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All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Time-series machine-learning error models for approximate solutions to parameterized dynamical systems

This work proposes a machine-learning framework for modeling the error incurred by approximate solutions to parameterized dynamical systems. In particular, we extend the machine-learning error models (MLEM) framework proposed in Ref. Freno and Carlberg (2019) to dynamical systems. The proposed Time-Series Machine-Learning Error Modeling (T-MLEM) method constructs a regression model that maps features – which comprise error indicators that are derived from standard a posteriori error-quantification techniques – to a random variable for the approximate-solution error at each time instance. The proposed framework considers a wide range of candidate features, regression methods, and additive noise models. We consider primarily recursive regression techniques developed for time-series modeling, including both classical time-series models (e.g., autoregressive models) and recurrent neural networks (RNNs), but also analyze standard non-recursive regression techniques (e.g., feed-forward neural networks) for comparative purposes. Finally, numerical experiments conducted on multiple benchmark problems illustrate that the long short-term memory (LSTM) neural network, which is a type of RNN, outperforms other methods and yields substantial improvements in error predictions over traditional approaches.

42 ENGINEERING↗
NASA NTRS2010

Parameterizing Coefficients of a POD-Based Dynamical System

A method of parameterizing the coefficients of a dynamical system based of a proper orthogonal decomposition (POD) representing the flow dynamics of a viscous fluid has been introduced. (A brief description of POD is presented in the immediately preceding article.) The present parameterization method is intended to enable construction of the dynamical system to accurately represent the temporal evolution of the flow dynamics over a range of Reynolds numbers. The need for this or a similar method arises as follows: A procedure that includes direct numerical simulation followed by POD, followed by Galerkin projection to a dynamical system has been proven to enable representation of flow dynamics by a low-dimensional model at the Reynolds number of the simulation. However, a more difficult task is to obtain models that are valid over a range of Reynolds numbers. Extrapolation of low-dimensional models by use of straightforward Reynolds-number-based parameter continuation has proven to be inadequate for successful prediction of flows. A key part of the problem of constructing a dynamical system to accurately represent the temporal evolution of the flow dynamics over a range of Reynolds numbers is the problem of understanding and providing for the variation of the coefficients of the dynamical system with the Reynolds number. Prior methods do not enable capture of temporal dynamics over ranges of Reynolds numbers in low-dimensional models, and are not even satisfactory when large numbers of modes are used. The basic idea of the present method is to solve the problem through a suitable parameterization of the coefficients of the dynamical system. The parameterization computations involve utilization of the transfer of kinetic energy between modes as a function of Reynolds number. The thus-parameterized dynamical system accurately predicts the flow dynamics and is applicable to a range of flow problems in the dynamical regime around the Hopf bifurcation. Parameter-continuation software can be used on the parameterized dynamical system to derive a bifurcation diagram that accurately predicts the temporal flow behavior.

Kalb, Virginia L.↗
DOE OSTI2024

DS-GL: Advancing Graph Learning via Harnessing the Power of Nature within Dynamic Systems

With the rapid digitization of the world, an increasing number of real-world applications are turning to nonEuclidean data, modeled as graphs. Due to their intrinsic high complexity and irregularity, learning from graph data demands tremendous computational power. Recently, CMOS-compatible Ising machines, i.e., dynamic systems composed of CMOS components, have emerged as a new approach that harnesses the inherent power of natural annealing within dynamic systems to efficiently resolve binary optimization problems and have been adopted for traditional graph computation, such as max-cut. However, when performing complex Graph Learning (GL) tasks, Ising machines face significant hurdles: (i) they are inherently binary and thus ill-suited for real-valued problems; (ii) their expensive all-to-all coupling network that guarantees effective natural annealing poses daunting scalability concerns. To address these challenges, this paper proposes a nature-powered graph learning framework dubbed DS-GL, which is the first effort to transform the process of solving graph learning problems into the natural annealing process within a parameterized dynamic system embodied as a CMOS chip. To tackle the two major hurdles, DS-GL first augments the Ising machine architecture to modify the self-reaction term of its Hamiltonian function from linear to quadratic, effectively serving as an energy regulator. This adjustment maintains the system’s original physical interpretation while enabling it to process continuous, real-valued data. Second, to address the scaling issue, DS-GL further upgrades the real-valued dense Ising machine by decomposing it into a mesh-based multi-PE dynamic system that supports efficient distributed spatial-temporal co-annealing across different PEs through sparse interconnects. By exploiting the inherent sparsity and component structures in real-world graphs, DS-GL is able to map complex graph learning tasks onto the scalable dynamic system while maintaining high accuracy. Evaluations with three diverse GL applications across six real-world datasets, including traffic flow and COVID-19 prediction, show that DS-GL can deliver from 102× to 106× speedups and 500× energy reduction over Graph Neural Networks on GPUs, with 5% - 20% accuracy enhancement.

Song, Ruibing↗
DOE OSTI2023

Probabilistic-learning-based stochastic surrogate model from small incomplete datasets for nonlinear dynamical systems

We consider a high-dimensional nonlinear computational model of a dynamical system, parameterized by a vector-valued control parameter, in the presence of uncertainties represented by an uncontrolled parameter modeled by a vector-valued random variable, and possibly with stochastic excitation. The objective is to construct a statistical surrogate model where the input is any deterministic value of the control parameter, and the output is a vector-valued observation of the computational model, which is a random vector whose probability measure is updated using a target dataset. To construct this statistical surrogate model, the stochastic response of the computational model must be built, which is a vector-valued time-discretized stochastic process in high dimension, depending on the control parameter. It is assumed that the computational cost of a single evaluation of the deterministic model is high. For the probabilistic updating, we consider a subset of the components of the observation of the computational model, defined as the “identification observation” of the computational model, for which a small target dataset is available. Therefore, the target dataset is associated with partial observability, corresponding to an incomplete data case. Given a prior probability model of the random control and uncontrolled parameters, a training dataset is constructed, consisting of realizations of the random triplet composed of the stochastic response, the random identification observation, and the random control parameter. Since the computational cost of a single evaluation of the deterministic model is assumed to be large, the training dataset is also of small size. The main challenges in this problem are the high dimensionality, partial observability leading to incomplete data in the target dataset for the identification observation of the computational model (which is not sufficient to identify the computational stochastic responses), and the availability of a small training dataset. To address these challenges, we propose a methodology based on statistical methods for constructing necessary reduced representations, direct probabilistic learning under constraints using probabilistic learning on manifolds (PLoM) constrained by the target dataset, and the use of a weak formulation of the Fourier transform of probability measures. Statistical conditioning is also employed to explore the learned dataset. The constructed predictive statistical surrogate model can be implemented in the context of online computation. Here, we apply this approach to a problem of nonlinear stochastic dynamics in high dimensions within the framework of deformable solids mechanics.

Engineering↗
DOE OSTI2021

Deep learning of parameterized equations with applications to uncertainty quantification

We propose a learning algorithm for discovering unknown parameterized dynamical systems by using observational data of the state variables. Our method is built upon and extends the recent work of discovering unknown dynamical systems, in particular those using deep neural network (DNN). We propose a DNN structure, largely based upon the residual network (ResNet), to not only learn the unknown form of the governing equation but also take into account the random effect embedded in the system, which is generated by the random parameters. Once the DNN model is successfully constructed, it is able to produce system prediction over longer term and for arbitrary parameter values. For uncertainty quantification, it allows us to conduct uncertainty analysis by evaluating solution statistics over the parameter space.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Nature-GL: A Revolutionary Learning Paradigm Unleashing Nature’s Power in Real-World Spatial-Temporal Graph Learning

Spatial-Temporal Graph Learning (ST-GL) is a prominent research area due to its unique capability to effectively learn real-world graphs. Applications of ST-GL pose stringent and various demands on not only real-time inference with low energy cost and high ac- curacy but also fast training. Unfortunately, as Moore’s Law approaches its limits and ST-GL model complexity drastically grows, the gap between digital hardware’s computational power and ST- GL application demands is widening. In response, this paper introduces Nature-GL, a nature-powered graph learning paradigm that exploits the principle of entropy increase to advance graph learning. In particular, Nature-GL transforms both the training and inference of real-valued ST-GL into electron-speed natural anneal- ing processes of a parameterized dynamical system that represents the target graphs. Experimental results across four real-world ap- plications with six datasets demonstrate that Nature-GL achieves orders-of-magnitude speedups in both training and inference, delivering higher accuracy compared to Graph Neural Networks.

Liu, Chuan [University of Rochester]↗
DOE OSTI2021

Non-intrusive nonlinear model reduction via machine learning approximations to low-dimensional operators

Abstract Although projection-based reduced-order models (ROMs) for parameterized nonlinear dynamical systems have demonstrated exciting results across a range of applications, their broad adoption has been limited by their intrusivity: implementing such a reduced-order model typically requires significant modifications to the underlying simulation code. To address this, we propose a method that enables traditionally intrusive reduced-order models to be accurately approximated in a non-intrusive manner. Specifically, the approach approximates the low-dimensional operators associated with projection-based reduced-order models (ROMs) using modern machine-learning regression techniques. The only requirement of the simulation code is the ability to export the velocity given the state and parameters; this functionality is used to train the approximated low-dimensional operators. In addition to enabling nonintrusivity, we demonstrate that the approach also leads to very low computational complexity, achieving up to $$10^3{\times }$$ 10 3 × in run time. We demonstrate the effectiveness of the proposed technique on two types of PDEs. The domain of applications include both parabolic and hyperbolic PDEs, regardless of the dimension of full-order models (FOMs).

42 ENGINEERING↗
NASA NTRS1989

Querying databases of trajectories of differential equations: Data structures for trajectories

One approach to qualitative reasoning about dynamical systems is to extract qualitative information by searching or making queries on databases containing very large numbers of trajectories. The efficiency of such queries depends crucially upon finding an appropriate data structure for trajectories of dynamical systems. Suppose that a large number of parameterized trajectories gamma of a dynamical system evolving in R sup N are stored in a database. Let Eta is contained in set R sup N denote a parameterized path in Euclidean Space, and let the Euclidean Norm denote a norm on the space of paths. A data structure is defined to represent trajectories of dynamical systems, and an algorithm is sketched which answers queries.

Grossman, Robert↗
NASA NTRS1988

An approach to simultaneous system design. II - Nonswitching gain and dynamic feedback compensation by algebraic geometric methods

This paper studies structured uncertainty problems in feedback system design, considers a compact parameterization of the space of linear dynamical systems and introduces 'base points' and 'critical points' as two algebraic-geometric objects that have significance in sensitivity and robustness studies, respectively. Using the Nevanlinna-Pick interpolation theory, the author obtains a necessary and sufficient condition for simultaneous stabilization of a structured one-parameter family of plants. A recent result due to Kharitonov, on the simultaneous stability of a parameterized family of polynomials, leads to a sufficiency condition for simultaneous stabilization of a structured multiparameter family of plants. Furthermore, the author considers 'simultaneous pole placement' of an r-tuple of plants as a means to arbitrarily tune the natural frequencies of a multimode linear dynamical system. The concept of 'nondegenerate' and 'twisted' r-tuples of plants is introduced as the pole placement problem is studied via Schubert enumerative geometry as an intersection problem on the associated Grassmannian. Various other design problems, viz., the strong stabilization problem and the dead beat control problem, are also considered.

Ghosh, Bijoy K.↗
NASA NTRSDate not supplied

Adaptive Optimization for System Performance and Combined Bernstein Polynomial, Optimal Reciprocal Collision Avoidance, Differential Dynamic Programming for Trajectory Replanning and Collision Avoidance for UAM Vehicles

The emerging urban air mobility (UAM) sector in aerospace is driving development of unconventional multi-modal vehicle configurations and autonomous flight. The combination of multi-modal vehicle dynamics, complex environment, requirements to deal with flight contingencies in an efficient and safe manner, as well as necessity for precise trajectory following and performance, are the driving influence behind adaptive optimization for system performance. We are interested in trajectory optimization algorithm that would system parameter estimation and identifying the optimal switching time between modes of hybrid dynamical systems. This presentation discusses a parameterized optimal control trajectory optimization algorithm that is an extended and generalized version of Differential Dynamic Programming (DDP), titled Parameterized Differential Dynamic Programming (PDDP). DDP is an efficient trajectory optimization algorithm relying on second order approximations of a system’s dynamics and cost function and has recently been applied to optimize systems with time invariant parameters. Experiments are presented applying PDDP to solve model predictive control (MPC) and moving horizon estimation (MHE) tasks simultaneously. In particular, PDDP is used to determine the optimal transition point between flight regimes of a complex urban air mobility (UAM) class vehicle exhibiting multiple phases of flight and to identify and compensate for actuation faults.

optimization↗
DOE OSTI2020

Data-driven recovery of hidden physics in reduced order modeling of fluid flows

In this article, we introduce a modular hybrid analysis and modeling (HAM) approach to account for hidden physics in reduced order modeling (ROM) of parameterized systems relevant to fluid dynamics. The hybrid ROM framework is based on using first principles to model the known physics in conjunction with utilizing the data-driven machine learning tools to model the remaining residual that is hidden in data. This framework employs proper orthogonal decomposition as a compression tool to construct orthonormal bases and Galerkin projection (GP) as a model to build the dynamical core of the system. Our proposed methodology hence compensates structural or epistemic uncertainties in models and utilizes the observed data snapshots to compute true modal coefficients spanned by these bases. The GP model is then corrected at every time step with a data-driven rectification using a long short-term memory (LSTM) neural network architecture to incorporate hidden physics. A Grassmannian manifold approach is also adopted for interpolating basis functions to unseen parametric conditions. The control parameter governing the system's behavior is thus implicitly considered through true modal coefficients as input features to the LSTM network. The effectiveness of the HAM approach is then discussed through illustrative examples that are generated synthetically to take hidden physics into account. Furthermore, our approach thus provides insights addressing a fundamental limitation of the physics-based models when the governing equations are incomplete to represent underlying physical processes.

42 ENGINEERING↗
NASA NTRS1991

Querying databases of trajectories of differential equations 2: Index functions

Suppose that a large number of parameterized trajectories (gamma) of a dynamical system evolving in R sup N are stored in a database. Let eta is contained R sup N denote a parameterized path in Euclidean space, and let parallel to center dot parallel to denote a norm on the space of paths. A data structures and indices for trajectories are defined and algorithms are given to answer queries of the following forms: Query 1. Given a path eta, determine whether eta occurs as a subtrajectory of any trajectory gamma from the database. If so, return the trajectory; otherwise, return null. Query 2. Given a path eta, return the trajectory gamma from the database which minimizes the norm parallel to eta - gamma parallel.

Grossman, Robert↗
DOE OSTI2026

ECLEIRS: Exact conservation law embedded identification of reduced states for parameterized nonlinear conservation laws from sparse and noisy data

Multi-query applications such as parameter estimation, uncertainty quantification and design optimization for parameterized partial differential equation (PDE) systems are expensive. While reduced/latent state dynamics approaches for parameterized PDEs offer a viable alternative, these approaches rely on high-quality data and struggle with highly sparse spatiotemporal noisy measurements typically obtained from experiments. Furthermore, there is no guarantee that these models satisfy governing physical conservation laws. In this article, we propose a reduced state dynamics approach, referred to as ECLEIRS, that embeds exact conservation in the solution and flux representation by utilizing a space-time divergence-free neural network formulation. We compare ECLEIRS with other reduced state dynamics approaches, those that do not enforce any physical constraints and those with physics-informed loss functions, for three shock-propagation problems: 1-D advection, 1-D Burgers and 2-D Euler equations. In conclusion, the numerical experiments conducted in this study demonstrate that ECLEIRS provides the most accurate prediction of dynamics for unseen parameters even in the presence of highly sparse and noisy data.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Aggregated DER_A Model Parameterization via Online Moving Horizon Estimation

Here, this paper introduces a methodology for parameterizing the DER_A model using a novel smooth mathematical representation, simplifying the process and preserving accuracy in modeling inverter-based generator (IBG). The methodology employs an online parameterization process that can operate in real-time. The model parameterization process is structured into five sequential steps, each targeting a specific aspect of the DER_A model through moving horizon estimation. This approach adapts to systems with varying voltage and frequency support requirements by selectively applying each step. Simulation results on systems with both known and unknown parameters validate the methodology’s effectiveness. The online moving horizon estimation technique accurately captures the dynamics of the overall system and ensures that the parameterized DER_A model closely mirrors the real system’s voltage, current, and power dynamics. The findings highlight the potential of this methodology to substantially improve and simplify the dynamic modeling of power systems, paving the way for more reliable and robust IBG and grid integration.

42 ENGINEERING↗
DOE OSTI2020

Impact analysis of DERs on bulk power system stability through the parameterization of aggregated DER_a model for real feeders

With an ever increasing percentage of distributed energy resources (DERs) connected behind the meter in the distribution system, it is becoming increasingly important to equip transmission planners with the visibility of DER dynamic performance in distribution system. Not having visibility of the disconnection of DERs with the occurrence of transmission events, could result in an erroneous view of the stability of bulk power system. Here, in this paper, a parameterized aggregated model (DER_a) serves as a representation of the distribution-level dynamics of real residential feeders, which is used for analysis of bulk power system stability. The parameters are obtained by executing dynamic Monte Carlo simulations. Faults are then induced at the substation level causing the DERs to trip which subsequently enables the parameterization of the low and high voltage breakpoints (v l0 , v l1 , v h0 , and v h1 ) of the DER_a model’s partial voltage trip block. These parameters are then utilized to study the effectiveness of the DER_a model to represent the behavior of the aggregated DERs’ response and their impact on the bulk power system. The case study shows the ability of the positive-sequence DER_a model to provide an accurate estimation of DERs that are susceptible to trip due to 3-$\phi$ and 1-$\phi$ faults of transmission.

14 SOLAR ENERGY↗