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At least 73 records · Page 4

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DOE OSTI2026

Efficient data-driven regression for reduced-order modeling of spatial pattern formation

We present an efficient data-driven regression approach for constructing reduced-order models (ROMs) of reaction-diffusion systems exhibiting pattern formation. The ROMs are learned non-intrusively from available training data of physically accurate numerical simulations. The method can be applied to general nonlinear systems through the use of polynomial model form, while not requiring knowledge of the underlying physical model, governing equations, or numerical solvers. The process of learning ROMs is posed as a low-cost least-squares problem in a reduced-order subspace identified via Proper Orthogonal Decomposition (POD). Numerical experiments on classical pattern-forming systems–including the Schnakenberg and Mimura–Tsujikawa models–demonstrate that higher-order surrogate models significantly improve prediction accuracy while maintaining low computational cost. The proposed method provides a flexible, non-intrusive model reduction framework, well suited for the analysis of complex spatio-temporal pattern formation phenomena.

Data-driven modeling↗

DOE OSTI2024

GX: a GPU-native gyrokinetic turbulence code for tokamak and stellarator design

GX is a code designed to solve the nonlinear gyrokinetic system for low-frequency turbulence in magnetized plasmas, particularly tokamaks and stellarators. In GX, our primary motivation and target is a fast gyrokinetic solver that can be used for fusion reactor design and optimization along with wide-ranging physics exploration. Here, this has led to several code and algorithm design decisions, specifically chosen to prioritize time to solution. First, we have used a discretization algorithm that is pseudospectral in the entire phase space, including a Laguerre–Hermite pseudospectral formulation of velocity space, which allows for smooth interpolation between coarse gyrofluid-like resolutions and finer conventional gyrokinetic resolutions and efficient evaluation of a model collision operator. Additionally, we have built GX to natively target graphics processors (GPUs), which are among the fastest computational platforms available today. Finally, we have taken advantage of the reactor-relevant limit of small $\rho _*$ by using the radially local flux-tube approach. In this paper we present details about the gyrokinetic system and the numerical algorithms used in GX to solve the system. We then present several numerical benchmarks against established gyrokinetic codes in both tokamak and stellarator magnetic geometries to verify that GX correctly simulates gyrokinetic turbulence in the small $\rho _*$. Moreover, we show that the convergence properties of the Laguerre–Hermite spectral velocity formulation are quite favourable for nonlinear problems of interest. Coupled with GPU acceleration, which we also investigate with scaling studies, this enables GX to be able to produce useful turbulence simulations in minutes on one (or a few) GPUs and higher fidelity results in a few hours using several GPUs. GX is open-source software that is ready for fusion reactor design studies.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2026

A Scalable Reduced‐Order Model for the Steady Navier–Stokes Equations

Scaling up new scientific technologies from laboratory to industry often involves demonstrating performance on a larger scale. Computer simulations can accelerate design and predictions in the deployment process, though traditional numerical methods are computationally intractable even for intermediate pilot plant scales. Recently, the component reduced order modeling method has been developed to tackle this challenge by combining projection reduced order modeling and discontinuous Galerkin domain decomposition. However, while many scientific or engineering applications involve nonlinear physics, this method has only been demonstrated for various linear systems. In this work, the component reduced order modeling method is extended to steady Navier–Stokes flow, with application to general nonlinear physics in view. The large‐scale, global domain is decomposed into a combination of small‐scale unit component. Linear subspaces for flow velocity and pressure are identified via proper orthogonal decomposition over sample snapshots collected from each small‐scale unit component. Velocity bases are augmented with a pressure supremizer to satisfy the inf–sup condition for stable pressure prediction. Two different nonlinear reduced order modeling methods are employed and compared for efficient evaluation of nonlinear advection: A third‐order tensor projection operator and the empirical quadrature procedure. The proposed method is demonstrated on the flow over arrays of five different unit objects, achieving a 23‐fold speedup with less than 4% relative error in domains up to 256 times larger than the unit components. Furthermore, a numerical experiment with the pressure supremizer strongly indicates the need for a supremizer for stable pressure prediction. A comparison between the tensorial approach and the empirical quadrature procedure revealed a slight advantage of the empirical quadrature procedure. The framework is compared with an alternating Schwarz‐based reduced‐order approach, demonstrating improved efficiency and robustness for the DG‐based global solver while retaining flexibility for sub‐scale iterative solvers. The method is further extended to a coupled advection–diffusion and Navier–Stokes system, illustrating its applicability to multi‐physics problems and its potential for more general, inter‐coupled nonlinear systems.

42 ENGINEERING↗

DOE OSTI2024

Real-time capable modeling of ICRF heating on NSTX and WEST via machine learning approaches

Abstract A real-time capable core Ion Cyclotron Range of Frequencies (ICRF) heating model on NSTX and WEST is developed. The model is based on two nonlinear regression algorithms, the random forest ensemble of decision trees and the multilayer perceptron neural network. The algorithms are trained on TORIC ICRF spectrum solver simulations of the expected flat-top operation scenarios in NSTX and WEST assuming Maxwellian plasmas. The surrogate models are shown to successfully capture the multi-species core ICRF power absorption predicted by the original model for the high harmonic fast wave and the ion cyclotron minority heating schemes while reducing the computational time by six orders of magnitude. Although these models can be expanded, the achieved regression scoring, computational efficiency and increased model robustness suggest these strategies can be implemented into integrated modeling frameworks for real-time control applications.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2026

Nonlinear saturation of ballooning modes in stellarators

Ballooning mode saturation is investigated in realistic stellarator configurations using the flux tube approach of Ham et al (2018 Plasma Phys. Control. Fusion 60 075017), Ham et al (2016 Phys. Rev. Lett. 116 235001). The method is adapted to account for the lack of exact force balance in stellarator equilibrium solvers that assume existence of nested flux surfaces. A variational approach for calculating flux tube energy is developed to overcome this force error problem in stellarator numerical equilibria. Saturated (equilibrium) flux tube states that cross 10%–20% of the plasma minor radius are shown to exist for linearly ballooning unstable profiles. It is shown that several features of the displaced flux tube structure in a full nonlinear MHD simulation of Wendelstein 7X are reproduced by our model. Saturated states are found in a compact stellarator equilibrium close but below the marginal ballooning linear instability, i.e. the unperturbed equilibrium is metastable. This suggests that edge-localized-mode-like explosive MHD behavior may be possible in stellarators.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2026

ReMU: regional minimal updating for model-based derivative-free optimization

Derivative-free optimization (DFO) problems are optimization problems where derivative information is unavailable or extremely difficult to obtain. Model-based DFO solvers have been applied extensively in scientific computing. Powell's NEWUOA (2004) [Powell, The NEWUOA software for unconstrained optimization without derivatives, in Large-Scale Nonlinear Optimization, Nonconvex Optimization and its Applications Vol. 83, G. Di Pillo and M. Roma, eds., Springer, 2006, pp. 255–297] and Wild's POUNDerS (2014) [Wild, Solving derivative-free nonlinear least squares problems with POUNDERS, in Advances and Trends in Optimization with Engineering Applications, T. Terlaky, M.F. Anjos, and S. Ahmed, eds., SIAM, 2017, pp. 529–540] explore the numerical power of the minimal norm Hessian (MNH) model for DFO and contributed to the open discussion on building better models with fewer data to achieve faster numerical convergence. Another decade later, we propose the regional minimal updating (ReMU) models, and extend the previous models into a broader class, including the H 2 norm models [Xie and Yuan, Least H 2 norm updating of quadratic interpolation models for derivative-free trust-region algorithms, IMA J. Numer. Anal. 46 (2025), pp. 21–50]. This paper shows motivation behind ReMU models, computational details, theoretical and numerical results on particular extreme points and the barycentre of ReMU's weight coefficient region, and the associated KKT matrix error and distance. Novel metrics, such as the truncated Newton step error, are proposed to numerically understand the new models' properties. A new algorithmic strategy, based on iteratively adjusting the ReMU model type, is also proposed, and shows numerical advantages by combining and switching between the barycentric model and the classic least Frobenius norm model in an online fashion.

derivative-free trust-region methods↗

DOE OSTI2024

Enhancing predictive capabilities in fusion burning plasmas through surrogate-based optimization in core transport solvers

Abstract This work presents the PORTALS framework (Rodriguez-Fernandez et al 2022 Nucl. Fusion 62 076036), which leverages surrogate modeling and optimization techniques to enable the prediction of core plasma profiles and performance with nonlinear gyrokinetic simulations at significantly reduced cost, with no loss of accuracy. The efficiency of PORTALS is benchmarked against standard methods, and its full potential is demonstrated on a unique, simultaneous 5-channel (electron temperature, ion temperature, electron density, impurity density and angular rotation) prediction of steady-state profiles in a DIII-D ITER Similar Shape plasma with GPU-accelerated, nonlinear CGYRO (Candy et al 2016 J. Comput. Phys. 324 73–93). This paper also provides general guidelines for accurate performance predictions in burning plasmas and the impact of transport modeling in fusion pilot plants studies.

Physics↗

DOE OSTI2026

SAM Finite Volume Method Development Status Update: GCR Application, Restart, and MultiApp

The System Analysis Module (SAM) is being developed as a modern system analysis code for advanced non-light-water-reactor safety analysis under the U.S. DOE NEAMS program. Previous feasibility studies have demonstrated that a staggered-grid finite volume method (SG-FVM), implemented under the MOOSE framework, can deliver more than an order of magnitude speedup over the existing continuous Galerkin finite element method (CG-FEM) solver for liquid-cooled, incompressible but thermally expandable flow systems. This work extends the previous effort to compressible, gas-cooled reactor applications, where pressure couples directly into the mass equation adding additional nonlinearity into the equation system. New code capabilities are implemented for pebble bed high-temperature gas-cooled reactor (PB-HTGR) analysis, including a pebble bed CoreChannel component, built-in pebble bed effective thermal conductivity model and channel-to-channel crossflow model. The capabilities are tested, benchmarked, and demonstrated for problems with increased level of model and physical complexities, including the HTTU effective thermal conductivity test, the SANA passive cooling test, and a demonstration case using the GPBR200 reactor design covering steady-state operation, DLOFC and PLOFC transients. Across all cases, the SG-FVM solver demonstrated strong robustness and efficiency, and the solutions agree well with reference results and data. The finding of this work proves that SG-FVM is a viable and efficient solver pathway for compressible, gas-cooled reactor system analysis in SAM. In addition, work has been done to successfully support SAM-FVM recover/restart code feature that is essential to reactor safety analysis applications, and MultiApp code feature that is essential to multi-scale and multi-physics simulations. In summary, this work continued from previous feasibility studies, and further demonstrated that the SG-FVM will serve as a strong foundation for SAM’s advanced solver algorithm for future deployment.

Zou, Ling↗

DOE OSTI2024

Sparsified time-dependent Fourier neural operators for fusion simulations

This paper presents a sparsified Fourier neural operator for coupled time-dependent partial differential equations (ST-FNO) as an efficient machine learning surrogate for fluid and particle-based fusion codes such as NIMROD (Non-Ideal Magnetohydrodynamics with Rotation - Open Discussion) and GTC (Gyrokinetic Toroidal Code). ST-FNO leverages the structures in the governing equations and utilizes neural operators to represent Green's function-like numerical operators in the corresponding numerical solvers. Once trained, ST-FNO can rapidly and accurately predict dynamics in fusion devices compared with first-principle numerical algorithms. In general, ST-FNO represents an efficient and accurate machine learning surrogate for numerical simulators for multi-variable nonlinear time-dependent partial differential equations, with the proposed architectures and loss functions. The efficacy of ST-FNO has been demonstrated using quiescent H-mode simulation data from NIMROD and kink-mode simulation data from GTC. The ST-FNO H-mode results show orders of magnitude reduction in memory and central processing unit usage in comparison with the numerical solvers in NIMROD when computing fields over a selected poloidal plane. The ST-FNO kink-mode results achieve a factor of 2 reduction in the number of parameters compared to baseline FNO models without accuracy loss.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2026

Improving ideal MHD equilibrium accuracy with physics-informed neural networks

We present a novel approach to compute three-dimensional magnetohydrodynamic equilibria with isotropic pressure profiles and nested surfaces by parametrizing Fourier modes with artificial neural networks (NNs). The full nonlinear global force residual of single equilibria across the volume in real space is then minimized with first order optimizers and compared to equilibria computed by conventional solvers. Already, we observe competitive computational cost to arrive at the same minimum residuals computable with existing codes. With increased computational cost, lower minima of the residual are computable with the NNs than with any other tested solver, establishing a new lower bound for the force residual. We use minimally complex NNs, and we expect significant improvements for solving not only single equilibria with NNs, but also for creating NN models valid over continuous distributions of equilibria.

ideal magnetohydrodynamics↗

DOE OSTI2025

Learning and discovering multiple solutions using physics-informed neural networks with random initialization and deep ensemble

In this work we explore the capability of physics-informed neural networks (PINNs) to discover multiple solutions. Many real-world phenomena governed by nonlinear differential equations (DEs), such as fluid flow, exhibit multiple solutions under the same conditions, yet capturing this solution multiplicity remains a significant challenge. A key difficulty lies in providing appropriate initial conditions or guesses, as widely used time-marching schemes and Newton’s method are highly sensitive to these choices when solving complex computational problems. While machine learning models, particularly PINNs, have shown promise in solving DEs, their ability to capture multiple solutions remains underexplored. In this work, we propose a simple and practical approach using PINNs to learn and discover multiple solutions. We first demonstrate that PINNs, when combined with random initialization and deep ensemble method—originally developed for uncertainty quantification—can effectively uncover multiple solutions to nonlinear ordinary and partial DEs. Although training large ensembles of PINNs may appear computationally demanding, this can be done efficiently using vectorization techniques supported by modern deep learning frameworks, allowing many networks to be trained simultaneously. Our approach highlights the critical role of initialization in shaping solution diversity, addressing an often-overlooked aspect of machine learning for scientific computing. Furthermore, we propose utilizing PINN-generated solutions as initial conditions or initial guesses for conventional numerical solvers to enhance accuracy and efficiency in capturing multiple solutions. Extensive numerical experiments, including the Allen–Cahn equation and cavity flow, where our approach successfully identifies both stable and unstable solutions, validate the effectiveness of our method. These findings establish a general and efficient framework for addressing solution multiplicity in nonlinear DEs.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

Grain2Mesh: Mesh Generation for Grain-Scale Nonlinear Elasticity Modeling

The nonlinear hysteretic behavior of rocks under cyclic loading is a crucial area of study in geomechanics. The macroscopic response of a variety of materials has been found to be contingent upon the behavior of the micro-scale structure. This project aims to develop a functional and maintainable software package for generating a multi-phase numerical mesh and accompanying simulation files for finite element modeling used in computational mechanics solvers. Meshes generated from images often lack key preprocessing that reduces noise and prevents mesh element distortion that can increase computational cost. By incorporating user feedback throughout, grain2mesh ensures a high-fidelity mesh that can be used to model grain-scale interactions such as shearing, crack propagation, and interfacial material contrast. Scientific applications of this software include material fracturing, stress-strain analysis for natural and engineered materials, and nonlinear meso-scale analysis.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2026

Integral Kernel Methods for Nonlinear Parabolic-Elliptic Systems

Nonlinear parabolic-elliptic systems arise in many physical, biological, and chemical phenomena such as chemotaxis, ion transport, self-gravitating particles, and Brownian vortices. Existing methods struggle with the strong coupling and high nonlinearity and nonlocality of some of these systems, especially the ill-conditioned, convection-dominated problems. To overcome numerical difficulties, current approaches rely on initial guesses, preconditioning, or iterative techniques with no convergence guarantees. They might suffer from poor scalability, large memory usage, and difficulty to parallelize. Inspired by the connection of parabolic-elliptic systems to stochastic processes, we introduce a novel meshless, monolithic, and fully explicit method that naturally encapsulates the elliptic and parabolic operators into a single step which updates each node deterministically with global information. By being fully quadrature-based, it avoids solving systems of discretized equations and does not utilize initial guesses or preconditioning, while requiring little memory and being easy to parallelize. We first derive the method in an integral kernel formulation with quadratic complexity in the number of integration nodes and then leverage kernel-independent fast multipole methods (FMM) to present a scalable algorithm with linear complexity. We provide numerical examples for the Poisson-Nernst-Planck equations in one, two, and three dimensions, together with the derivation of the integral kernel for each case. Furthermore, the examples demonstrate the fast convergence and scalability of the FMM-accelerated algorithm, as well as its suitability for convection-dominated problems, making it competitive against traditional PDE solvers.

PDE systems↗

DOE OSTI2025

Alternative mixed integer linear programming optimization for joint job scheduling and data allocation in grid computing

This paper presents a novel approach to the joint optimization of job scheduling and data allocation in grid computing environments. We formulate this joint optimization problem as a mixed integer quadratically constrained program. To tackle the nonlinearity in the constraint, we alternatively fix a subset of decision variables and optimize the remaining ones via Mixed Integer Linear Programming (MILP). We solve the MILP problem at each iteration via an off-the-shelf MILP solver. Our experimental results show that our method significantly outperforms existing heuristic methods, employing either independent optimization or joint optimization strategies. We have also verified the generalization ability of our method over grid environments with various sizes and its high robustness to the algorithm setting.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2026

Quantum kinetic modeling of KEEN waves in a warm-dense regime

We report the first fully kinetic, quantum study of kinetic electrostatic electron nonlinear (KEEN) waves, showing that quantum diffraction systematically erodes the classical trapping mechanism, narrows harmonic locking to the fundamental, and hastens post-drive decay. Electrons are evolved with a second-order Strang-split 1D1V Wigner–Poisson solver that couples conservative semi-Lagrangian WENO advection to an analytic Fourier space update for the non-local Wigner term, while ions remain classical. We focus on collisionless dynamics in a weakly coupled regime, providing a controlled baseline before collisional extensions. Short, frequency-tuned ponderomotive pulses drive KEEN formation in a uniform Maxwellian plasma; as the dimensionless quantum parameter H rises from the classical limit to values relevant to warm-dense matter, doped semiconductors, and 2D electron systems, the drive threshold increases, higher harmonics are damped, trapped electron vortices diffuse, and the subplasma electrostatic energy relaxes to a lower stationary level, as confirmed by continuous wavelet analysis. These microscopic changes carry macroscopic weight. Ignition-scale capsules now compress matter to regimes where the electron de Broglie wavelength rivals the Debye length, making classical kinetic descriptions insufficient. By extending KEEN physics into this quantum domain, our results offer a potential diagnostic of non-equilibrium electron dynamics for next-generation inertial-confinement designs and high-energy-density platforms, indicating that predictive fusion modeling may benefit from the integration of kinetic fidelity with quantum effects.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2024

Development of Segregated Thermal-Hydraulics Solvers in MOOSE

The simulation of fluid flows is an essential part of the design and analysis of nuclear systems. Algorithms able to simulate flows at different fidelity levels are available in the Multiphysics Object-Oriented Simulation Environment (MOOSE) and MOOSE-based applications such as Pronghorn \cite{novak2018pronghorn}, Pronghorn-Subchannel, RELAP-7, and SAM. Currently, significant effort is being invested in the development of coarse-mesh Computational Fluid Dynamics (CFD) capabilities within MOOSE and Pronghorn for the simulation of Generation IV nuclear reactors. Traditionally, the solution algorithms in MOOSE have relied on Newton or quasi-Newton methods (such as the preconditioned Jacobian-free Newton-Krylov method) where residuals and Jacobians (or approximations thereof) are constructed. Both Newton and quasi-Newton methods require the solution of a linear system at each nonlinear Newton iteration with the Jacobian as the system matrix. The Jacobian contains blocks originating from all variables in the problem (i.e., for thermal-hydraulics at least pressure, velocities, and temperature). Due to the formulation of the problem in a general multiphysics setting on unstructured mesh, creating a good preconditioner for the linear system can be challenging, thus many fluid applications have utilized direct solver-based methods such as LU factorization. However, with increasing system size and complexity in multi-dimensional problems, the direct solution of linear systems becomes computationally expensive both in execution time and and memory. For this reason, recent effort has focused on adapting segregated solution algorithms for CFD problems in MOOSE. These algorithms use fixed-point iteration between segregated systems whose assembly and preconditioning are easier those of the monolithic system. Initial results show that the segregated solution algorithm outperforms the monolithic approach in terms of memory usage and for large 3D problems in terms of CPU time as well.

42 ENGINEERING↗

DOE OSTI2024

Development of Segregated Thermal-Hydraulics Solvers in MOOSE

42 ENGINEERING↗

DOE OSTI2026

Conditional diffusion machine-learning framework for mapping valence electron distribution from convergent beam electron diffraction

Quantitative convergent beam electron diffraction (CBED) enables determination of aspherical valence electron distributions through refinement of low-order structure factors, which are highly sensitive to chemical bonding and charge density variations. However, conventional quantitative CBED (QCBED) requires solving a highly nonlinear inverse problem with many coupled parameters, and computationally intensive dynamical diffraction calculations, making it time-consuming and difficult to apply to complex systems. More broadly, reconstructing charge density and orbital electron distribution from diffraction data has long been a central challenge in both x-ray and electron crystallography. Here, in this study, we introduce an artificial-intelligence (AI)-based framework that replaces traditional refinement with a data-driven inverse solver. Using a large synthetic CBED dataset generated by Bloch-wave simulations, we train a conditional diffusion model to directly infer crystal structural parameters and multipole density formalism parameters, and hence valence electron distributions, from CBED patterns alone. By learning from forward simulations across realistic parameter space, the model effectively solves the inverse problem. Compared with direct regression approaches, the diffusion-based framework provides posterior parameter distributions for rigorous uncertainty quantification while preserving quantitative fidelity and reducing analysis time by orders of magnitude. By eliminating the need for external single-crystal x-ray diffraction data and complex nonlinear refinement, this approach enables practical, high-throughput, and in situ quantitative CBED, enabling real-time mapping of valence electron distributions and their correlation with functional responses in quantum and energy materials.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗