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DOE OSTI2020

Non-autoregressive time-series methods for stable parametric reduced-order models

Advection-dominated dynamical systems, characterized by partial differential equations, are found in applications ranging from weather forecasting to engineering design where accuracy and robustness are crucial. There has been significant interest in the use of techniques borrowed from machine learning to reduce the computational expense and/or improve the accuracy of predictions for these systems. These rely on the identification of a basis that reduces the dimensionality of the problem and the subsequent use of time series and sequential learning methods to forecast the evolution of the reduced state. Often, however, machine-learned predictions after reduced-basis projection are plagued by issues of stability stemming from incomplete capture of multiscale processes as well as due to error growth for long forecast durations. To address these issues, we have developed a non-autoregressive time series approach for predicting linear reduced-basis time histories of forward models. In particular, we demonstrate that non-autoregressive counterparts of sequential learning methods such as long short-term memory (LSTM) considerably improve the stability of machine-learned reduced-order models. Further, we evaluate our approach on the inviscid shallow water equations and show that a non-autoregressive variant of the standard LSTM approach that is bidirectional in the principal component directions obtains the best accuracy for recreating the nonlinear dynamics of partial observations. Moreover-and critical for many applications of these surrogates-inference times are reduced by three orders of magnitude using our approach, compared with both the equation-based Galerkin projection method and the standard LSTM approach.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Linear embedding of nonlinear dynamical systems and prospects for efficient quantum algorithms

The simulation of large nonlinear dynamical systems, including systems generated by discretization of hyperbolic partial differential equations, can be computationally demanding. Such systems are important in both fluid and kinetic computational plasma physics. This motivates exploring whether a future error-corrected quantum computer could perform these simulations more efficiently than any classical computer. In this work, we describe a method for mapping any finite nonlinear dynamical system to an infinite linear dynamical system (embedding) and detail three specific cases of this method that correspond to previously studied mappings. Then we explore an approach for approximating the resulting infinite linear system with finite linear systems (truncation). Using a number of qubits only logarithmic in the number of variables of the nonlinear system, a quantum computer could simulate truncated systems to approximate output quantities if the nonlinearity is sufficiently weak. Other aspects of the computational efficiency of the three detailed embedding strategies are also discussed.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

On the mechanism of ionization oscillations in Hall thrusters

Low-frequency ionization oscillations involving plasma and neutral density (breathing modes) are the most violent perturbations in Hall thrusters for electric propulsion. Because of its simplicity, the zero-dimensional (0D) predator–prey model of two nonlinearly coupled ordinary differential equations for plasma and neutral density has often been used for the characterization of such oscillations and scaling estimates. In this work, we investigate the properties of its continuum analog, the one-dimensional (1D) system of two nonlinearly coupled equations in partial derivatives (PDEs) for plasma and neutral density. This is a more general model, of which the standard 0D predator–prey model is a special limit case. We show that the 1D model is stable and does not show any oscillations for the boundary conditions relevant to Hall thrusters and the uniform ion velocity. We then propose a reduced 1D model based on two coupled PDEs for plasma and neutral densities that is unstable and exhibit oscillations if the ion velocity profile with the near-the-anode back-flow (toward the anode) region is used. Comparisons of the reduced model with the predictions of the full model that takes into account the self-consistent plasma response show that the main properties of the breathing mode are well captured. In particular, it is shown that the frequency of the breathing mode oscillations is weakly dependent on the final ion velocity but shows a strong correlation with the width of the ion back-flow region.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2021

A multi-sheath model for highly nonlinear plasma wakefields

An improved description for nonlinear plasma wakefields with phase velocities near the speed of light is presented and compared against fully kinetic particle-in-cell simulations. These wakefields are excited by intense particle beams or lasers pushing plasma electrons radially outward, creating an ion bubble surrounded by a sheath of electrons characterized by the source term S≡−1enp(ρ−Jz/c), where ρ and Jz are the charge and axial current densities, respectively. Previously, the sheath source term was described phenomenologically with a positive-definite function, resulting in a positive definite wake potential. In reality, the wake potential is negative at the rear of the ion column which is important for self-injection and accurate beam loading models. To account for this, we introduce a multi-sheath model in which the source term, S, of the plasma wake can be negative in regions outside the ion bubble. Using this model, we obtain a new expression for the wake potential and a modified differential equation for the bubble radius. Numerical results obtained from these equations are validated against particle-in-cell simulations for unloaded and loaded wakes. The new model provides accurate predictions of the shape and duration of trailing bunch current profiles that flatten plasma wakefields. It is also used to design a trailing bunch for a desired longitudinally varying loaded wakefield. We present beam loading results for laser wakefields and discuss how the model can be improved for laser drivers in future work. Finally, we discuss differences between the predictions of the multi- and single-sheath models for beam loading.

Dalichaouch, T. N. (ORCID:0000000247350150)↗

DOE OSTI2021

Machine learning methods for probabilistic locked-mode predictors in tokamak plasmas

A rotating tokamak plasma can interact resonantly with the external helical magnetic perturbations, also known as error fields. This can lead to locking and then to disruptions. We leverage machine learning (ML) methods to predict the locking events. We use a coupled third-order nonlinear ordinary differential equation model to represent the interaction of the magnetic perturbation and the plasma rotation with the error field. This model is sufficient to describe qualitatively the locking and unlocking bifurcations. Here, we explore using ML algorithms with the simulation data and experimental data, focusing on the methods that can be used with sparse datasets. These methods lead to the possibility of the avoidance of locking in real-time operations. We describe the operational space in terms of two control parameters: the magnitude of the error field and the rotation frequency associated with the momentum source that maintains the plasma rotation. The outcomes are quan- tified by order parameters that completely characterize the state, whether locked or unlocked. We use unsupervised ML methods to classify locked/unlocked states and note the usefulness of a certain normalization of the order parameters. Three supervised ML classifiers are used in suite to estimate the probability of locking in the region of control parameter space with hysteresis, i.e., the set of control parameters for which both locked and unlocked states can exist. The results show that a neural network gives the best estimate of the locking probability. An analogy of the present locking model with the van der Waals equation of state is also provided.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2022

Discovering causal structure with reproducing-kernel Hilbert space ε -machines

We merge computational mechanics’ definition of causal states (predictively equivalent histories) with reproducing-kernel Hilbert space (RKHS) representation inference. The result is a widely applicable method that infers causal structure directly from observations of a system’s behaviors whether they are over discrete or continuous events or time. A structural representation—a finite- or infinite-state kernel ϵ-machine—is extracted by a reduced-dimension transform that gives an efficient representation of causal states and their topology. In this way, the system dynamics are represented by a stochastic (ordinary or partial) differential equation that acts on causal states. We introduce an algorithm to estimate the associated evolution operator. Paralleling the Fokker–Planck equation, it efficiently evolves causal-state distributions and makes predictions in the original data space via an RKHS functional mapping. We demonstrate these techniques, together with their predictive abilities, on discrete-time, discrete-value infinite Markov-order processes generated by finite-state hidden Markov models with (i) finite or (ii) uncountably infinite causal states and (iii) continuous-time, continuous-value processes generated by thermally driven chaotic flows. The method robustly estimates causal structure in the presence of varying external and measurement noise levels and for very high-dimensional data.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2022

Learning the temporal evolution of multivariate densities via normalizing flows

In this work, we propose a method to learn multivariate probability distributions using sample path data from stochastic differential equations. Specifically, we consider temporally evolving probability distributions (e.g., those produced by integrating local or nonlocal Fokker–Planck equations). Here, we analyze this evolution through machine learning assisted construction of a time-dependent mapping that takes a reference distribution (say, a Gaussian) to each and every instance of our evolving distribution. If the reference distribution is the initial condition of a Fokker–Planck equation, what we learn is the time-T map of the corresponding solution. Specifically, the learned map is a multivariate normalizing flow that deforms the support of the reference density to the support of each and every density snapshot in time. We demonstrate that this approach can approximate probability density function evolutions in time from observed sampled data for systems driven by both Brownian and Lévy noise. We present examples with two- and three-dimensional, uni- and multimodal distributions to validate the method.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Multi-fidelity modeling to predict the rheological properties of a suspension of fibers using neural networks and Gaussian processes

Unveiling the rheological properties of fiber suspensions is of paramount interest to many industrial applications. There are multiple factors, such as fiber aspect ratio and volume fraction, that play a significant role in altering the rheological behavior of suspensions. Three-dimensional (3D) numerical simulations of coupled differential equations of the suspension of fibers are computationally expensive and time-consuming. Machine learning algorithms can be trained on the available data and make predictions for the cases where no numerical data are available. However, some widely used machine learning surrogates, such as neural networks, require a relatively large training dataset to produce accurate predictions. Multi-fidelity models, which combine high-fidelity data from numerical simulations and less expensive lower fidelity data from resources such as simplified constitutive equations, can pave the way for more accurate predictions. Here, we focus on neural networks and the Gaussian processes with two levels of fidelity, i.e., high and low fidelity networks, to predict the steady-state rheological properties, and compare them to the single-fidelity network. High-fidelity data are obtained from direct numerical simulations based on an immersed boundary method to couple the fluid and solid motion. The low-fidelity data are produced by using constitutive equations. Multiple neural networks and the Gaussian process structures are used for the hyperparameter tuning purpose. Results indicate that with the best choice of hyperparameters, both the multi-fidelity Gaussian processes and neural networks are capable of making predictions with a high level of accuracy with neural networks demonstrating marginally better performance.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2022

Temperature distribution in a laser-heated diamond anvil cell as described by finite element analysis

Finite element analysis (FEA) is a powerful tool for numerically solving partial differential equations over complex geometries and is thus useful for analyzing heat transport in laser-heated diamond anvil cell (LHDAC) experiments. Our models expand on previously published simulations by calculating the volume-averaged temperatures of both the sample and insulation/pressure media under steady-state heating to determine the thermal pressure of the hot sample. Our goal is to produce an accurate relationship between the measured surface temperature of the absorbing sample and the temperature of the transparent insulating media, which is used to determine thermal pressure but susceptible to steep temperature gradients. We find that in doing so, our FEA models of temperature within the pressure/insulation media can differ from simplified estimates of temperature gradients by more than a factor of 2. We also explore temperature-dependent and temperature-independent thermal conductivity models and find that the volume-averaged temperatures differ by up to a factor of 1.3, forcing the predicted thermal pressures determined to also differ by up to a factor of 1.5 at a temperature of 2000 K at 50 GPa for neon. Higher temperatures exacerbate this difference. We also find that unintentional asymmetric sample insertion and sample heating, which are common in LHDAC experiments, do not have a first-order effect on volume-averaged temperatures. The FEA models, available in both Python and FlexPDE, are versatile across different sample geometries, materials, and heat source laser shapes.

Farah, Frederick↗

DOE OSTI2022

Time-series forecasting using manifold learning, radial basis function interpolation, and geometric harmonics

We address a three-tier numerical framework based on nonlinear manifold learning for the forecasting of high-dimensional time series, relaxing the “curse of dimensionality” related to the training phase of surrogate/machine learning models. At the first step, we embed the high-dimensional time series into a reduced low-dimensional space using nonlinear manifold learning (local linear embedding and parsimonious diffusion maps). Then, we construct reduced-order surrogate models on the manifold (here, for our illustrations, we used multivariate autoregressive and Gaussian process regression models) to forecast the embedded dynamics. Finally, we solve the pre-image problem, thus lifting the embedded time series back to the original high-dimensional space using radial basis function interpolation and geometric harmonics. The proposed numerical data-driven scheme can also be applied as a reduced-order model procedure for the numerical solution/propagation of the (transient) dynamics of partial differential equations (PDEs). In conclusion, we assess the performance of the proposed scheme via three different families of problems: (a) the forecasting of synthetic time series generated by three simplistic linear and weakly nonlinear stochastic models resembling electroencephalography signals, (b) the prediction/propagation of the solution profiles of a linear parabolic PDE and the Brusselator model (a set of two nonlinear parabolic PDEs), and (c) the forecasting of a real-world data set containing daily time series of ten key foreign exchange rates spanning the time period 3 September 2001–29 October 2020.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2023

Quantum computing for fusion energy science applications

This is a review of recent research exploring and extending present-day quantum computing capabilities for fusion energy science applications. We begin with a brief tutorial on both ideal and open quantum dynamics, universal quantum computation, and quantum algorithms. Then, we explore the topic of using quantum computers to simulate both linear and nonlinear dynamics in greater detail. Because quantum computers can only efficiently perform linear operations on the quantum state, it is challenging to perform nonlinear operations that are generically required to describe the nonlinear differential equations of interest. In this work, we extend previous results on embedding nonlinear systems within linear systems by explicitly deriving the connection between the Koopman evolution operator, the Perron–Frobenius evolution operator, and the Koopman–von Neumann evolution (KvN) operator. We also explicitly derive the connection between the Koopman and Carleman approaches to embedding. Extension of the KvN framework to the complex-analytic setting relevant to Carleman embedding, and the proof that different choices of complex analytic reproducing kernel Hilbert spaces depend on the choice of Hilbert space metric are covered in the appendixes. Finally, we conclude with a review of recent quantum hardware implementations of algorithms on present-day quantum hardware platforms that may one day be accelerated through Hamiltonian simulation. We discuss the simulation of toy models of wave–particle interactions through the simulation of quantum maps and of wave–wave interactions important in nonlinear plasma dynamics.

Joseph, I. (ORCID:0000000255400840)↗

DOE OSTI2024

Probabilistic locked mode predictor in the presence of a resistive wall and finite island saturation in tokamaks

We present a framework for estimating the probability of locking to an error field in a rotating tokamak plasma. This leverages machine learning methods trained on data from a mode-locking model, including an error field, resistive magnetohydrodynamics modeling of the plasma, a resistive wall, and an external vacuum region, leading to a fifth-order ordinary differential equation (ODE) system. It is an extension of the model without a resistive wall introduced by Akçay et al. [Phys. Plasmas 28, 082106 (2021)]. Tearing mode saturation by a finite island width is also modeled. We vary three pairs of control parameters in our studies: the momentum source plus either the error field, the tearing stability index, or the island saturation term. The order parameters are the time-asymptotic values of the five ODE variables. Normalization of them reduces the system to 2D and facilitates the classification into locked (L) or unlocked (U) states, as illustrated by Akçay et al., [Phys. Plasmas 28, 082106 (2021)]. This classification splits the control space into three regions: L̂, with only L states; Û, with only U states; and a hysteresis (hysteretic) region Ĥ, with both L and U states. In regions L̂ and Û, the cubic equation of torque balance yields one real root. Region Ĥ has three roots, allowing bifurcations between the L and U states. The classification of the ODE solutions into L/U is used to estimate the locking probability, conditional on the pair of the control parameters, using a neural network. We also explore estimating the locking probability for a sparse dataset, using a transfer learning method based on a dense model dataset.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2025

A compartmentalized model of multiphase chemical kinetics

There are significant challenges in predicting multiphase chemical kinetics due to the complex coupling of reaction and mass transport across a phase boundary (i.e., interface). Here, we describe a framework for predicting multiphase kinetics that embeds the elementary kinetic steps of reaction, solvation, and diffusion into a coarse grain spatial description of two phases. The model is constructed to bridge the short-timescale interfacial dynamics observed in molecular simulations with the longer timescales observed in kinetic experiments. A simple set of governing differential equations is derived, which, when solved numerically or analytically, yield accurate predictions of multiphase kinetics in microdroplets. Although the equations are formulated for gas-liquid reactions, the underlying conceptual framework is general and can be applied to transformations in other two-phase systems (solid-liquid, liquid-liquid, etc.).

Chemical kinetics and dynamics↗

DOE OSTI2026

Neural operator transformers capture bifurcating drift-wave turbulence in fusion plasma simulations

Self-consistent modeling of turbulence-driven transport is critical for optimizing confinement in magnetically confined fusion plasmas, such as tokamaks and stellarators. In particular, capturing the long-term co-evolution of turbulence, flow, and background plasma profiles remains computationally challenging. Direct numerical simulation of these multiscale, highly nonlinear processes is often demanding and impractical for real-time control or design optimization. To address this bottleneck, we investigate transformer-based neural operator partial differential equation surrogates for emulating the dynamics of drift-wave turbulence bifurcation mediated by zonal flows, using the modified Hasegawa–Wakatani (MHW) model as a prototypical system. We find that the finetuned neural operator model has excellent performance in capturing the multi-spatiotemporal-scales of MHW turbulence bifurcation and is robust to testing on rare and out-of-distribution dynamics. Specifically, we demonstrate that a single unified model accurately predicts both quasi-steady-state turbulence and a wide range of dynamical transition processes, such as nonlinear saturation, spontaneous suppression of turbulence, and the emergence of macroscopic zonal flows, over time horizons vastly exceeding the local turbulence correlation time. This computationally efficient approach establishes a strong foundation for fast, AI-based modeling of complex, multiscale phenomena in magnetized fusion plasmas.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2023

Fluctuating hydrodynamics and the Rayleigh–Plateau instability

The Rayleigh–Plateau instability occurs when surface tension makes a fluid column become unstable to small perturbations. At nanometer scales, thermal fluctuations are comparable to interfacial energy densities. Consequently, at these scales, thermal fluctuations play a significant role in the dynamics of the instability. These microscopic effects have previously been investigated numerically using particle-based simulations, such as molecular dynamics (MD), and stochastic partial differential equation–based hydrodynamic models, such as stochastic lubrication theory. In this paper, we present an incompressible fluctuating hydrodynamics model with a diffuse-interface formulation for binary fluid mixtures designed for the study of stochastic interfacial phenomena. An efficient numerical algorithm is outlined and validated in numerical simulations of stable equilibrium interfaces. We present results from simulations of the Rayleigh–Plateau instability for long cylinders pinching into droplets for Ohnesorge numbers of Oh = 0.5 and 5.0. Both stochastic and perturbed deterministic simulations are analyzed and ensemble results show significant differences in the temporal evolution of the minimum radius near pinching. Short cylinders, with lengths less than their circumference, were also investigated. As previously observed in MD simulations, we find that thermal fluctuations cause these to pinch in cases where a perturbed cylinder would be stable deterministically. Finally, we show that the fluctuating hydrodynamics model can be applied to study a broader range of surface tension–driven phenomena.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2026

A Statistician’s Overview of Physics-Informed Neural Networks for Spatio-Temporal Data

The recent success of deep neural network models with physical constraints (so-called, Physics-Informed Neural Networks, PINNs) has led to renewed interest in the incorporation of mechanistic information in predictive models. Statisticians and others have long been interested in this problem, which has led to several practical and innovative solutions dating back decades. In this overview, we focus on the problem of data-driven prediction and inference of dynamic spatio-temporal processes that include mechanistic information, such as would be available from partial differential equations, with a strong focus on the quantification of uncertainty associated with data, process, and parameters. Here, we give a brief review of several paradigms and focus our attention on Bayesian implementations given they naturally accommodate uncertainty quantification. We then show that it is straight-forward to include the Bayesian PINN (B-PINN) within the Bayesian hierarchical model (BHM) framework that has long been considered for modeling dynamic spatio-temporal processes. Such a BHM-PINN is illustrated via a simulation study in which a latent nonlinear Burgers’ equation PDE governs the dynamics of Poisson distributed spatio-temporal data. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

Bayesian↗

DOE OSTI2020

On the Derivation of Quasi-Newton Formulas for Optimization in Function Spaces

Newton’s method is usually preferred when solving optimization problems due to its superior convergence properties compared to gradient-based or derivative-free optimization algorithms. However, deriving and computing second-order derivatives needed by Newton’s method often is not trivial and, in some cases, not possible. In such cases quasi-Newton algorithms are a great alternative. In this paper, we provide a new derivation of well-known quasi-Newton formulas in an infinite-dimensional Hilbert space setting. Furthermore, it is known that quasi-Newton update formulas are solutions to certain variational problems over the space of symmetric matrices. In this paper, we formulate similar variational problems over the space of bounded symmetric operators in Hilbert spaces. By changing the constraints of the variational problem we obtain updates (for the Hessian and Hessian inverse) not only for the Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton method but also for Davidon–Fletcher–Powell (DFP), Symmetric Rank One (SR1), and Powell-Symmetric-Broyden (PSB). In addition, for an inverse problem governed by a partial differential equation (PDE), we derive DFP and BFGS “structured” secant formulas that explicitly use the derivative of the regularization and only approximates the second derivative of the misfit term. We show numerical results that demonstrate the desired mesh-independence property and superior performance of the resulting quasi-Newton methods.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Fusion Fuel Cycle Inventory Reduction Studies Using a Processing-Time–Based Discrete-Time Interval Model

Developing a Fusion Pilot Plant (FPP) design that minimizes risks due to tritium in-process inventory (IPI) is an important concern for the operation of commercial devices. This becomes even more of concern since an FPP will be breeding more tritium than is burned in the reactor for sustainability. The IPI is the tritium moving through the system that is not in the storage and delivery subsystem. Here a process model that solves time-dependent differential equations based on processing times was used to investigate the reduction of the IPI of a potential fuel cycle design. The impact of new and more efficient technologies such as direct internal recycling (DIR), metal foil pumps, continuous pumping, improved isotope separation, and hydrogen separating continuous pumps on IPI was investigated by adjusting subsystem processing times and material flow streams. It was shown that any of the insertions of DIR studied in this paper caused a reduction in the total IPI of the system and proved to be the optimal way to reduce the IPI in the system. Fuel cycle modifications near the torus, such as a coupled DIR and improved pumping systems, produced the largest reductions in tritium inventory.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗