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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 127 records · Page 7
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Preconditioning for Hyper-reduction in Reduced Order Models

Many projection-based reduced order models (pROM) that utilize the governing equation and data to accelerate physical simulations can be applied to nonlinear dynamical systems. To avoid full order model (FOM) scale update for each time step, hyperreduction techniques are developed to sample high dimensional nonlinear terms. Our study aims to investigate if preconditioning the least-squares problem used for the nonlinear approximation can improve the robustness of the condition number of the problem while achieving high accuracy. In our study, we use the row-normalization matrix motivated by the Christoffel function as the preconditioner and solve the corresponding weighted least-squares problem. Numerical results for Lagrangian hydrodynamics examples are analyzed to explore how the preconditioner works compared to existing hyper-reduction techniques.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Divide and conquer: Learning chaotic dynamical systems with multistep penalty neural ordinary differential equations

Forecasting high-dimensional dynamical systems is a fundamental challenge in various fields, such as geosciences and engineering. Neural Ordinary Differential Equations (NODEs), which combine the power of neural networks and numerical solvers, have emerged as a promising algorithm for forecasting complex nonlinear dynamical systems. However, classical techniques used for NODE training are ineffective for learning chaotic dynamical systems. In this work, we propose a novel NODE-training approach that allows for robust learning of chaotic dynamical systems. Here, our method addresses the challenges of non-convexity and exploding gradients associated with underlying chaotic dynamics. Training data trajectories from such systems are split into multiple, non-overlapping time windows. In addition to the deviation from the training data, the optimization loss term further penalizes the discontinuities of the predicted trajectory between the time windows. The window size is selected based on the fastest Lyapunov time scale of the system. Multi-step penalty(MP) method is first demonstrated on Lorenz equation, to illustrate how it improves the loss landscape and thereby accelerates the optimization convergence. MP method can optimize chaotic systems in a manner similar to least-squares shadowing with significantly lower computational costs. Our proposed algorithm, denoted the Multistep Penalty NODE, is applied to chaotic systems such as the Kuramoto-Sivashinsky equation, the two-dimensional Kolmogorov flow, and ERA5 reanalysis data for the atmosphere. It is observed that MP-NODE provide viable performance for such chaotic systems, not only for short-term trajectory predictions but also for invariant statistics that are hallmarks of the chaotic nature of these dynamics.

Chaotic dynamical systems↗
DOE OSTI2026

Direct Nonlinear Approximation for Security Region Boundary of Integrated Energy Systems: A Polynomial Chaos Expansion Solution

The strong interdependence of electricity, gas, and heating systems can facilitate fault propagation within integrated energy systems (IESs), posing significant challenges to secure operation. This paper proposes a polynomial chaos expansion (PCE)-based approximation method to accurately characterize the IES security region boundary (IES–SRB). By integrating the Karush-Kuhn-Tucker conditions with PCE theory, the IES-SRB approximation problem is reformulated as a set of nonlinear equations concerning the approximation coefficients. Using the Galerkin projection method, these equations are further transformed into a system of projection equations that govern the polynomial approximation coefficients in the IES-SRB approximation. To reduce computational complexity while maintaining high approximation accuracy, a piecewise polynomial approximation method is proposed. Numerical studies on the E39-G20-H6 and E118-G96-H52 IES test systems demonstrate that the proposed method can accurately and effectively construct IES security regions.

Wu, Chenghao [Northeast Electric Power University]↗
DOE OSTI2024

Learning nonlinear operators in latent spaces for real-time predictions of complex dynamics in physical systems

Abstract Predicting complex dynamics in physical applications governed by partial differential equations in real-time is nearly impossible with traditional numerical simulations due to high computational cost. Neural operators offer a solution by approximating mappings between infinite-dimensional Banach spaces, yet their performance degrades with system size and complexity. We propose an approach for learning neural operators in latent spaces, facilitating real-time predictions for highly nonlinear and multiscale systems on high-dimensional domains. Our method utilizes the deep operator network architecture on a low-dimensional latent space to efficiently approximate underlying operators. Demonstrations on material fracture, fluid flow prediction, and climate modeling highlight superior prediction accuracy and computational efficiency compared to existing methods. Notably, our approach enables approximating large-scale atmospheric flows with millions of degrees, enhancing weather and climate forecasts. Here we show that the proposed approach enables real-time predictions that can facilitate decision-making for a wide range of applications in science and engineering.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Structural inference of networked dynamical systems with universal differential equations

Networked dynamical systems are common throughout science in engineering; e.g., biological networks, reaction networks, power systems, and the like. For many such systems, nonlinearity drives populations of identical (or near-identical) units to exhibit a wide range of nontrivial behaviors, such as the emergence of coherent structures (e.g., waves and patterns) or otherwise notable dynamics (e.g., synchrony and chaos). Here, we seek to infer (i) the intrinsic physics of a base unit of a population, (ii) the underlying graphical structure shared between units, and (iii) the coupling physics of a given networked dynamical system given observations of nodal states. These tasks are formulated around the notion of the Universal Differential Equation, whereby unknown dynamical systems can be approximated with neural networks, mathematical terms known a priori (albeit with unknown parameterizations), or combinations of the two. We demonstrate the value of these inference tasks by investigating not only future state predictions but also the inference of system behavior on varied network topologies. The effectiveness and utility of these methods are shown with their application to canonical networked nonlinear coupled oscillators.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Data-driven learning of Mori–Zwanzig operators for isotropic turbulence

Developing reduced-order models for turbulent flows, which contain dynamics over a wide range of scales, is an extremely challenging problem. In statistical mechanics, the Mori–Zwanzig (MZ) formalism provides a mathematically exact procedure for constructing reduced-order representations of high-dimensional dynamical systems, where the effects due to the unresolved dynamics are captured in the memory kernel and orthogonal dynamics. Turbulence models based on MZ formalism have been scarce due to the limited knowledge of the MZ operators, which originates from the difficulty in deriving MZ kernels for complex nonlinear dynamical systems. In this work, we apply a recently developed data-driven learning algorithm, which is based on Koopman's description of dynamical systems and Mori's linear projection operator, on a set of fully resolved isotropic turbulence datasets to extract the Mori–Zwanzig operators. With data augmentation using known turbulence symmetries, the extracted Markov term, memory kernel, and orthogonal dynamics are statistically converged and the generalized fluctuation–dissipation relation can be verified. The properties of the memory kernel and orthogonal dynamics, and their dependence on the choices of observables are investigated to address the modeling assumptions that are commonly used in MZ-based models. A series of numerical experiments are then constructed using the extracted kernels to evaluate the memory effects on prediction. The results show that the prediction errors are strongly affected by the choice of observables and can be further reduced by including the past history of the observables in the memory kernel.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Direct Estimation of Parameters in ODE Models Using WENDy: Weak-Form Estimation of Nonlinear Dynamics

Abstract We introduce the Weak-form Estimation of Nonlinear Dynamics (WENDy) method for estimating model parameters for non-linear systems of ODEs. Without relying on any numerical differential equation solvers, WENDy computes accurate estimates and is robust to large (biologically relevant) levels of measurement noise. For low dimensional systems with modest amounts of data, WENDy is competitive with conventional forward solver-based nonlinear least squares methods in terms of speed and accuracy. For both higher dimensional systems and stiff systems, WENDy is typically both faster (often by orders of magnitude) and more accurate than forward solver-based approaches. The core mathematical idea involves an efficient conversion of the strong form representation of a model to its weak form, and then solving a regression problem to perform parameter inference. The core statistical idea rests on the Errors-In-Variables framework, which necessitates the use of the iteratively reweighted least squares algorithm. Further improvements are obtained by using orthonormal test functions, created from a set of $$C^{\infty }$$ C ∞ bump functions of varying support sizes.We demonstrate the high robustness and computational efficiency by applying WENDy to estimate parameters in some common models from population biology, neuroscience, and biochemistry, including logistic growth, Lotka-Volterra, FitzHugh-Nagumo, Hindmarsh-Rose, and a Protein Transduction Benchmark model. Software and code for reproducing the examples is available at https://github.com/MathBioCU/WENDy .

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

HomPINNs: Homotopy physics-informed neural networks for learning multiple solutions of nonlinear elliptic differential equations

Physics-informed neural networks (PINNs) based machine learning is an emerging framework for solving nonlinear differential equations. However, due to the implicit regularity of neural network structure, PINNs can only find the flattest solution in most cases by minimizing the loss functions. In this paper, we combine PINNs with the homotopy continuation method, a classical numerical method to compute isolated roots of polynomial systems, and propose a new deep learning framework, named homotopy physics-informed neural networks (HomPINNs), for solving multiple solutions of nonlinear elliptic differential equations. The implementation of an HomPINN is a homotopy process that is composed of the training of a fully connected neural network, named the starting neural network, and training processes of several PINNs with different tracking parameters. The starting neural network is to approximate a starting function constructed by the trivial solutions, while other PINNs are to minimize the loss functions defined by boundary condition and homotopy functions, varying with different tracking parameters. These training processes are regraded as different steps of a homotopy process, and a PINN is initialized by the well-trained neural network of the previous step, while the first starting neural network is initialized using the default initialization method. Finally, several numerical examples are presented to show the efficiency of our proposed HomPINNs, including reaction-diffusion equations with a heart-shaped domain.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Toward performance-portable PETSc for GPU-based exascale systems

The Portable Extensible Toolkit for Scientific computation (PETSc) library delivers scalable solvers for nonlinear time-dependent differential and algebraic equations and for numerical optimization. The PETSc design for performance portability addresses fundamental GPU accelerator challenges and stresses flexibility and extensibility by separating the programming model used by the application from that used by the library, and it enables application developers to use their preferred programming model, such as Kokkos, RAJA, SYCL, HIP, CUDA, or OpenCL, on upcoming exascale systems. Furthermore, a blueprint for using GPUs from PETSc-based codes is provided, and case studies emphasize the flexibility and high performance achieved on current GPU-based systems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Symmetry-breaking-induced rare fluctuations in a time-delay dynamic system

Inspired by the experimental and numerical findings, we study the dynamic instabilities of two coupled nonlinear delay differential equations that are used to describe the coherent oscillations between the top and bottom boundary layers in turbulent Rayleigh–Bénard convection. By introducing two sensitivity parameters for the instabilities of the top and bottom boundary layers, we find three different types of solutions, namely in-phase single-period oscillations, multi-period oscillations and chaos. The chaos solution contains rare but large amplitude fluctuations. The statistical properties of these fluctuations are consistent with those observed in the experiment for the massive eruption of thermal plumes, which causes random reversals of the large-scale circulation in turbulent Rayleigh–Bénard convection. Finally, our study thus provides new insights into the origin of rare massive eruptions and sudden changes of large-scale flow pattern that are often observed in convection systems of geophysical and astrophysical scales.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Inference of neutrino flavor evolution through data assimilation and neural differential equations

The evolution of neutrino flavor in dense environments such as core-collapse supernovae and binary compact object mergers constitutes an important and unsolved problem. Its solution has potential implications for the dynamics and heavy-element nucleosynthesis in these environments. In this paper, we build upon recent work to explore inference-based techniques for the estimation of model parameters and neutrino flavor evolution histories. Here, we combine data assimilation, ordinary differential equation solvers, and neural networks to craft an inference approach tailored for nonlinear dynamical systems. Using this architecture, and a simple two-neutrino-beam, two-flavor model, we compare the performances of nine different optimization algorithms and expand upon previous assessments of the efficacy of inference for tackling problems in flavor evolution. We find that employing this new architecture, together with evolutionary optimization algorithms, accurately captures flavor histories in the small-scale model and allows us to quickly explore both model parameters and initial flavor content. In future work we plan to extend these inference techniques to large numbers of neutrinos.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2025

Understanding plasma turbulence through exact coherent structures

Plasma turbulence is a key challenge in understanding transport phenomena in magnetically confined plasmas. This work presents a generalized framework to analyze plasma turbulence that utilizes periodic orbit theory. In periodic orbit theory, doubly periodic solutions (coherent structures) of the governing equation(s) serve as building blocks of the considered turbulent dynamics. To illustrate the concept and method, the particularly simple Kuramoto–Sivashinsky (referred to here as LMRT for the original authors: LaQuey, Mahajan, Rutherford, and Tang) trapped-ion mode toy model is used. By applying numerical optimization techniques to the LMRT equation, we extract coherent spacetime patterns that represent the library of allowable fundamental structures of the equation. These structures provide a framework to systematically describe turbulence as a composition of recurrent solutions, revealing an underlying order within chaotic plasma motion. Although illustrated here using the simplified LMRT model for clarity, this framework provides a general strategy that can be extended to more complex and realistic models of plasma turbulence, including gyrokinetic systems. This offers a new method for predicting and potentially controlling transport processes in fusion plasmas by providing a bridge between nonlinear dynamical systems theory and plasma physics in the form of a generalized framework with which to analyze and understand spatially extended nonlinear partial differential equations.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2020

Efficient and Robust Dynamic Simulation of Power Systems With Holomorphic Embedding

Dynamic simulation is vitally important in power system analysis, but traditional approaches based on numerical integration over small time steps are time-consuming. Also, the Newton-Raphson method suffers from difficulty in convergence when solving nonlinear algebraic equations. In this paper, we propose a novel dynamic simulation approach based on holomorphic embedding. By obtaining a high-order approximation of system dynamics, it achieves a much larger time step and thus enhances the computational efficiency significantly. In addition, the new approach avoids non-convergence issues in solving algebraic equations, which improves robustness. The approach includes flexible modeling of synchronous generators and controllers, and we propose a method for modeling generator coordinate transformations. The approach is tested on the IEEE 39-bus, 10-generator system and a Polish 2383-bus, 327-generator system. The results demonstrate promising computational efficiency and satisfactory numerical robustness for the analysis of large-scale power systems.

42 ENGINEERING↗
DOE OSTI2021

Probabilistic learning on manifolds constrained by nonlinear partial differential equations for small datasets

A novel extension of the Probabilistic Learning on Manifolds (PLoM) is presented. It makes it possible to synthesize solutions to a wide range of nonlinear stochastic boundary value problems described by partial differential equations (PDEs) for which a stochastic computational model (SCM) is available and which depend on a vector-valued random control parameter. The cost of a single numerical evaluation of this SCM is assumed to be such that only a limited number of points can be computed for constructing the training dataset (small data). Each point of the training dataset is made up of realizations from a vector-valued stochastic process (the stochastic solution) and the associated random control parameter on which it depends. The presented PLoM constrained by PDE allows for generating a large number of learned realizations of the stochastic process and its corresponding random control parameter. These learned realizations are generated so as to minimize the vector-valued random residual of the PDE in the mean-square sense. Appropriate novel methods are developed to solve this challenging problem. Three applications are presented. The first one is a simple uncertain nonlinear dynamical system with a nonstationary stochastic excitation. The second one concerns the 2D nonlinear unsteady Navier–Stokes equations for incompressible flows in which the Reynolds number is the random control parameter. Here, the last one deals with the nonlinear dynamics of a 3D elastic structure with uncertainties. The results obtained make it possible to validate the PLoM constrained by stochastic PDE but also provide further validation of the PLoM without constraint.

Machine learning↗
DOE OSTI2020

Nonlinear dynamic analysis of solution multiplicity of buoyancy ventilation in a typical underground structure

Buoyancy ventilation is widely used in underground buildings, such as underground hydropower stations. Multiple solutions of buoyancy ventilation may exist in those underground structures. In this study, we developed a transient model comprising an ordinary differential equation system to describe buoyancy ventilation patterns in typical two-zone underground structures. Additionally, the accuracy of the model was validated. Nonlinear dynamical analysis was conducted to study multiple steady-state airflow. According to mathematical derivation, the configuration of one local heat source at the bottom corner introduces two stable solutions. The criterion to determine the stability and existence of solutions for more general scenarios was developed. Using this criterion, we obtained the multiple steady states of any two-zone underground buildings for different stack height ratios and the strength ratios of the heat sources. Furthermore, this criterion can be adopted for the design of buoyancy ventilation or natural smoke ventilation systems. Designers can change the height ratio of the stack or the heat ratio of two zones to induce the desired ventilation patterns. Finally, a case study was conducted with field measurements to demonstrate the use of the nonlinear dynamical analysis method to investigate the multiple steady states of buoyancy ventilation. Through the case study, we validated that the proposed criterion could produce the same result as the nonlinear dynamical analysis.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
DOE OSTI2025

PETSc/TAO developments for GPU-based early exascale systems

The Portable Extensible Toolkit for Scientific Computation (PETSc) library provides scalable solvers for nonlinear time-dependent differential and algebraic equations and for numerical optimization via the Toolkit for Advanced Optimization (TAO). PETSc is used in dozens of scientific fields and is an important building block for many simulation codes. During the U.S. Department of Energy’s Exascale Computing Project, the PETSc team has made substantial efforts to enable efficient utilization of the massive fine-grain parallelism present within exascale compute nodes and to enable performance portability across exascale architectures. We recap some of the challenges that designers of numerical libraries face in such an endeavor, and then discuss the many developments we have made, which include the addition of new GPU backends, features supporting efficient on-device matrix assembly, better support for asynchronicity and GPU kernel concurrency, and new communication infrastructure. In conclusion, we evaluate the performance of these developments on some pre-exascale systems as well as the early exascale systems Frontier and Aurora, using compute kernel, communication layer, solver, and mini-application benchmark studies, and then close with a few observations drawn from our experiences on the tension between portable performance and other goals of numerical libraries.

Exascale Computing Project (ECP)↗
DOE OSTI2021

Hamiltonian systems, Toda lattices, solitons, Lax pairs on weighted Z -graded graphs

In this study, we consider discrete one-dimensional nonlinear equations and present the procedure of lifting them to Z -graded graphs. We identify conditions that allow one to lift one-dimensional solutions to solutions on graphs. In particular, we prove the existence of solitons for static potentials on graded fractal graphs. We also show that even for a simple example of a topologically interesting graph, the corresponding non-trivial Lax pairs and associated unitary transformations do not lift to a Lax pair on the Z -graded graph.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

Chapter 4 - Recent Advances in Identification of Differential Equations from Noisy Data: IDENT Review

Differential equations and numerical methods are extensively used to model various real-world phenomena in science and engineering. With modern developments, we aim to find the underlying differential equation from a single observation of time-dependent data. If we assume that the differential equation is a linear combination of various linear and nonlinear differential terms, then the identification problem can be formulated as solving a linear system. The goal then reduces to finding the optimal coefficient vector that best represents the time derivative of the given data. We review some recent works on the identification of differential equations. We find some common themes for the improved accuracy: (i) The formulation of linear system with proper denoising is important, (ii) how to utilize sparsity and model selection to find the correct coefficient support needs careful attention, and (iii) there are ways to improve the coefficient recovery. We present an overview and analysis of recent developments on the topic.

97 MATHEMATICS AND COMPUTING↗