARMS-DNI effectively mitigates the substantial DNI overestimation present in the conventional NSRDB across surface sites, particularly in conditions categorized as cloudy overcast. Consequently, this innovative model substantially enhances the overall accuracy of the NSRDB.
DNI↗
Modelling the impact of host galaxy dust on type Ia supernova distance measurements
Type Ia Supernovae (SNe Ia) are a critical tool in measuring the accelerating expansion of the universe. Recent efforts to improve these standard candles have focused on incorporating the effects of dust on distance measurements with SNe Ia. In this paper, we use the state-of-the-art Dark Energy Survey 5 year sample to evaluate two different families of dust models: empirical extinction models derived from SNe Ia data and physical attenuation models from the spectra of galaxies. In this work, we use realistic simulations of SNe Ia to forward-model different models of dust and compare summary statistics in order to test different assumptions and impacts on SNe Ia data. Among the SNe Ia-derived models, we find that a logistic function of the total-to-selective extinction R V best recreates the correlations between supernova distance measurements and host galaxy properties, though an additional 0.02 mag of grey scatter is needed to fully explain the scatter in SNIa brightness in all cases. These empirically derived extinction distributions are highly incompatible with the physical attenuation models from galactic spectral measurements. From these results, we conclude that SNe Ia must either preferentially select extreme ends of galactic dust distributions, or that the characterization of dust along the SNe Ia line-of-sight is incompatible with that of galactic dust distributions.
79 ASTRONOMY AND ASTROPHYSICS↗
Two-loop contributions of axion-like particles to electromagnetic and chromomagnetic form factors
Abstract Axions and axion-like particles emerge in many models for physics beyond the Standard Model. Thus, they have gained increasing research interest in both experimental and theoretical physics apart from their original proposition as a solution to the strong CP -problem. Among other aspects it has recently been shown that ALPs can potentially provide a solution to the long-lasting discrepancy between theory and experiment of the anomalous magnetic moment of the muon. Provided that the ALP has flavor-violating couplings to leptons, they can also mediate flavor-violating decays like $$\mu \rightarrow e\gamma $$ μ → e γ . Both processes are mediated through related form factors that we compute to two-loop order. We further show numerical implications of our calculations and how they might affect constraints on ALP couplings derived from experiments.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
Finding Real Uncertainties From Physical Simulations
Modeling strong gravitational lenses is computationally expensive for the complex data from modern and next-generation cosmic surveys. Deep learning has emerged as a promising approach for finding lenses and predicting lensing parameters, such as the Einstein radius. Mean-variance Estimators (MVEs) are a common approach for obtaining aleatoric (data) uncertainties from a neural network prediction. However, neural networks have not been demonstrated to perform well on out-of-domain target data successfully - e.g., when trained on simulated data and applied to real, observational data. In this work, we perform the first study of the efficacy of MVEs in combination with unsupervised domain adaptation (UDA) on strong lensing data. The source domain data is noiseless, and the target domain data has noise mimicking modern cosmology surveys. We find that adding UDA to MVE increases the accuracy on the target data by a factor of about two over an MVE model without UDA. Including UDA also permits much more well-calibrated aleatoric uncertainty predictions. Advancements in this approach may enable future applications of MVE models to real observational data.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
Dynamic probabilistic risk assessment and game theory for cyber security risk analysis in nuclear power plants
Nuclear Power Plants and energy systems have become more prone to cyber-attacks with their digitalization and the increased use of smart equipment. Hence, it is important to quantify the risk associated with cyber-attacks in such systems. Dynamic Probabilistic Risk Assessment which involves studying the evolution of a system due to random events and operator and attacker actions during a cyber-attack by employing a physics-based model of the system is a suitable framework to quantify cybersecurity risk in nuclear power plants. In addition to the plant dynamics, it is also important to model the strategies of the attackers and plant operators for an effective cybersecurity risk assessment. Game theory provides a set of necessary tools to model such strategic interactions. In this research, a framework that integrates dynamic probabilistic risk assessment with game theory for cybersecurity risk analysis in nuclear power plants is presented. The mathematical formulation is derived based on the theory of continuous event trees. We propose a game theory based action model, that utilizes physics-based rewards to define the strategies of attackers and operators at every decision epoch. As a case study, the risk associated with cyber-attacks on the digital components in the secondary side of a pressurized water reactor is studied using a reduced order model. A set of attacker actions and a set of operator actions are defined for the system. The operator and attacker interactions were modelled using simultaneous game, their action policies were computed using the concept of mixed strategy Nash equilibrium and the evolution of the system was studied.
97 MATHEMATICS AND COMPUTING↗
Accelerating Computational Materials Discovery with Machine Learning and Cloud High-Performance Computing: from Large-Scale Screening to Experimental Validation
High-throughput computational materials discovery has promised significant acceleration of the design and discovery of new materials for many years. Despite a surge in interest and activity, the constraints imposed by large-scale computational resources present a significant bottleneck. Furthermore, examples of large-scale computational discovery carried through experimental validation remain scarce, especially for materials with product applicability. In this paper, we demonstrate how this vision became reality by first combining state-of-the-art artificial intelligence (AI) models and traditional physics-based models on cloud high performance computing (HPC) resources to quickly navigate through more than 32 million candidates and predict around half a million potentially stable materials. Focusing on solid-state electrolytes for battery applications, our discovery pipeline further identified 18 promising candidates with new compositions and rediscovered a decade’s worth of collective knowledge in the field as a byproduct. By employing around one thousand virtual machines in the cloud, this process took less than 80 hours. We then synthesized and experimentally characterized the structures and conductivities of our top candidates, the Na x Li 3-x YCl 6 (0.5 ≤ x ≤ 2.5) series, demonstrating the potential of these compounds to serve as solid electrolytes. Additional candidate materials are currently under experimental investigation that could offer more examples of the computational discovery of new phases of Li- and Na-conducting solid electrolytes. We believe this unprecedented approach of synergistically integrating AI models and cloud HPC not only accelerates materials discovery but also showcases the potency of AI-guided experimentation in unlocking transformative scientific breakthroughs with real-world applications.
36 MATERIALS SCIENCE↗
Ab initio leading order effective potential for elastic proton scattering based on the symmetry-adapted no-core shell model
Calculating microscopic optical potentials for elastic scattering at intermediate energies from light nuclei in an ab initio fashion within the Watson expansion has been established within the last few years. Based on the Watson expansion of the multiple scattering series, we employ a nonlocal translationally invariant nuclear density derived within the symmetry-adapted no-core shell model (SA-NCSM) framework from a chiral next-to-next-to-leading order (NNLO) nucleon-nucleon interaction and the very same interaction for a consistent full-folding calculation of the effective (optical) potential for nucleon-nucleus scattering for medium-heavy nuclei. The leading order effective (optical) folding potential is computed by integrating over a translationally invariant SA-NCSM one-body scalar density, spin-projected momentum distribution, and the Wolfenstein amplitudes 𝐴, 𝐶, and 𝑀. The resulting nonlocal potentials serve as input for a momentum space Lippmann-Schwinger equation. In the SA-NCSM, the model space is systematically up-selected using Sp(3,ℝ) symmetry considerations. For the light nucleus of 6 He, we establish a systematic selection scheme in the SA-NCSM for scattering observables. Then, we apply this scheme to calculations of scattering observables, such as differential cross sections, analyzing powers, and spin rotation functions for elastic proton scattering from 20 Ne and 40 Ca in the energy regime between 65 and 200 MeV, and compare to available data. Furthermore, our calculations show that the leading order effective nucleon-nucleus potential in the Watson expansion of multiple scattering theory obtained from an up-selected SA-NCSM model space describes 40 Ca elastic scattering observables reasonably well to about 60 degrees in the center-of-mass frame, which coincides roughly with the validity of the NNLO chiral interaction used to calculate both the nucleon-nucleon amplitudes and the one-body scalar and spin nuclear densities.
Ab initio calculations↗
Commissioning of the Mu2e tracker DAQ, planning for the Vertical Slice Test and pre-pattern recognition studies
The primary objective of the Mu2e experiment at Fermilab is to search for the neutrino-less coherent $\mu \rightarrow e$ conversion in the field of an aluminum nucleus ($\mu^- \text{Al} \rightarrow e^- \text{Al}$). The signature of this process is a monochromatic Conversion Electron (CE) with an energy of approximately 104.97 MeV \cite{bartoszek2015mu2e}. Within the Standard Model (SM), the branching ratio for this process, including neutrino masses and oscillation, is expected to be less than $\mathcal{O}(10^{-50})$. This value is far beyond current experimental capabilities. However, models of physics beyond the SM predict much higher relative rates, approaching an observable level. The SINDRUM II experiment set an upper limit on muon conversion at $7 \times 10^{-13}$ (90\% CL) on Au target \cite{SINDRUMII:2006dvw}, and the Mu2e collaboration aims to improve this limit by four orders of magnitude. Observing this process would provide a clear evidence of physics beyond the Standard Model. A brief discussion of the theoretical and experimental aspects is provided in Chapter \ref{intr}. Mu2e adopts a sophisticated experimental setup to achieve its goals, further described in Chapter \ref{mu2echapter}. The central part of the Mu2e detector is the tracker, that consists of 18 tracking stations. The tracker must provide excellent momentum resolution, approximately 1 MeV/c, to distinguish the monochromatic CE signal from the background. To minimize the energy losses, a straw tube tracker will be used \cite{bobbb}. Chapter \ref{chaptertrk} provides an overview of the straw tracker design and its working principles. This Thesis presents a comprehensive study of the Mu2e tracker, covering complementary aspects from initial commissioning to optimization and first steps of the calibration processes. My work at Fermilab has been focused on the complete Data Acquisition (DAQ) testing from both hardware and software perspectives. I was involved in the commissioning of the Mu2e DAQ system and the Vertical Slice Test (VST) of the tracker. The VST encompasses the entire testing chain, from the straws to the readout, and to processed data on disk. I was also focused on the offline analysis, especially on pre-pattern recognition studies, to explore the best methods for identifying $\delta$-electrons during the data taking. Chapter \ref{commissioning} details the commissioning of the tracker DAQ system, emphasizing the importance of understanding of the readout process before the data acquisition. This includes validating the readout logic and firmware through Monte Carlo simulations to confirm functionality and buffering, monitoring the quality of the data from the tracker preamplifiers and front-end electronics, and assessing overall DAQ performance to ensure reliability during future calibration and data-taking. Chapter \ref{planning} discusses the initial steps towards the tracker calibration. The ultimate goal is to perform a time calibration of the first assembled station of the tracker using cosmic muons, aiming for a longitudinal hit position resolution better than 4 cm. This involves determining the signal propagation times and channel-to-channel delays. I performed a Monte Carlo study to determine the impact of the station orientation on the quality of the calibration, in particular on the cosmic track reconstruction, focusing on potential biases that could arise. These studies provide essential insights into the operation, optimization, and calibration of the Mu2e tracker system. Given the high data volume expected during Mu2e operations, estimated at approximately 7 PBytes per year, optimizing memory usage and minimizing CPU consumption are critical. A significant challenge lies in effectively flagging $\delta$-electron hits, which are the primary source of hits in the tracker, without compromising the efficiency of CE hit detection and track reconstruction. A detailed study of pre-pattern recognition and a thorough comparison of two $\delta$-electron flagging algorithms is provided in Chapter \ref{delta}. In Chapter \ref{conclusions}, the findings are concisely summarized, offering a comprehensive synthesis of the research and emphasizing the key insights derived from this study.
43 PARTICLE ACCELERATORS↗
Urban weather modeling using WRF: linking physical assumptions, code implementation, and observational needs
The Weather Research and Forecasting (WRF) model includes urban schemes that simulate the influence of urban surfaces on the atmosphere using parameterizations for flux, and radiative exchanges. Three core schemes – the Bulk urban parameterization, Single-Layer Urban Canopy Model (SLUCM), and Multi-Layer Urban Canopy Model (MLUCM) – represent increasing levels of complexity. Although the parameterizations within these urban schemes are described in the literature, their specific implementation remains poorly documented, thus slowing down model development efforts. This manuscript provides a roadmap to the three urban schemes in WRF version 4.5.2, presenting equations using the same symbols as in the model code, along with references to code lines, and including graphics and explanations that connect the code to its physical foundations. Our thorough review of the urban parameterizations implemented in WRF version 4.5.2 highlighted a handful of parameters that may introduce discontinuities in simulations: (i) in the SLUCM, a 1 mm h −1 rain rate threshold is employed to switch between two minimum moisture availability parameterizations, thus impacting latent heat flux calculations; (ii) in the SLUCM a threshold is used to partition shortwave radiation into direct and diffuse components; (iii) in all three urban schemes, the bulk Richardson number is employed to select the similarity function, which influences the vertical distribution of heat and momentum. We also identified a highly simplified treatment of the radiative balance on roof surfaces. The implications of these simplifications can be assessed through targeted observations across relevant conditions, including varying precipitation rates, cloud cover, and transitions between stability regimes. Furthermore, the widespread application of the Monin-Obukhov similarity theory in these urban schemes warrants model evaluation under highly stable and unstable conditions and in heterogeneous urban settings with variable land cover and building heights on scales finer than model resolution. To address these challenges, we offer guidance on observational strategies, emphasizing the need for multi-parametric measurements to capture potential compensating biases and multi-height measurements that align with the levels where quantities are diagnostic and prognosed in the model (i.e., the lowest atmospheric level of the WRF model). Finally, our inspection of the code revealed implementation bugs that have now been corrected in WRF versions 4.6.0 and 4.6.1. Sensitivity tests over the Atlanta urban area show that these corrections affect surface temperatures, underscoring the importance of performing rigorous documentation and verification of the implementation of parameterizations in model code.
54 ENVIRONMENTAL SCIENCES↗
Development of physics-consistent conditional diffusion model to overcome data scarcity in critical heat flux
Deep generative modeling provides a powerful pathway to overcome data scarcity in energy-related applications where experimental data are often limited. By learning the underlying probability distribution of the training dataset, deep generative models, such as the diffusion model, can generate high-fidelity synthetic samples that statistically resemble the training data. Such synthetic data generation can significantly enrich the size and diversity of the available training data, and more importantly, improve the robustness of downstream machine learning models in predictive tasks. The objective of this paper is to investigate the effectiveness of diffusion models for overcoming data scarcity in nuclear energy applications. By leveraging a public dataset on critical heat flux which covers a wide range of commercial nuclear reactor operational conditions, we developed a diffusion model that can generate an arbitrary amount of synthetic samples. Since a vanilla diffusion model can only generate samples randomly, we also developed a conditional diffusion model capable of generating targeted critical heat flux data under user-specified thermal-hydraulic conditions. The performance of the diffusion model was evaluated based on its ability to capture empirical feature distributions and pair-wise correlations, as well as to maintain physical consistency. The results showed that both the diffusion model and conditional diffusion model can successfully generate realistic and physics-consistent critical heat flux data. Furthermore, uncertainty quantification results demonstrate that the conditional diffusion model is highly effective in augmenting critical heat flux data while maintaining acceptable levels of uncertainty.
22 GENERAL STUDIES OF NUCLEAR REACTORS↗
Data‐driven variational method for discrepancy modeling: Dynamics with small‐strain nonlinear elasticity and viscoelasticity
Abstract The effective inclusion of a priori knowledge when embedding known data in physics‐based models of dynamical systems can ensure that the reconstructed model respects physical principles, while simultaneously improving the accuracy of the solution in the previously unseen regions of state space. This paper presents a physics‐constrained data‐driven discrepancy modeling method that variationally embeds known data in the modeling framework. The hierarchical structure of the method yields fine scale variational equations that facilitate the derivation of residuals which are comprised of the first‐principles theory and sensor‐based data from the dynamical system. The embedding of the sensor data via residual terms leads to discrepancy‐informed closure models that yield a method which is driven not only by boundary and initial conditions, but also by measurements that are taken at only a few observation points in the target system. Specifically, the data‐embedding term serves as residual‐based least‐squares loss function, thus retaining variational consistency. Another important relation arises from the interpretation of the stabilization tensor as a kernel function, thereby incorporating a priori knowledge of the problem and adding computational intelligence to the modeling framework. Numerical test cases show that when known data is taken into account, the data driven variational (DDV) method can correctly predict the system response in the presence of several types of discrepancies. Specifically, the damped solution and correct energy time histories are recovered by including known data in the undamped situation. Morlet wavelet analyses reveal that the surrogate problem with embedded data recovers the fundamental frequency band of the target system. The enhanced stability and accuracy of the DDV method is manifested via reconstructed displacement and velocity fields that yield time histories of strain and kinetic energies which match the target systems. The proposed DDV method also serves as a procedure for restoring eigenvalues and eigenvectors of a deficient dynamical system when known data is taken into account, as shown in the numerical test cases presented here.
Masud, Arif↗