Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “ordinary differential equation”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 469 records · Page 26
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Multirate Exponential Rosenbrock Methods

In this paper we propose a novel class of methods for high-order accurate integration of multirate systems of ordinary differential equation initial-value problems. The proposed methods construct multirate schemes by approximating the action of matrix φ functions within explicit exponential Rosenbrock (ExpRB) methods, thereby called multirate ExpRB (MERB) methods. They consist of the solution to a sequence of modified “fast” initial-value problems, which may themselves be approximated through subcycling any desired initial-value problem solver. In addition to proving how to construct MERB methods from certain classes of ExpRB methods, we provide rigorous convergence analysis of these methods and derive efficient MERB schemes of orders 2 through 6 (the highest-order infinitesimal multirate methods to date). Lastly, we then present numerical simulations to confirm these theoretical convergence rates and to compare the efficiency of MERB methods against other recently introduced high-order multirate methods.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Data-Driven Closures and Assimilation for Stiff Multiscale Random Dynamics

Here, we introduce a data-driven and physics-informed framework for propagating uncertainty in stiff, multiscale random ordinary differential equations (RODEs) driven by correlated (colored) noise. Unlike systems subjected to Gaussian white noise, a deterministic equation for the joint probability density function (PDF) of RODE state variables does not exist in closed form. Moreover, such an equation would require as many phase-space variables as there are states in the RODE system. To alleviate this curse of dimensionality, we instead derive exact, albeit unclosed, reduced-order PDF (RoPDF) equations for low-dimensional observables/quantities of interest. The unclosed terms take the form of state-dependent conditional expectations, which are directly estimated from data at sparse observation times. However, for systems exhibiting stiff, multiscale dynamics, data sparsity introduces regression discrepancies that compound during RoPDF evolution. This is overcome by introducing a kinetic-like defect term to the RoPDF equation, which is learned by assimilating in sparse, low-fidelity RoPDF estimates. Two assimilation methods are considered, namely nudging and deep neural networks, which are successfully tested against Monte Carlo simulations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

ARKODE: A Flexible IVP Solver Infrastructure for One-step Methods

We describe the ARKODE library of one-step time integration methods for ordinary differential equation (ODE) initial-value problems (IVPs). In addition to providing standard explicit and diagonally implicit Runge–Kutta methods, ARKODE supports one-step methods designed to treat additive splittings of the IVP, including implicit-explicit (ImEx) additive Runge–Kutta methods and multirate infinitesimal (MRI) methods. We present the role of ARKODE within the SUNDIALS suite of time integration and nonlinear solver libraries, the core ARKODE infrastructure for utilities common to large classes of one-step methods, as well as its use of “time stepper” modules enabling easy incorporation of novel algorithms into the library. Numerical results show example problems of increasing complexity, highlighting the algorithmic flexibility afforded through this infrastructure, and include a larger multiphysics application leveraging multiple algorithmic features from ARKODE and SUNDIALS.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Crowd Dynamics: Modeling and Control of Multiagent Systems

This review aims to present recent developments in modeling and control of multiagent systems. A particular focus is set on crowd dynamics characterized by complex interactions among agents, also called social interactions, and large-scale systems. Specifically, in a crowd each individual agent interacts with a field generated by the other agents and the environment. These systems can be modeled at the microscopic scale by ordinary differential equations, while an alternative description at the mesoscopic scale is given by a partial differential equation for the propagation of the probability density of the agents. Control actions can be applied at the individual level as well as at the level of the corresponding fields. This article presents and compares different control types, and the specific application to multilane, multiclass traffic is developed in some detail, showing the main tools at work in a hybrid setting with relevant impacts on autonomous driving.

33 ADVANCED PROPULSION SYSTEMS↗
DOE OSTI2020

Whole cell simulator

A whole cell model (WCM) is a comprehensive multi-scale computational model representing all the known biochemical processes in a cell. It relies on a variety of intracellular pathway models and omics data. This computational framework enables seamless integration of diverse simulation methods used in WCM such as stochastic simulation algorithm (SSA), ordinary differential equations (ODE), flux balance analysis (FBA), and logic-based approaches. These methods run simultaneously, not only for whole pathways but also for subsets of reactions. Furthermore, we aim to enable dynamic switching between the methods when beneficial. Is

GEORGAKOUDIS, GIORGIS↗
DOE OSTI2022

Latent Space Dynamics Identification

LaSDI is a data-driven physical simulation software that forms a latent space for a given high-fidelity model and discovers a set of ordinary differential equations for the latent space dynamics. It allows a fast and accurate solution process, which is useful for multi-query decision making applications, such as design optimization and uncertainty quantification. The performance of the LaSDI framework is demonstrated on four different problems, i.e., 1D and 2D Burgers equations, nonlinear heat conduction, and radial advection problems. Both linear and nonlinear compression techniques, such as neural network and proper orthogonal decomposition, are used to form a latent space. A concept of local dynamics identification procedure is introduced to enable a parametric model, which enhances the accuracy level over a given parameter space.

Fries, William↗
DOE OSTI2021

pnnl/neuromancer

Dynamics-based deep learning methods to modernize current scientific computing methods. Neuromancer is currently capable of solving inverse problems for a system of ordinary differential equations. The functionality includes system identification and constrained optimal control of unknown or partially known ODEs.

Skomski, Elliott↗
DOE OSTI2021

pnnl/neural_ODE_ICLR2020

We show how to model discrete ordinary differential equations (ODE) with algebraic nonlinearities as deep neural networks with varying degrees of prior knowledge. We derive the stability guarantees of the network layers based on the implicit constraints imposed on the weight's eigenvalues. Moreover, we show how to use barrier methods to generically handle additional inequality constraints. We demonstrate the prediction accuracy of learned neural ODEs evaluated on open-loop simulations compared to ground truth dynamics with bi-linear terms.

Tuor, Aaron↗
DOE OSTI2021

yuwangcn/C4_dynamic_model

A dynamic systems model of C4 photosynthesis was developed based on the previous NADP-ME metabolic model for maize (Wang et al., 2014 ab). The NADP-ME metabolic model is an ordinary differential equation model including all individual steps in C4 photosynthetic carbon metabolism. Here, the model is extended to include posttranslational regulation and temperature response of enzyme activities, dynamic stomata conductance, and leaf energy balance. This model is written in Matlab (R2019a) Steady-state and dynamic gas exchange data for maize (B73), sugarcane (CP88-1762) and sorghum (Tx430) were measured in a greenhouse in Urbana, IL from July 25, 2019 through August 8, 2019. Data include: CO2 response curves Light response curves Photosynthetic induction curves measured in the transition from darkness to high light (1800 μmol m-2 s-1), data logged every 1 min. Photosynthetic induction in the transition from darkness to high light (1800 μmol m-2 s-1) to determine the kinetics of rubisco activation in these C4 crops (τ_Rubisco), data logged every 10 s. Gas exchange under fluctuating light. After dark adaptation, the leaves undergo three light change steps, light intensity was set as 1800 µmol m-2 s-1, 200 µmol m-2 s-1 and 1800 µmol m-2 s-1 for each 1800 s step.

Wang, Yu↗
DOE OSTI2025

ORMATEX

The Oak Ridge Matrix Exponential (ORMATEX) software library contains methods to compute the matrix exponential and the action of the matrix exponential on a vector. Additionally, this package contains the related methods for the phi-functions which commonly appear in a wide class of exponential time integration methods. Krylov methods are provided to evaluate the matrix exponential-vector and phi-vector products for cases where the matrix is large and sparse. Utilizing these methods, ORMATEX implements performant exponential integrators for large systems of coupled ordinary differential equations (ODEs). The exponential time integration routines in ORMATEX are particularly suitable to large, stiff systems of equations. These routines may be utilized as a competitive alternative to classical implicit and explicit time integration schemes for certain classes of differential equations where the problem stiffness can be predominately explained by the linear terms.

Gurecky, William [Oak Ridge National Laboratory (O↗
DOE OSTI2026

ssys

`ssys` is a Python toolkit for exact algebraic transformation of ordinary differential equation (ODE) models into canonical S-system or Generalized Mass Action (GMA) form. Given a model in Antimony or SBML format, `ssys` produces a mathematically equivalent representation. The transformation introduces auxiliary variables as needed to decompose a broad class of nonlinearities into products of power-law terms. The recast is exact: the original and transformed systems have identical dynamics on the invariant constraint manifold defined by auxiliary variable definitions, given consistent initial conditions.

Hlavacek, William [Los Alamos National Laboratory]↗
DOE OSTI2025

Order conditions for nonlinearly partitioned Runge-Kutta methods

Recently, a new class of nonlinearly partitioned Runge–Kutta (NPRK) methods was proposed for nonlinearly partitioned systems of autonomous ordinary differential equations y' = F(y, y). The target class of problems are those in which different scales, stiffnesses, or physics are coupled in a nonlinear way, wherein the desired partition cannot be written in a classical additive or component-wise fashion. Here we use a rooted-tree analysis to derive full-order conditions for NPRKM methods, where M denotes the number of nonlinear partitions. Due to the nonlinear coupling and thereby the mixed product differentials, it turns out that the standard node-colored rooted tree analysis used in analyzing ODE integrators does not naturally apply. Instead we develop a new edge-colored rooted-tree framework to address the nonlinear coupling. The resulting order conditions are enumerated, are provided directly for up to fourth order with M = 2 and third order with M = 3, and are related to existing order conditions of additive and partitioned RK methods. We conclude with an example that shows how the nonlinear order conditions can be used to obtain an embedded estimate of the state-dependent nonlinear coupling strength in a dynamical system.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Diffusion-Model-Assisted Supervised Learning of Generative Models for Density Estimation

Here, we present a supervised learning framework of training generative models for density estimation. Generative models, including generative adversarial networks (GANs), normalizing flows, and variational auto-encoders (VAEs), are usually considered as unsupervised learning models, because labeled data are usually unavailable for training. Despite the success of the generative models, there are several issues with the unsupervised training, e.g., requirement of reversible architectures, vanishing gradients, and training instability. To enable supervised learning in generative models, we utilize the score-based diffusion model to generate labeled data. Unlike existing diffusion models that train neural networks to learn the score function, we develop a training-free score estimation method. This approach uses mini-batch-based Monte Carlo estimators to directly approximate the score function at any spatial-temporal location in solving an ordinary differential equation (ODE), corresponding to the reverse-time stochastic differential equation (SDE). This approach can offer both high accuracy and substantial time savings in neural network training. Once the labeled data are generated, we can train a simple, fully connected neural network to learn the generative model in the supervised manner. Compared with existing normalizing flow models, our method does not require the use of reversible neural networks and avoids the computation of the Jacobian matrix. Compared with existing diffusion models, our method does not need to solve the reverse-time SDE to generate new samples. As a result, the sampling efficiency is significantly improved. We demonstrate the performance of our method by applying it to a set of 2D datasets as well as real data from the University of California Irvine (UCI) repository.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

TChem v2.0 - A Software Toolkit for the Analysis of Complex Kinetic Models

TChem is an open source software library for solving complex computational chemistry problems and analyzing detailed chemical kinetic models. The software provides support for: complex kinetic models for gas-phase and surface chemistry; thermodynamic properties based on NASA polynomials; species production/consumption rates; stable time integrator for solving stiff time ordinary differential equations; and, reactor models such as homogenous gas-phase ignition (with analytical Jacobian matrices), continuously stirred tank reactor, plug-flow reactor. This toolkit builds upon earlier versions that were written in C and featured tools for gas-phase chemistry only. The current version of the software was completely refactored in C++, uses an object-oriented programming model, and adopts Kokkos as its portability layer to make it ready for the next generation computing architectures i.e., multi/many core computing platforms with GPU accelerators. We have expanded the range of kinetic models to include surface chemistry and have added examples pertaining to Continuously Stirred Tank Reactors (CSTR) and Plug Flow Reactor (PFR) models to complement the homogenous ignition examples present in the earlier versions. To exploit the massive parallelism available from modern computing platforms, the current software interface is designed to evaluate samples in parallel, which enables large scale parametric studies, e.g. for sensitivity analysis and model calibration.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

CSPlib - A Software Toolkit for the Analysis of Dynamical Systems and Chemical Kinetic Models

CSPlib is an open source software library for analyzing general ordinary differential equation (ODE) systems and detailed chemical kinetic ODE systems. It relies on the computational singular perturbation (CSP) method for the analysis of these systems. The software provides support for: General ODE models (gODE model class) for computing source terms and Jacobians for a generic ODE system; TChem model (ChemElemODETChem model class) for computing source term, Jacobian, other necessary chemical reaction data, as well as the rates of progress for a homogenous batch reactor using an elementary step detailed chemical kinetic reaction mechanism. This class relies on the TChem [2] library; A set of functions to compute essential elements of CSP analysis (Kernel class). This includes computations of the eigensolution of the Jacobian matrix, CSP basis vectors and co-vectors, time scales (reciprocals of the magnitudes of the Jacobian eigenvalues), mode amplitudes, CSP pointers, and the number of exhausted modes. This class relies on the Tines library; A set of functions to compute the eigensolution of the Jacobian matrix using Tines library GPU eigensolver; A set of functions to compute CSP indices (Index Class). This includes participation indices and both slow and fast importance indices.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Mixing Cell Model: A One-Dimensional Numerical Model for Assessment of Water Flow and Contaminant Transport in the Unsaturated Zone

This report describes the Mixing Cell Model code, a one dimensional model for water flow and solute transport in the unsaturated zone under steady state or transient flow conditions. The model is based on the principles and assumptions underlying mixing-cell model formulations. The unsaturated zone is discretized into a series of independent mixing cells. Each cell may have unique hydrologic, lithologic, and sorptive properties. Ordinary differential equations describe the material (water and solute) balance within each cell. Water-flow equations are derived from the continuity equation, assuming that unit gradient conditions exist at all times in each cell. Pressure gradients are considered implicitly through model discretization. Unsaturated hydraulic conductivity and moisture contents are determined by the material specific moisture-characteristic curves. Solute-transport processes include explicit treatment of advective processes, first order chain decay, and linear sorption reactions. Dispersion is addressed through implicit and explicit dispersion. Implicit dispersion is an inherent feature of all mixing-cell models and originates from the formulation of the problem in terms of mass balance around fully mixed volume elements. Expressions are provided that relate implicit dispersion to the physical dispersion of the system. Two FORTRAN codes were developed to solve the water flow and solute-transport equations: (1) the Mixing Cell Model for Flow (MCMF) solves transient water-flow problems and (2) the Mixing Cell Model for Transport (MCMT) solves the solute-transport problem. The transient water-flow problem is typically solved first by estimating the water flux through each cell in the model domain as a function of time using the MCMF code. These data are stored in either ASCII or binary files that are later read by the solute transport code (MCMT). Code output includes solute pore water concentrations, water and solute inventories in each cell and at each specified output time, and water and solute fluxes through each cell and specified output time. Computer run times for coupled transient water flow and solute transport were typically several seconds on a 2 GHz Intel Pentium IV desktop computer. The model was benchmarked against analytical solutions and finite element approximations to the partial differential equations (PDE) describing unsaturated flow and transport. Differences between the maximum solute flux estimated by the mixing cell model and the PDE models were typically less than two percent. This revision includes an option for a fixed concentration lower boundary condition for diffusive fluxes for versions 020321 and later.

54 ENVIRONMENTAL SCIENCES↗