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DOE OSTI2024

Cost-efficient finite-volume high-order schemes for compressible magnetohydrodynamics

We present an efficient dimension-by-dimension finite-volume method which solves the adiabatic magnetohydrodynamics equations at high discretization order, using the constrained-transport approach on Cartesian grids. Results are presented up to tenth order of accuracy. The algorithmic architecture of this method is very close to that of commonly employed second-order schemes: it requires only one reconstructed value per face for each computational cell, independently of the scheme's order. This property is highly beneficial for the numerical efficiency. It results from reusing the required values already available in neighboring grid cells, in contrast to standard algorithms that require a number of reconstructions and evaluations which increases with the scheme's order of accuracy. At a given resolution, these high-order schemes present significantly less numerical dissipation than commonly employed lower-order approaches. Thus, results of comparable accuracy are achievable at a substantially coarser resolution, yielding overall performance gains. We also present a way to include physical dissipative terms: viscosity, magnetic diffusivity and cooling functions, respecting the finite-volume and constrained-transport frameworks. Benefits of this method are shown through applications in turbulent flows.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Evaluation of Hybrid FPOG Applications in Regulated and Deregulated Markets Using HERON

Recent changes in the U.S. energy market, such as low natural gas prices and increased electricity production for variable renewable energy (VRE) sources, have led to an economic crisis for existing light-water reactor (LWR) nuclear power plants (NPP). Many owners and operators of LWRs have elected to decommission these plants rather than continue using them as consistent sources of clean baseload power. This has led to exploration of various possibilities to increase the economic viability of these units, including market restructuring to monetize benefits LWRs already provide to the grid through ancillary markets, load following and economic dispatch, and possible integration of secondary systems directly to the NPP for production of additional products through technologies such as hydrogen electrolysis or water desalination. Previous studies have considered the technologies associated with these Integrated Energy Systems (IES) activities, and the analysis of markets for these secondary products. To analyze the economic viability of various system configurations including IES, especially given the uncertainty surrounding load demand, electricity prices, and the availability of VRE resources, the stochastic technoeconomic analysis package HERON (Heuristic Energy Resource Optimization Network) was released earlier this year as an extension of the risk analysis framework RAVEN (Risk Analysis Virtual Environment). HERON focuses foremost on making the complex uncertainty quantification analysis tools approachable for energy systems analysts, also providing general dispatch optimization algorithms for those workflows. HERON continues to be improved and tested as a significant part of the IES viability analyses performed in this work. HERON is not a capacity expansion model. To consider market and grid energy system development in a variety of scenarios, HERON is best used in coupling with modelling tools such as US-REGEN, which sacrifice some of the uncertainty analysis and resolution of HERON's modelling for the ability to efficiently predict the change in the grid energy system's profile due to economic drivers over decades. HERON can then use this information to explore the economic viability of introducing changes to the predicted outcomes, such as the introduction of an IES. In this work, experts at EPRI using US-REGEN provide six projection scenarios for use in HERON stochastic technoeconomic analysis (STEA) in considering the options available for increasing LWR economic viability through introduction of a hydrogen-centric IES using a high-temperature steam electrolysis plant (HTSE), hydrogen storage, and a constant-rate contracted hydrogen consumer. The results obtained are differential in nature; they do not report expected profits for any configuration, but rather report on the possible increase in the NPV of a configuration with respect to a baseline no-IES configuration. Due to the uncertainty captured in the variable net load of the systems, there is likewise uncertainty in the mean values reported. We consider this viability both in terms of a regulated market, where the energy producers and IES are owned and operated by single entity, as well as a deregulated market, where the IES chooses its bid for electricity generation and is then dispatched by the grid system operator. Results indicate that for deregulated markets, the inclusion of the IES is often statistically beneficial. This is especially true in policies that are not favorable towards nuclear, as nuclear is less often dispatched and is forced to deal with frequent idle capacity. In the nominal case as well as the case of carbon tax policies, inclusion of the IES clearly benefited the economic performance of the NPP. In the regulated case, however, there was a trend towards minimizing the IES, likely due to the optimal sizing performed by US-REGEN of the NPP within the system as well as the lack of penalty for idle capacity at the NPP in the regulated market analyses.

99 GENERAL AND MISCELLANEOUS↗

DOE OSTI2025

Solving sparse finite element problems on neuromorphic hardware

The finite element method (FEM) is one of the most important and ubiquitous numerical methods for solving partial differential equations (PDEs) on computers for scientific and engineering discovery. Applying the FEM to larger and more detailed scientific models has driven advances in high-performance computing for decades. Here we demonstrate that scalable spiking neuromorphic hardware can directly implement the FEM by constructing a spiking neural network that solves the large, sparse, linear systems of equations at the core of the FEM. We show that for the Poisson equation, a fundamental PDE in science and engineering, our neural circuit achieves meaningful levels of numerical accuracy and close to ideal scaling on modern, inherently parallel and energy-efficient neuromorphic hardware, specifically Intel’s Loihi 2 neuromorphic platform. We illustrate extensions to irregular mesh geometries in both two and three dimensions as well as other PDEs such as linear elasticity. Our spiking neural network is constructed from a recurrent network model of the brain’s motor cortex and, in contrast to black-box deep artificial neural network-based methods for PDEs, directly translates the well-understood and trusted mathematics of the FEM to a natively spiking neuromorphic algorithm.

Applied mathematics↗

DOE OSTI2022

Performance Analysis of Speculative Parallel Adaptive Local Timestepping for Conservation Laws

Stable simulation of conservation laws, such as those used to model fluid dynamics and plasma physics applications, requires the satisfaction of the so-called Courant-Friedrichs-Lewy condition. By allowing regions of the mesh to advance with different timesteps that locally satisfy this stability constraint, significant work reduction can be attained when compared to a time integration scheme using a single timestep size. However, parallelizing this algorithm presents considerable difficulty. Since the stability condition depends on the state of the system, dependencies become dynamic and potentially non-local. In this article, we present an adaptive local timestepping algorithm using an optimistic (Timewarp-based) parallel discrete event simulation. We introduce waiting heuristics to limit misspeculation and a semi-static load balancing scheme to eliminate load imbalance as parts of the mesh require finer or coarser timesteps. Last, we outline an interface for separating the physics of the specific conservation law from the temporal integration allowing for productive adoption of our proposed algorithm. We present a misspeculation study for three conservation laws, demonstrating both the productivity of the local timestepping API, for which 74% of the lines of code are reused across different conservation laws, and the robustness of the waiting heuristics—at most 1.5% of element updates are rolled back. Our performance studies demonstrate up to a 2.8× speedup versus a baseline unoptimized local timestepping approach, a 4x improvement in per-node throughput compared to an MPI parallelization of synchronous timestepping, and scalability up to 3,072 cores on NERSC’s Cori Haswell partition.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Where did the tumor start? An inverse solver with sparse localization for tumor growth models

In this work, we present a numerical scheme for solving an inverse problem for parameter estimation in tumor growth models for glioblastomas, a form of aggressive primary brain tumor. The growth model is a reaction–diffusion partial differential equation (PDE) for the tumor concentration. We use a PDE-constrained optimization formulation for the inverse problem. The unknown parameters are the reaction coefficient (proliferation), the diffusion coefficient (infiltration), and the initial condition field for the tumor PDE. Segmentation of magnetic resonance imaging (MRI) scans drive the inverse problem where segmented tumor regions serve as partial observations of the tumor concentration. Like most cases in clinical practice, we use data from a single time snapshot. Moreover, the precise time relative to the initiation of the tumor is unknown, which poses an additional difficulty for inversion. We perform a frozen-coefficient spectral analysis and show that the inverse problem is severely ill-posed. We introduce a biophysically motivated regularization on the structure and magnitude of the tumor initial condition. In particular, we assume that the tumor starts at a few locations (enforced with a sparsity constraint on the initial condition of the tumor) and that the initial condition magnitude in the maximum norm is equal to one. We solve the resulting optimization problem using an inexact quasi-Newton method combined with a compressive sampling algorithm for the sparsity constraint. Our implementation uses PETSc and AccFFT libraries. We conduct numerical experiments on synthetic and clinical images to highlight the improved performance of our solver over a previously existing solver that uses standard two-norm regularization for the calibration parameters. The existing solver is unable to localize the initial condition. Our new solver can localize the initial condition and recover infiltration and proliferation. In clinical datasets (for which the ground truth is unknown), our solver results in qualitatively different solutions compared to the two-norm regularized solver.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

A new form of the Saint-Venant equations for variable topography

The solution stability of river models using the one-dimensional (1D) Saint-Venant equations can be easily undermined when source terms in the discrete equations do not satisfy the Lipschitz smoothness condition for partial differential equations. Although instability issues have been previously noted, they are typically treated as model implementation issues rather than as underlying problems associated with the form of the governing equations. This study proposes a new reference slope form of the Saint-Venant equations to ensure smooth slope source terms and eliminate one source of potential numerical oscillations. It is shown that a simple algebraic transformation of channel geometry provides a smooth reference slope while preserving the correct cross-section flow area and the total Piezometric pressure gradient that drives the flow. The reference slope method ensures the slope source term in the governing equations is Lipschitz continuous while maintaining all the underlying complexity of the real-world geometry. The validity of the mathematical concept is demonstrated with the open-source Simulation Program for River Networks (SPRNT) model in a series of artificial test cases and a simulation of a small urban creek. Validation comparisons are made with analytical solutions and the Hydrologic Engineering Center's River Analysis System (HEC-RAS) model. The new method reduces numerical oscillations and instabilities without requiring ad hoc smoothing algorithms.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2021

Consistent coupling algorithms for coupled core-edge simulations of plasma turbulence

Two schemes for coupling gyrokinetic simulations of microturbulence in tokamaks are proposed. The first scheme is based on an additive Schwarz domain decomposition. In this work, we show that, because the goal of turbulence is long-time averages of the dynamics rather than temporal accuracy, the iteration to self-consistency across domains, which is typically required by Schwarz schemes, can be avoided, thereby accelerating the computation. Second, we propose a coupling scheme that relies entirely on the addition of source terms, leaving the boundary conditions arbitrary. The practical motivations for such a scheme are discussed, and forms of the source terms that ensure consistency and stability are derived. The schemes are tested on a nonlinear, one-dimensional model problem, and the first scheme is further tested on the Hasegawa–Wakatani model.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2021

Nominal and adversarial synthetic PMU data for standard IEEE test systems

GridSTAGE (Spatio-Temporal Adversarial scenario GEneration) is a framework for the simulation of adversarial scenarios and the generation of multivariate spatio-temporal data in cyber-physical systems. GridSTAGE is developed based on Matlab and leverages Power System Toolbox (PST) where the evolution of the power network is governed by nonlinear differential equations. Using GridSTAGE, one can create several event scenarios that correspond to several operating states of the power network by enabling or disabling any of the following: faults, AGC control, PSS control, exciter control, load changes, generation changes, and different types of cyber-attacks. Standard IEEE bus system data is used to define the power system environment. GridSTAGE emulates the data from PMU and SCADA sensors. The rate of frequency and location of the sensors can be adjusted as well. Detailed instructions on generating data scenarios with different system topologies, attack characteristics, load characteristics, sensor configuration, control parameters are available in the Github repository - https://github.com/pnnl/GridSTAGE. There is no existing adversarial data-generation framework that can incorporate several attack characteristics and yield adversarial PMU data. The GridSTAGE framework currently supports simulation of False Data Injection attacks (such as a ramp, step, random, trapezoidal, multiplicative, replay, freezing) and Denial of Service attacks (such as time-delay, packet-loss) on PMU data. Furthermore, it supports generating spatio-temporal time-series data corresponding to several random load changes across the network or corresponding to several generation changes. A Koopman mode decomposition (KMD) based algorithm to detect and identify the false data attacks in real-time is proposed in https://ieeexplore.ieee.org/document/9303022. Machine learning-based predictive models are developed to capture the dynamics of the underlying power system with a high level of accuracy under various operating conditions for IEEE 68 bus system. The corresponding machine learning models are available at https://github.com/pnnl/grid_prediction.

99 GENERAL AND MISCELLANEOUS↗

DOE OSTI2022

A survey of unsupervised learning methods for high-dimensional uncertainty quantification in black-box-type problems

Constructing surrogate models for uncertainty quantification (UQ) on complex partial differential equations (PDEs) having inherently high-dimensional O(10 n ), n ≥ 2, stochastic inputs (e.g., forcing terms, boundary conditions, initial conditions) poses tremendous challenges. The “curse of dimensionality” can be addressed with suitable unsupervised learning techniques used as a pre-processing tool to encode inputs onto lower-dimensional subspaces while retaining its structural information and meaningful properties. In this work, we review and investigate thirteen dimension reduction methods including linear and nonlinear, spectral, blind source separation, convex and non-convex methods and utilize the resulting embeddings to construct a mapping to quantities of interest via polynomial chaos expansions (PCE). Here, we refer to the general proposed approach as manifold PCE (m-PCE), where manifold corresponds to the latent space resulting from any of the studied dimension reduction methods. To investigate the capabilities and limitations of these methods we conduct numerical tests for three physics-based systems (treated as black-boxes) having high-dimensional stochastic inputs of varying complexity modeled as both Gaussian and non-Gaussian random fields to investigate the effect of the intrinsic dimensionality of input data. We demonstrate both the advantages and limitations of the unsupervised learning methods and we conclude that a suitable m-PCE model provides a cost-effective approach compared to alternative algorithms proposed in the literature, including recently proposed expensive deep neural network-based surrogates and can be readily applied for high-dimensional UQ in stochastic PDEs.

42 ENGINEERING↗

DOE OSTI2020

PETSc Users Manual (Rev. 3.13)

This manual describes the use of PETSc for the numerical solution of partial differential equations and related problems on high-performance computers. The Portable, Extensible Toolkit for Scientific Computation (PETSc) is a suite of data structures and routines that provide the building blocks for the implementation of large-scale application codes on parallel (and serial) computers. PETSc uses the MPI standard for all message-passing communication. PETSc includes an expanding suite of parallel linear solvers, nonlinear solvers, and time integrators that may be used in application codes written in Fortran, C, C++, and Python. PETSc provides many of the mechanisms needed within parallel application codes, such as parallel matrix and vector assembly routines. The library is organized hierarchically, enabling users to employ the level of abstraction that is most appropriate for a particular problem. By using techniques of object-oriented programming, PETSc provides enormous flexibility for users. PETSc is a sophisticated set of software tools; as such, for some users it initially has a much steeper learning curve than a simple subroutine library. In particular, for individuals without some computer science background, experience programming in C, C++, python, or Fortran and experience using a debugger such as gdb or dbx, it may require a significant amount of time to take full advantage of the features that enable efficient software use. However, the power of the PETSc design and the algorithms it incorporates may make the efficient implementation of many application codes simpler than \rolling them" yourself.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Volumetric Rendering on Wavelet-Based Adaptive Grid

Numerical modeling of physical phenomena frequently involves processes across a wide range of spatial and temporal scales. In the last two decades, the advancements in wavelet-based numerical methodologies to solve partial differential equations, combined with the unique properties of wavelet analysis to resolve localized structures of the solution on dynamically adaptive computational meshes, make it feasible to perform large-scale numerical simulations of a variety of physical systems on a dynamically adaptive computational mesh that changes both in space and time. Volumetric visualization of the solution is an essential part of scientific computing, yet the existing volumetric visualization techniques do not take full advantage of multi-resolution wavelet analysis and are not fully tailored for visualization of a compressed solution on the wavelet-based adaptive computational mesh. Our objective is to explore the alternatives for the visualization of time-dependent data on space-time varying adaptive mesh using volume rendering while capitalizing on the available sparse data representation. Two alternative formulations are explored. The first one is based on volumetric ray casting of multi-scale datasets in wavelet space. Rather than working with the wavelets at the finest possible resolution, a partial inverse wavelet transform is performed as a preprocessing step to obtain scaling functions on a uniform grid at a user-prescribed resolution. As a result, a solution in physical space is represented by a superposition of scaling functions on a coarse regular grid and wavelets on an adaptive mesh. An efficient and accurate ray casting algorithm is based just on these coarse scaling functions. Additional details are added during the ray tracing by taking an appropriate number of wavelets into account based on support overlap with the interpolation point, wavelet coefficient magnitude, and other characteristics, such as opacity accumulation (front to back ordering) and deviation from frontal viewing direction. The second approach is based on complementing of wavelet-based adaptive mesh to the traditional Adaptive Mesh Refinement (AMR) mesh. Both algorithms are illustrated and compared to the existing volume visualization software for Rayleigh-Benard thermal convection and electron density data sets in terms of rendering time and visual quality for different data compression of both wavelet-based and AMR adaptive meshes.

Vezolainen, Alexei V.↗

DOE OSTI2023

PETSc/TAO Users Manual (Rev. 3.19)

This manual describes the use of the Portable, Extensible Toolkit for Scientific Computation (PETSc) and the Toolkit for Advanced Optimization (TAO) for the numerical solution of partial differential equations and related problems on high-performance computers. PETSc/TAO is a suite of data structures and routines that provide the building blocks for the implementation of large-scale application codes on parallel (and serial) computers. PETSc uses the MPI standard for all distributed memory communication. PETSc/TAO includes a large suite of parallel linear solvers, nonlinear solvers, time integrators, and opti mization that may be used in application codes written in Fortran, C, C++, and Python (via petsc4py; see Getting Started). PETSc provides many of the mechanisms needed within parallel application codes, such as parallel matrix and vector assembly routines. The library is organized hierarchically, enabling users to employ the level of abstraction that is most appropriate for a particular problem. By using techniques of object-oriented programming, PETSc provides enormous flexibility for users. PETSc is a sophisticated set of software tools; as such, for some users it initially has a much steeper learning curve than packages such as MATLAB or a simple subroutine library. In particular, for individuals without some computer science background, experience programming in C, C++, python, or Fortran and experience using a debugger such as gdb or lldb, it may require a significant amount of time to take full advantage of the features that enable efficient software use. However, the power of the PETSc design and the algorithms it incorporates may make the efficient implementation of many application codes simpler than “rolling them” yourself. For many tasks a package such as MATLAB is often the best tool; PETSc is not intended for the classes of problems for which effective MATLAB code can be written. There are several packages, built on PETSc, that may satisfy your needs without requiring directly using PETSc. We recommend reviewing these packages functionality before starting to code directly with PETSc. PETSc can be used to provide a “MPI parallel linear solver” in an otherwise sequential, or OpenMP parallel code. This approach cannot provide extremely large improvements in the application time by utilizing large numbers of MPI processes but can still improve the performance. Certainly all parts of a previously sequential code need not be parallelized but the matrix generation portion must be parallelized to expect true scalability to large numbers of MPI processes. See PCMPI for details on how to utilize the PETSc MPI linear solver server. Since PETSc is under continued development, small changes in usage and calling sequences of routines will occur. PETSc has been supported for twenty-five years; see mailing list information on our website for information on contacting support.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Grad–Shafranov equilibria via data-free physics informed neural networks

A large number of magnetohydrodynamic (MHD) equilibrium calculations are often required for uncertainty quantification, optimization, and real-time diagnostic information, making MHD equilibrium codes vital to the field of plasma physics. In this paper, we explore a method for solving the Grad–Shafranov equation by using physics-informed neural networks (PINNs). For PINNs, we optimize neural networks by directly minimizing the residual of the partial differential equation as a loss function. We show that PINNs can accurately and effectively solve the Grad–Shafranov equation with several different boundary conditions, making it more flexible than traditional solvers. This method is flexible as it does not require any mesh and basis choice, thereby streamlining the computational process. We also explore the parameter space by varying the size of the model, the learning rate, and boundary conditions to map various tradeoffs such as between reconstruction error and computational speed. Additionally, we introduce a parameterized PINN framework, expanding the input space to include variables such as pressure, aspect ratio, elongation, and triangularity in order to handle a broader range of plasma scenarios within a single network. Parameterized PINNs could be used in future work to solve inverse problems such as shape optimization.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2026

Statistics of base polytopes in F-theory

We propose a new statistical ensemble of toric bases for elliptic Calabi-Yaus used in F-theory models, by focusing on only the convex hull of the base, i.e., the base polytope. This physically motivated coarse-graining greatly simplifies the combinatorial complexity of the part of the 4d F-theory landscape with toric bases. We develop a Monte Carlo approach that randomly samples the base polytopes within fixed boxes, with proper statistical weights. We first apply the algorithm to the set of 2d base polytopes, generating an enlarged set of toric 2d bases that include certain types of codimension-two (4,6) points, and we validate our approach against exact numbers. We then explore the set of 3d base polytopes which fit in a set of “maximal” 3d boxes, and estimate the total number of inequivalent 3d base polytopes to be 10 85 –10 90 . We provide statistical data such as the distribution of non-Higgsable gauge groups on these bases. Amusingly, a similar method can also be applied to generate reflexive polytopes in various dimensions. In both the reflexive and base polytope cases, the number of relevant polytopes obeys a Gaussian distribution as a function of the number of vertices, which can be understood in terms of other results on random polytopes in the math literature.

Differential and algebraic geometry↗

DOE OSTI2022

kynema-fmb [SWR-23-07]

Kynema-FMB (FKA: Kynema) is an open-source performance portable flexible multibody (FMB) dynamics solver designed for time-domain simulations. While originally tailored for wind turbine structural dynamics, the formulation and implementation are those of a general flexible-multidbody dynamics solver that can readily be applied to a wide range of systems. Kynema was designed with a narrow focus, namely to provide a lightweight, fast, accurate FMD solver for coupling to computational-fluid-dynamics (CFD) codes, especially the CFD codes in the Kynema suite, for fluid-structure-interaction (FSI) simulations. Kynema-FMB is equipped to model systems that can be represented as a collection of beams and rigid bodies that are connected through constraints. Degrees of freedom are defined in the inertial/global frame of reference and include displacements and rotations (formally as rotation matrices, but stored as quaternions). The underlying formulation is built on a Lie-group time integrator designed for index-3 differential-algebraic equations, which is second-order accurate in time (Bruls et al., 2012). Beam models are based on geometrically exact beam theory and are discretized as high-order spectral finite elements similar to those in BeamDyn (Wang et al., 2017). The governing equations for a FMD system like a wind turbine constitute a highly nonlinear system of constrained partial-differential equations. Kynema-FMB uses analytical Jacobians in the nonlinear-system solves in each time step. Linear systems use sparse storage and several third-party sparse-linear-system solvers are enabled. Ill conditioning of linear systems is mitigated with preconditioning described in Bottasso et al, 2008. Kynema-FMB is integrated with a simple open-source controller (ROSCO). There is an application programming interface (API) for coupling to geometry-resolved CFD (like that in Sharma et al., 2023) and actuator-force CFD (like that in Kuhn et al., 2025). In the latter, for actuator-line models, Kynema-FMB includes an internal blade-element solver that depends on user-provided lookup tables for coefficients of lift and drag, i.e., aerodynamic polars. Kynema-FMB is written in C++ and leverages Kokkos and Kokkos-Kernels (KokkosEcosystem) as its performance portability layer enabling simulations on both CPU and GPU systems. The repository is equipped with extensive automated testing at the unit and regression/system levels. The following describes the high-level development objectives conceived for Kynema: *Kynema will follow modern software development best practices, including test-driven development (TDD), version control, hierarchical automated testing, and continuous integration (CI) for a robust development environment. *The core data structures are memory efficient and enable vectorization and parallelization at multiple levels. *Data structures are data-oriented to exploit methods for accelerated computing including high utilization of chip resources (e.g., single instruction multiple data (SIMD) instruction sets) and parallelization using GP-GPUs. *The computational algorithms incorporate robust open-source libraries for mathematical operations, resource allocation, and data management. *The API design considers multiple stakeholder needs and ensure integration with existing and future ecosystems for data science, machine learning, and AI. *Kynema-FMB is written in modern C++ and leverages Kokkos as its performance-portability library with inspiration from the kynema stack.

Sprague, MichaelA.↗

DOE OSTI2024

A Comprehensive Analysis of PINNs for Power System Transient Stability

The integration of machine learning in power systems, particularly in stability and dynamics, addresses the challenges brought by the integration of renewable energies and distributed energy resources (DERs). Traditional methods for power system transient stability, involving solving differential equations with computational techniques, face limitations due to their time-consuming and computationally demanding nature. This paper introduces physics-informed Neural Networks (PINNs) as a promising solution for these challenges, especially in scenarios with limited data availability and the need for high computational speed. PINNs offer a novel approach for complex power systems by incorporating additional equations and adapting to various system scales, from a single bus to multi-bus networks. Our study presents the first comprehensive evaluation of physics-informed Neural Networks (PINNs) in the context of power system transient stability, addressing various grid complexities. Additionally, we introduce a novel approach for adjusting loss weights to improve the adaptability of PINNs to diverse systems. Our experimental findings reveal that PINNs can be efficiently scaled while maintaining high accuracy. Furthermore, these results suggest that PINNs significantly outperform the traditional ode45 method in terms of efficiency, especially as the system size increases, showcasing a progressive speed advantage over ode45.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

A Comprehensive Analysis of PINNs for Power System Transient Stability

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2025

A Scalable Interior‐Point Gauss–Newton Method for PDE‐Constrained Optimization With Bound Constraints

Here, we present a scalable approach to solve a class of partial differential equation (PDE)‐constrained optimization problems with bound constraints. This approach utilizes a robust full‐space interior‐point (IP)‐Gauss–Newton optimization method. To cope with the poorly‐conditioned IP‐Gauss–Newton saddle‐point linear systems that need to be solved approximately, once per optimization step, we propose two spectrally related preconditioners. These preconditioners leverage the limited informativeness of data in regularized PDE‐constrained optimization problems. A block Gauss–Seidel preconditioner is proposed for the GMRES‐based solution of the IP‐Gauss–Newton linear systems. It is shown, for a large‐class of PDE‐ and bound‐constrained optimization problems, that the spectrum of the block Gauss–Seidel preconditioned IP‐Gauss–Newton matrix is asymptotically independent of discretization and is not impacted by the ill‐conditioning that notoriously plagues interior‐point methods. We exploit symmetry of the IP‐Gauss–Newton linear systems and propose a regularization and log‐barrier Hessian preconditioner for the preconditioned conjugate gradient (PCG)‐based solution of the equivalent IP‐Gauss–Newton–Schur complement linear systems. The eigenvalues of the block Gauss–Seidel preconditioned IP‐Gauss–Newton matrix, that are not equal to one, are identical to the eigenvalues of the regularization and log‐barrier Hessian preconditioned Schur complement matrix. The scalability of the approach is demonstrated on two example problems. The numerical solution of these optimization problems is shown to require a discretization independent number of IP‐Gauss–Newton linear solves. Furthermore, the linear systems are solved in a discretization and IP ill‐conditioning independent number of preconditioned Krylov subspace iterations. The parallel scalability of the preconditioner, achieved via algebraic multigrid component solvers when applicable, and the aforementioned algorithmic scalability permits a parallel scalable means to compute solutions of a large class of PDE‐ and bound‐constrained problems.

PDE-constrained optimization↗