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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 415 records · Page 23
All sourcesNASA NTRSDOE OSTI
DOE OSTI2015

Representing ductile damage with the dual domain material point method

In this study, we incorporate a ductile damage material model into a computational framework based on the Dual Domain Material Point (DDMP) method. As an example, simulations of a flyer plate experiment involving ductile void growth and material failure are performed. The results are compared with experiments performed on high purity tantalum. We also compare the numerical results obtained from the DDMP method with those obtained from the traditional Material Point Method (MPM). Effects of an overstress model, artificial viscosity, and physical viscosity are investigated. Our results show that a physical bulk viscosity and overstress model are important in this impact and failure problem, while physical shear viscosity and artificial shock viscosity have negligible effects. A simple numerical procedure with guaranteed convergence is introduced to solve for the equilibrium plastic state from the ductile damage model.

36 MATERIALS SCIENCE↗
DOE OSTI2025

HPC-Driven Modeling with ML-Based Surrogates for Magnon-Photon Dynamics in Hybrid Quantum System

Here, we introduce a hybrid computational framework that merges HPC-based numerical solvers with physics-informed ML surrogates for efficient modeling of magnon-photon interactions. By running short-duration, high-fidelity Maxwell-LLG simulations and feeding their results into an ML model, we substantially cut simulation time while achieving accurate predictions across larger spatiotemporal domains.

Accuracy↗
NASA NTRS2016

Uncertainty Reduction using Bayesian Inference and Sensitivity Analysis: A Sequential Approach to the NASA Langley Uncertainty Quantification Challenge

This paper presents a computational framework for uncertainty characterization and propagation, and sensitivity analysis under the presence of aleatory and epistemic un- certainty, and develops a rigorous methodology for efficient refinement of epistemic un- certainty by identifying important epistemic variables that significantly affect the overall performance of an engineering system. The proposed methodology is illustrated using the NASA Langley Uncertainty Quantification Challenge (NASA-LUQC) problem that deals with uncertainty analysis of a generic transport model (GTM). First, Bayesian inference is used to infer subsystem-level epistemic quantities using the subsystem-level model and corresponding data. Second, tools of variance-based global sensitivity analysis are used to identify four important epistemic variables (this limitation specified in the NASA-LUQC is reflective of practical engineering situations where not all epistemic variables can be refined due to time/budget constraints) that significantly affect system-level performance. The most significant contribution of this paper is the development of the sequential refine- ment methodology, where epistemic variables for refinement are not identified all-at-once. Instead, only one variable is first identified, and then, Bayesian inference and global sensi- tivity calculations are repeated to identify the next important variable. This procedure is continued until all 4 variables are identified and the refinement in the system-level perfor- mance is computed. The advantages of the proposed sequential refinement methodology over the all-at-once uncertainty refinement approach are explained, and then applied to the NASA Langley Uncertainty Quantification Challenge problem.

Uncertainty↗
DOE OSTI2015

Consistent Feature Extraction From Vector Fields: Combinatorial Representations and Analysis Under Local Reference Frames

This dissertation presents research on addressing some of the contemporary challenges in the analysis of vector fields—an important type of scientific data useful for representing a multitude of physical phenomena, such as wind flow and ocean currents. In particular, new theories and computational frameworks to enable consistent feature extraction from vector fields are presented. One of the most fundamental challenges in the analysis of vector fields is that their features are defined with respect to reference frames. Unfortunately, there is no single “correct” reference frame for analysis, and an unsuitable frame may cause features of interest to remain undetected, thus creating serious physical consequences. This work develops new reference frames that enable extraction of localized features that other techniques and frames fail to detect. As a result, these reference frames objectify the notion of “correctness” of features for certain goals by revealing the phenomena of importance from the underlying data. An important consequence of using these local frames is that the analysis of unsteady (time-varying) vector fields can be reduced to the analysis of sequences of steady (timeindependent) vector fields, which can be performed using simpler and scalable techniques that allow better data management by accessing the data on a per-time-step basis. Nevertheless, the state-of-the-art analysis of steady vector fields is not robust, as most techniques are numerical in nature. The residing numerical errors can violate consistency with the underlying theory by breaching important fundamental laws, which may lead to serious physical consequences. This dissertation considers consistency as the most fundamental characteristic of computational analysis that must always be preserved, and presents a new discrete theory that uses combinatorial representations and algorithms to provide consistency guarantees during vector field analysis along with the uncertainty visualization of unavoidable discretization errors. Together, the two main contributions of this dissertation address two important concerns regarding feature extraction from scientific data: correctness and precision. The work presented here also opens new avenues for further research by exploring more-general reference frames and more-sophisticated domain discretizations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Validating quantum-classical programming models with tensor network simulations

The exploration of hybrid quantum-classical algorithms and programming models on noisy near-term quantum hardware has begun. As hybrid programs scale towards classical intractability, validation and benchmarking are critical to understanding the utility of the hybrid computational model. In this paper, we demonstrate a newly developed quantum circuit simulator based on tensor network theory that enables intermediate-scale verification and validation of hybrid quantum-classical computing frameworks and programming models. We present our tensor-network quantum virtual machine (TNQVM) simulator which stores a multi-qubit wavefunction in a compressed (factorized) form as a matrix product state, thus enabling single-node simulations of larger qubit registers, as compared to brute-force statevector simulators. Our simulator is designed to be extensible in both the tensor network form and the classical hardware used to run the simulation (multicore, GPU, distributed). The extensibility of the TNQVM simulator with respect to the simulation hardware type is achieved via a pluggable interface for different numerical backends (e.g., ITensor and ExaTENSOR numerical libraries). We demonstrate the utility of our TNQVM quantum circuit simulator through the verification of randomized quantum circuits and the variational quantum eigensolver algorithm, both expressed within the eXtreme-scale ACCelerator (XACC) programming model.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2013

GEOS Code Development Road Map

GEOS is a massively parallel computational framework designed to enable HPC-based simulations of subsurface reservoir stimulation activities with the goal of optimizing current operations and evaluating innovative stimulation methods. GEOS will enable coupling of different solvers associated with the various physical processes occurring during reservoir stimulation in unique and sophisticated ways, adapted to various geologic settings, materials and stimulation methods. The overall architecture of the framework includes consistent data structures and will allow incorporation of additional physical and materials models as demanded by future applications. Along with predicting the initiation, propagation and reactivation of fractures, GEOS will also generate a seismic source term that can be linked with seismic wave propagation codes to generate synthetic microseismicity at surface and downhole arrays. Similarly, the output from GEOS can be linked with existing fluid/thermal transport codes. GEOS can also be linked with existing, non-intrusive uncertainty quantification schemes to constrain uncertainty in its predictions and sensitivity to the various parameters describing the reservoir and stimulation operations. We anticipate that an implicit-explicit 3D version of GEOS, including a preliminary seismic source model, will be available for parametric testing and validation against experimental and field data by Oct. 1, 2013.

02 PETROLEUM↗
DOE OSTI2017

A Monte Carlo model for 3D grain evolution during welding

Welding is one of the most wide-spread processes used in metal joining. However, there are currently no open-source software implementations for the simulation of microstructural evolution during a weld pass. Here we describe a Potts Monte Carlo based model implemented in the SPPARKS kinetic Monte Carlo computational framework. The model simulates melting, solidification and solid-state microstructural evolution of material in the fusion and heat-affected zones of a weld. The model does not simulate thermal behavior, but rather utilizes user input parameters to specify weld pool and heat-affect zone properties. Weld pool shapes are specified by Bezier curves, which allow for the specification of a wide range of pool shapes. Pool shapes can range from narrow and deep to wide and shallow representing different fluid flow conditions within the pool. Surrounding temperature gradients are calculated with the aide of a closest point projection algorithm. Furthermore, the model also allows simulation of pulsed power welding through time-dependent variation of the weld pool size. Example simulation results and comparisons with laboratory weld observations demonstrate microstructural variation with weld speed, pool shape, and pulsed-power.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Prediction of Probabilistic Shock Initiation Thresholds of Energetic Materials Through Evolution of Thermal-Mechanical Dissipation and Reactive Heating

The ignition threshold of an energetic material (EM) quantifies the macroscopic conditions for the onset of self-sustaining chemical reactions. The threshold is an important theoretical and practical measure of material attributes that relate to safety and reliability. Historically, the thresholds are measured experimentally. In this work, we present a new Lagrangian computational framework for establishing the probabilistic ignition thresholds of heterogeneous EM out of the evolutions of coupled mechanical-thermal-chemical processes using mesoscale simulations. Furthermore, the simulations explicitly account for microstructural heterogeneities, constituent properties, and interfacial processes and capture processes responsible for the development of material damage and the formation of hotspots in which chemical reactions initiate. The specific mechanisms tracked include viscoelasticity, viscoplasticity, fracture, post-fracture contact, frictional heating, heat conduction, reactive chemical heating, gaseous product generation, and convective heat transfer. To determine the ignition threshold, the minimum macroscopic loading required to achieve self-sustaining chemical reactions with a rate of reactive heat generation exceeding the rate of heat loss due to conduction and other dissipative mechanisms is determined. Probabilistic quantification of the processes and the thresholds are obtained via the use of statistically equivalent microstructure sample sets (SEMSS). The predictions are in agreement with available experimental data.

36 MATERIALS SCIENCE↗
DOE OSTI2023

Neural Network Enhanced RKPM for Electrochemical-Mechanical Coupled Damage Modeling of Energy Storage Materials

Energy storage materials undergo significant charge cycling, which makes understanding their reliability and durability fundamental in predicting performance and service life. Strong electrochemical-mechanical coupling and highly anisotropic material properties contribute to the formation and propagation of micro-cracking, largely along material interfaces and grain boundaries. For Li-ion batteries, for example, lithium moving between electrodes during charging and discharging process causes expansion and contraction of grains, and the strongly anisotropic and nonlinearly [Li]-dependent grain material properties can cause grains to expand into and contract away from each other, leading to chemo-mechanical cracking. In the first part of this work, a RKPM based computational framework for solving the coupled solid-phase lithium conservation with Fickian diffusion and the lithium concentration dependent anisotropic mechanical problem subjected to a highly nonlinear Butler-Volmer boundary condition is introduced. The choice of RKPM completeness conditions for lithium concentration and mechanical deformation fields, and the variational consistency condition for the domain integration of the coupled problem is first determined. In the second part of this work, a neural network-enhanced reproducing kernel particle method (NN-RKPM) [1] is leveraged to accurately capture damage and crack propagation throughout the material, by learning the location, orientation, and sharpness of discontinuity while allowing for a coarser nodal distribution than that is necessary for capturing sharp solution transitions using traditional mesh-based methods. NN-RKPM is used to inform how crack opening and closure in turn affect the coupled chemical equations and material microstructure.

damage modeling↗
DOE OSTI2017

Improving Uintah's Scalability Through the Use of Portable Kokkos-Based Data Parallel Tasks

The University of Utah’s Carbon Capture Multidisciplinary Simulation Center (CCMSC) is using the Uintah Computational Framework to predict performance of a 1000 MWe ultra-supercritical clean coal boiler. The center aims to utilize the Intel Xeon Phi-based DOE systems, Theta and Aurora, through the Aurora Early Science Program by using the Kokkos C++ library to enable node-level performance portability. This paper describes infrastructure advancements and portability improvements made possible by the integration of Kokkos within Uintah. This integration marks a step towards consolidating Uintah’s MPI+PThreads and MPI+CUDA hybrid parallelism approaches into a single MPI+Kokkos approach. Scalability results are presented that compare serial and data parallel task execution models for a challenging radiative heat transfer calculation, central to the center’s predictive boiler simulations. Here, these results demonstrate both good strong-scaling characteristics to 256 Knights Landing (KNL) processors on the NSF Stampede system, and show the KNL-based calculation to compete with prior GPU-based results for the same calculation.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Multi-modal Energy-optimal Trip Scheduling in Real-time (METS-R) for Transportation Hubs (Final Report)

This report summarizes the work performed under the award number EE0008524. The project develops the Multi-modal Energy-optimal Trip Scheduling in Real-time (METS-R) platform as the next-generation transportation solution based on autonomous electric vehicles (AEV) serving passenger trips from and to urban transportation hubs, to substantially reduce transportation energy consumption. Extensive data collection and analyses were first conducted to understand the demand patterns and energy consumption of hub-based on-road trips. Then, a data-driven framework that consists of an analytical module and a simulation module was proposed. For the analytical module, five planning + operation tools were developed to support the planning and energy-efficient operations of urban AEV services: the charging station planning that robotically allocates charging supplies based on the stationary charging demand distribution; the transit planning and demand adaptive scheduling model that efficiently generates\ candidate transit routes from hubs to other places and dynamically adjusts the transit time table to fit the current demand; the online energy-efficient routing that learns the energy-optimal paths from observations of link-level energy consumption in real-time; the hub-based ridesharing that matches trip requests together with account for the uncertainty of future trip demand and vehicle supply; and finally, the integrated demand prediction and anomaly detection pipeline that leverages the flight/train time table and support other planning/operation tools. To demonstrate the performance of these tools, a scalable high-performance agent-based simulator was built. We divided the urban space into multiple service zones where each zone was considered as an agent for passenger generation and vehicle charging. Two types of AEV agents were coded to model two types of mobility services: AEV taxi and AEV transit. For the AEV taxi, the team implemented the functions of pickup/drop-off passengers, energy-efficient routing, ridesharing, fleet rebalancing, and recharging. For the AEV bus, the team implemented the functions of demand-adaptive route scheduling, passenger boarding, and recharging. A high-performance computing framework was introduced to receive various profiling information (such as link energy updates, vehicle speed) from the simulator instances and communicate the operational commands back to the instances. The numerical experiments show that each of the proposed operational algorithms can reduce energy consumption and improve system efficiency. Furthermore, there exists the need to collectively consider multiple planning + operational strategies as multiple strategies can influence each other in terms of performance impacts. Recommendations for future work related to AEV planning and simulation are discussed.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗
DOE OSTI2014

Noise Decoherence and Errors from Entanglement-function Theory for Quantum Computing

A significant problem in quantum computing is the development of physical realizations of algorithms that are robust against noise. One way to examine and mitigate noise would be to simulate large sets of qubits coupling to the external environment on classical computers. This is extremely challenging as quantum information processing is in some sense tied to computing resources that scale exponentially with the number of computing elements (qubits). In this LDRD, we set the foundation for a computational framework potentially allowing simulations of 1000s of qubits vs. 10s now possible. Exact wave-function-based methods demand exponentially increasing resources with system size. The method proposed, entanglement-functional theory (EFT), requires vastly fewer resources. The crucial step is to map the information contained in the wave-functions into a simpler object with associated 1.) auxiliary gate operations and 2.) entanglement functionals of this object. This is similar to the Time-dependent Density Functional Theory (TDDFT) approach that has revolutionized chemistry and materials science. Instead of dealing with the exponentially large wave-function, EFT works with a polynomially large set of projections (the density) that are easily manipulated through unitary operations. For a given set of quantum gates, an isomorphism exists that relates the sequence of events to the time-dependent density. A system of entangled qubits can be simulated at drastically reduced cost relative to existing state-of-the-art vector-state simulation codes.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

A HPC “Cyber Wind Facility” Incorporating Fully-Coupled CFD/CSD for Turbine-Platform-Wake Interactions with the Atmosphere and Ocean

The central aims of the DOE-supported “Cyber Wind Facility” project center on the recognition that wind turbines over land and ocean generate power from atmospheric winds that are inherently turbulent and strongly varying, both spatially over the rotor disk and in temporally as the rotating blades pass through atmospheric eddies embedded within the mean wind. The daytime unstable atmospheric boundary layer (ABL) is particularly variable in space time as solar heating generates buoyancy-driven motions that interact with strong mean shear in the ABL “surface layer,” the lowest 200 - 300 m where wind turbines reside in farms. With the “Cyber Wind Facility” (CWF) program we initiate a research and technology direction in which “cyber data” are generated from “computational experiments” within a “facility” akin to a wind tunnel, but with true space-time atmospheric turbulence that drive utility-scale wind turbines at full-scale Reynolds numbers. With DOE support we generated the key “modules” within a computational framework to create a first generation Cyber Wind Facility (CWF) for single wind turbines in the daytime ABL---both over land where the ABL globally unstable and over water with closer-to-neutral atmospheric conditions but with time response strongly affected by wave-induced forcing of the wind turbine platform (here a buoy configuration). The CWF program has significantly improved the accuracy of actuator line models, evaluated with the Cyber Wind Facility in full blade-boundary-layer-resolved mode. The application of the CWF made in this program showed the existence of important ramp-like response events that likely contribute to bearing fatigue failure on the main shaft and that the advanced ALM method developed here captures the primary nonsteady response characteristics. Long-time analysis uncovered distinctive key dynamics that explain primary mechanisms that underlie potentially deleterious load transients. We also showed that blade bend-twist coupling plays a central role in the elastic responses of the blades to atmospheric turbulence, impacting turbine power.

17 WIND ENERGY↗
DOE OSTI2018

Machine-Learning Enabled New Insights into the Coil-to-Globule Transition of Thermosensitive Polymers Using a Coarse-Grained Model

In this paper, we present a computational framework that integrates coarse-grained (CG) molecular dynamics (MD) simulations and a data-driven machine-learning (ML) method to gain insights into the conformations of polymers in solutions. We employ this framework to study conformational transition of a model thermosensitive polymer, poly(N-isopropylacrylamide) (PNIPAM). Here, we have developed the first of its kind, a temperature-independent CG model of PNIPAM that can accurately predict its experimental lower critical solution temperature (LCST) while retaining the tacticity in the presence of an explicit water model. The CG model was extensively validated by performing CG MD simulations with different initial conformations, varying the radius of gyration of chain, the chain length, and the angle between the adjacent monomers of the initial configuration of PNIPAM (total simulation time = 90 μs). Moreover, for the first time, we utilize the nonmetric multidimensional scaling (NMDS) method, a data-driven ML approach, to gain further insights into the mechanisms and pathways of this coil-to-globule transition by analyzing CG MD simulation trajectories. In conclusion, NMDS analysis provides entirely new insights and shows multiple metastable states of PNIPAM during its coil-to-globule transition above the LCST.

36 MATERIALS SCIENCE↗
DOE OSTI2015

Complik: A versatile tool for automatizing the representation of material data in MCNP models - 14412

At the Paul Scherrer Institut, the CASMO/SIMULATE code system is used as reference scheme for core analyses of the Swiss reactors. Besides, Monte Carlo (MC) based neutron transport codes are in parallel being established both for out-of-core applications and full core analyses, in order to provide higher fidelity solutions to problems beyond the modelling capabilities of the standard core modelling methods. In this perspective, a necessity is to be able to conduct the MC calculations at any operating conditions, requiring thus proper initialisation of the thermal-hydraulic feedback as well as at any burnup, prompting the need for a detailed modelling of all material zones in the irradiated fuel assemblies. To tackle the later challenge, the development of the COMPLINK tool, aimed at initialising large-scale MC models of burned core configurations, using as basis upstream 2-D assembly/ 3-D core analyses results, was initiated and is presented in this paper. A main feature is the capability to extend in an automatic manner, a user given MC model in order to represent material data with a higher resolution, keeping track of the location of such materials also within nested repeated structures. This capability prevents the user from the burden of developing specific computational framework for a target application. Since the same functionalities are required to analyse out-of-core spent fuel managements systems, a first verification of COMPLINK is illustrated on the basis of a criticality safety evaluation (CSE) with MCNP6.1 of a Swiss PWR wet storage pool loaded with fresh and burned fuel assemblies. Starting from CASMO-5 depletion calculations at each axial layer of the depleted fuel assemblies and reconstructed based on realistic irradiation histories, COMPLINK is then applied to load the isotopic compositions into the material zones of the corresponding MCNP6.1 assembly models. On this basis, the CSE analyses are conducted with focus on investigating axial and radial modelling effects. All in all, the obtained results are found to be well in line with the impact on CSE expected at the considered burnup levels, providing thus confidence in the correct implementation of the COMPLINK functionalities and confirming its effectiveness as well as efficiency for the initialization of all material zones in large-scale MC models of complex and strongly heterogeneous burned configurations. (authors)

42 ENGINEERING↗
DOE OSTI2022

Metal dissolution from first principles: Potential-dependent kinetics and charge transfer

Metal dissolution and its inverse process are integral to both corrosion and electrodeposition; however, many mechanistic details regarding the dissolution process are challenging to decipher. These include how ion dissolution kinetics and charge transfer are influenced by the competition between metal and solvent interactions under an electrode potential. Here, in this work, we introduce a computational framework based on density functional theory with grand-canonical treatment of electrons to directly predict the potential energy landscape for metal dissolution at a constant potential. Using aluminum as an example, we demonstrate that dissolution kinetics is governed by competing kinetics between two physical processes associated with metal–metal bond breaking and ion-migration within the electrical double layer, respectively. We identify a kinetic transition between regimes dominated by each of these processes and show that this transition depends on the operating electrode potential, among other key factors. It is further found that kinetics and thermodynamics of these processes can be described with a simple, one-parameter Marcus-theory-type model. Beyond offering new understanding of charge transfer during dissolution, our simulation protocol provides a recipe for directly predicting other important quantities in electrochemical reactions from first principles that are difficult to measure, such as the symmetry factor.

36 MATERIALS SCIENCE↗
DOE OSTI2023

High-throughput ab initio design of atomic interfaces using InterMatch

Forming a hetero-interface is a materials-design strategy that can access an astronomically large phase space. However, the immense phase space necessitates a high-throughput approach for an optimal interface design. Here we introduce a high-throughput computational framework, InterMatch, for efficiently predicting charge transfer, strain, and superlattice structure of an interface by leveraging the databases of individual bulk materials. Specifically, the algorithm reads in the lattice vectors, density of states, and the stiffness tensors for each material in their isolated form from the Materials Project. From these bulk properties, InterMatch estimates the interfacial properties. We benchmark InterMatch predictions for the charge transfer against experimental measurements and supercell density-functional theory calculations. We then use InterMatch to predict promising interface candidates for doping transition metal dichalcogenide MoSe 2 . Finally, we explain experimental observation of factor of 10 variation in the supercell periodicity within a few microns in graphene/α-RuCl 3 by exploring low energy superlattice structures as a function of twist angle using InterMatch. We anticipate our open-source InterMatch algorithm accelerating and guiding ever-growing interfacial design efforts. Moreover, the interface database resulting from the InterMatch searches presented in this paper can be readily accessed online.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2026

Semi-Analytical Hierarchical Bayesian Inference of Nonlinear Model Structure in Stochastic Dynamics: Applied to Compartmental Models of Infectious Diseases

A Bayesian computational framework for parsimonious inference in stochastic nonlinear dynamical systems is presented. This framework enables the concurrent estimation of system states, time-varying parameters, time-invariant parameters, and the optimal sparsity structure of the model parameters. Because differential equation-based models are often simplified mechanistic or phenomenological representations, robust inference from noisy measurement data requires explicit treatment of model error and uncertainty. Model error and time-varying parameters can be represented as random processes, enabling inference while making minimal assumptions about the underlying sources of discrepancy and variability. Adopting stochastic differential equation representations affords the model significant flexibility, but can also render it susceptible to overfitting during statistical inversion, where the inferred model may track noise rather than the underlying signal. To alleviate the effects of overfitting and to enable the discovery of the optimal sparse representation of the time-invariant parameters, a Bayesian sparse learning algorithm is embedded within the framework. This sparse learning framework adopts an approximate hierarchical Bayesian setting defined by a series of semi-analytical expressions. The model structure inference framework is validated using a stochastic compartmental model for tracking and forecasting active cases of an infectious disease. Compartmental models describe population-level infectious disease dynamics through interactions among population fractions grouped by disease state. Mathematically, such models consist of a system of coupled ordinary differential equations. This example adopts an expressive compartmental model that includes multiple possible interactions between disease states, motivated by early uncertainty surrounding COVID-19 reinfection dynamics and their implications for long-term epidemic forecasting. The sparse learning exercise permits the inference of a priori unknown epidemiological dynamics from simulated public health data, discovering the nested compartmental model that optimizes the trade-off between average data-fit and model complexity. It is shown that inducing sparsity among the model parameters eliminates redundant interactions between compartments, equivalently revealing the optimal coupling structure between differential equations.

97 MATHEMATICS AND COMPUTING↗