Time-Accurate and highly-Stable Explicit operators for stiff differential equations
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Bassenne, Maxime↗
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Efficient Simulation of Stiff Stochastic Differential Equations.
Abstract not provided.
Han, Xiaoying↗
Modeling fast diffusion processes in time integration of stiff stochastic differential equations.
Abstract not provided.
Han, Xiaoying↗
Stiff neural ordinary differential equations
Neural Ordinary Differential Equations (ODEs) are a promising approach to learn dynamical models from time-series data in science and engineering applications. This work aims at learning neural ODEs for stiff systems, which are usually raised from chemical kinetic modeling in chemical and biological systems. We first show the challenges of learning neural ODEs in the classical stiff ODE systems of Robertson’s problem and propose techniques to mitigate the challenges associated with scale separations in stiff systems. We then present successful demonstrations in stiff systems of Robertson’s problem and an air pollution problem. The demonstrations show that the usage of deep networks with rectified activations, proper scaling of the network outputs as well as loss functions, and stabilized gradient calculations are the key techniques enabling the learning of stiff neural ODEs. The success of learning stiff neural ODEs opens up possibilities of using neural ODEs in applications with widely varying time-scales, such as chemical dynamics in energy conversion, environmental engineering, and life sciences.
97 MATHEMATICS AND COMPUTING↗
Reinforcement Learning-Based Approach for EMT Automation of Large-Scale PV Plants
In the pursuit of efficient and precise modeling of large-scale power systems, particularly utility-scale photovoltaic (PV) plants, Electromagnetic Transient (EMT) simulations play a crucial role. As utility-scale PV plants increase in size and complexity, traditional computational methods become inadequate, necessitating more advanced techniques. This paper highlights the progressive efforts made to accelerate EMT simulations. A novel continuous reinforcement learning (RL) strategy is explored to automate the differentiation and categorization of stiff and non-stiff differential algebraic equations (DAEs). The use of stiff and non-stiff integration methods applied to relevant parts of the DAEs assists with the speed-up of the simulations. The paper details the data acquisition, development and offline training of the RL model, leading to its validation that demonstrates a high precision in optimizing simulation methods. The proposed RL promises to significantly enhance the efficacy of EMT simulations, offering a robust framework for the future of power system analysis.
Xia, Qianxue↗
Physics-Informed Neural Network Solution of Point Kinetics Equations for a Nuclear Reactor Digital Twin
A digital twin (DT) for nuclear reactor monitoring can be implemented using either a differential equations-based physics model or a data-driven machine learning model. The challenge of a physics-model-based DT consists of achieving sufficient model fidelity to represent a complex experimental system, whereas the challenge of a data-driven DT consists of extensive training requirements and a potential lack of predictive ability. We investigate the performance of a hybrid approach, which is based on physics-informed neural networks (PINNs) that encode fundamental physical laws into the loss function of the neural network. We develop a PINN model to solve the point kinetic equations (PKEs), which are time-dependent, stiff, nonlinear, ordinary differential equations that constitute a nuclear reactor reduced-order model under the approximation of ignoring spatial dependence of the neutron flux. The PINN model solution of PKEs is developed to monitor the start-up transient of Purdue University Reactor Number One (PUR-1) using experimental parameters for the reactivity feedback schedule and the neutron source. The results demonstrate strong agreement between the PINN solution and finite difference numerical solution of PKEs. We investigate PINNs performance in both data interpolation and extrapolation. For the test cases considered, the extrapolation errors are comparable to those of interpolation predictions. Extrapolation accuracy decreases with increasing time interval.
22 GENERAL STUDIES OF NUCLEAR REACTORS↗
Enhancing Drinking Water Quality Modeling: Leveraging Physics Informed Neural Networks for Learning with Imperfect Reaction Models and Partial Data
Chemical kinetics models, typically formulated as systems of ordinary or partial differential equations, are valuable tools for simulating drinking water quality. However, these models often face inaccuracies due to discrepancies between the laboratory and the real-world conditions, as well as limitations in experimental analytical methods, hindering the accurate representation of the true underlying chemical mechanisms. In this study, we propose a Physics Informed Neural Network (PINN), using the eXtreme Theory of Functional Connections, to improve the prediction of chemical concentrations over time. The PINN method accounts for imperfect chemical models and incorporates partial data to improve predictions. Focusing on reactions describing water disinfection residual and disinfectant byproduct formation, which are crucial for public health and regulatory compliance, we demonstrate that the PINN model is able to accurately predict the concentrations of chemical species across various pH values. Notably, the model extends its accuracy to predict concentrations of chemical species not originally included in its training data. The developed method can be extended to a variety of chemical systems, offering a wide array of potential applications.
13 HYDRO ENERGY↗
Dynamic modelling of slip in a wind turbine spherical roller main bearing
This paper considers the problem of the dynamic modelling of macro slip in spherical roller bearings. By revisiting the fundamental physics which drive these systems, potential issues in existing models have been identified. Furthermore, in pure rolling conditions it was found that governing differential equations become "stiff", requiring the use of implicit methods of time integration. The problem of individual roller macro slip in a wind turbine main bearing is then investigated using a simplified representation of system dynamics. Model results indicate clear links between slip/friction and the operational strategy of the wind turbine, as well as significantly higher frictional effects in the downwind main bearing row. Due to modelling simplifications, these results should not yet be considered conclusive, with further work required.
17 WIND ENERGY↗
A physics-constrained neural ordinary differential equations approach for robust learning of stiff chemical kinetics
The high computational cost associated with solving for detailed chemistry poses a significant challenge for predictive computational fluid dynamics (CFD) simulations of turbulent reacting flows. While deep learning techniques have been explored to develop faster surrogate models, they often fail to integrate reliably with CFD solvers. This instability arises because traditional deep learning approaches optimize for training error without ensuring compatibility with ordinary differential equation (ODE) solvers, resulting in accumulation of errors over time. Recently, neuralODE (NODE) based approaches have been shown to be a promising technique to emulate and accelerate detailed chemistry computations. Here, in the present work, we extend this NODE framework for stiff chemical kinetics by incorporating mass conservation constraints directly into the loss function during training. This ensures that the total mass as well as the individual elemental species masses are conserved in an a-posteriori manner. Proof-of-concept studies are performed with the novel physics-constrained NODE (PC-NODE) approach for homogeneous autoignition of hydrogen-air mixture over a range of composition and thermodynamic conditions. It is demonstrated that the PC-NODE framework not only improves the physical consistency of the resulting data-driven model with respect to mass conservation criteria, but also improves training efficiency. PC-NODE is shown to achieve 2–100× speedup relative to the hydrogen-air detailed chemical mechanism depending on the type of the ODE solver (implicit or explicit) used during autoregressive inference tests. Lastly, a-posteriori studies are performed wherein the trained PC-NODE model is coupled with a CFD solver. It is shown that higher accuracy is achieved with PC-NODE relative to the purely data-driven NODE approach. Moreover, PC-NODE also exhibits robustness and generalizability to unseen initial conditions from within (interpolative capability) as well as outside (extrapolative capability) the training regime.
computational combustion↗
A stiff order condition theory for Runge–Kutta methods applied to semilinear ODEs
Classical convergence theory of Runge–Kutta methods assumes that the time step is small relative to the Lipschitz constant of the ordinary differential equation (ODE). For stiff problems, that assumption is often violated, and a problematic degradation in accuracy, known as order reduction, can arise. Methods with high stage order, e.g., Gauss–Legendre and Radau, are known to avoid order reduction, but they must be fully implicit. For the broad class of semilinear ODEs, which consist of a stiff linear term and non-stiff nonlinear term, we show that weaker conditions suffice. Here, our new semilinear order conditions are formulated in terms of orthogonality relations and can be enumerated by rooted trees. Finally, we prove global error bounds that hold uniformly with respect to stiffness of the linear term.
Mathematics and Computing↗
Physics informed deep neural network embedded in a chemical transport model for the Amazon rainforest
Secondary organic aerosols (SOA) are fine particles in the atmosphere, which interact with clouds, radiation and affect the Earth’s energy budget. SOA formation involves chemistry in gas phase, aqueous aerosols, and clouds. Simulating these chemical processes involve solving a stiff set of differential equations, which are computationally expensive steps for three-dimensional chemical transport models. Deep neural networks (DNNs) are universal function approximators that could be used to represent the complex nonlinear changes in aerosol physical and chemical processes; however, key challenges such as generalizability to extended time periods, preservation of mass balance, simulating sparse model outputs, and maintaining physical constraints have limited their use in atmospheric chemistry. Here, we develop an approach of using a physics-informed DNN that overcomes previous such challenges and demonstrates its applicability for the chemical formation processes of isoprene epoxydiol SOA (IEPOX-SOA) over the Amazon rainforest. The DNN is trained with data generated by simulating IEPOX-SOA over the entire atmospheric column, using the Weather Research and Forecasting Model coupled with Chemistry (WRF-Chem). The trained DNN is then embedded within WRF-Chem to replace the computationally expensive default solver of IEPOX-SOA formation. The trained DNN predictions generalizes well with the default model simulation of the IEPOX-SOA mass concentrations and its size distribution (20 size bins) over several days of simulations in both dry and wet seasons. The embedded DNN reduces the computational expense of WRF-Chem by a factor of 2. Our approach shows promise in terms of application to other computationally expensive chemistry solvers in climate models.
54 ENVIRONMENTAL SCIENCES↗
TChem v2.0 - A Software Toolkit for the Analysis of Complex Kinetic Models
TChem is an open source software library for solving complex computational chemistry problems and analyzing detailed chemical kinetic models. The software provides support for: complex kinetic models for gas-phase and surface chemistry; thermodynamic properties based on NASA polynomials; species production/consumption rates; stable time integrator for solving stiff time ordinary differential equations; and, reactor models such as homogenous gas-phase ignition (with analytical Jacobian matrices), continuously stirred tank reactor, plug-flow reactor. This toolkit builds upon earlier versions that were written in C and featured tools for gas-phase chemistry only. The current version of the software was completely refactored in C++, uses an object-oriented programming model, and adopts Kokkos as its portability layer to make it ready for the next generation computing architectures i.e., multi/many core computing platforms with GPU accelerators. We have expanded the range of kinetic models to include surface chemistry and have added examples pertaining to Continuously Stirred Tank Reactors (CSTR) and Plug Flow Reactor (PFR) models to complement the homogenous ignition examples present in the earlier versions. To exploit the massive parallelism available from modern computing platforms, the current software interface is designed to evaluate samples in parallel, which enables large scale parametric studies, e.g. for sensitivity analysis and model calibration.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Dynamic Catalysis Fundamentals: I. Fast calculation of limit cycles in dynamic catalysis
Dynamic catalysis—the forced oscillation of catalytic reaction coordinate potential energy surfaces (PES)—has recently emerged as a promising method for the acceleration of heterogeneously-catalyzed reactions. Theoretical study of enhancement of rates and supra-equilibrium product yield via dynamic catalysis has, to-date, been severely limited by onerous computational demands of forward integration of stiff, coupled ordinary differential equations (ODEs) that are necessary to quantitatively describe periodic cycling between PESs. Here, we establish a new approach that reduces, by ≳108×, the computational cost of finding the time-averaged rate at dynamic steady state (i.e. the limit cycle for linear and nonlinear systems of kinetic equations). Our developments are motivated by and conceived from physical and mathematical insight drawn from examination of a simple, didactic case study for which closed-form solutions of rate enhancement are derived in explicit terms of periods of oscillation and elementary step rate constants. Generalization of such closed-form solutions to more complex catalytic systems is achieved by introducing a periodic boundary condition requiring the dynamic steady state solution to have the same periodicity as the kinetic oscillations and solving the corresponding differential equations by linear algebra or Newton-Raphson-based approaches. The methodology is well-suited to extension to non-linear systems for which we detail the potential for multiple solutions or solutions with different periodicities. For linear and non-linear systems alike, the acute decrement in computational expense enables rapid optimization of oscillation waveforms and, consequently, accelerates understanding of the key catalyst properties that enable maximization of reaction rates, conversions, and selectivities during dynamic catalysis.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
TINES - Time Integration, Newton and Eigen Solver v. 1.0
SAND2021-1505 O. TINES is an open source software providing math infrastructure for solving many stiff time ordinary differential equations (ODEs) and/or differential algebraic equations (DAEs) using a batch hierarchical parallelism. The code is written using a parallel programming model (i.e., Kokkos) to future-proof the next generation parallel computing platforms such as GPU accelerators. This code is developed to support Exascale Catalytic Chemistry (ECC) Project. The code provides fundamental math helpers that can aid other research projects. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.
Kim, Kyungjoo↗
The role of stiffness in training and generalization of ResNets
Neural ordinary differential equations (NODEs) have recently regained popularity as large-depth limits of a large class of neural networks. In particular, residual neural networks (ResNets) are equivalent to an explicit Euler discretization of an underlying NODE, where the transition from one layer to the next is one time step of the discretization. The relationship between continuous and discrete neural networks has been of particular interest. Notably, analysis from the ordinary differential equation viewpoint can potentially lead to new insights for understanding the behavior of neural networks in general. In this work, we take inspiration from differential equations to define the concept of stiffness for a ResNet via the interpretation of a ResNet as the discretization of a NODE. Here, we then examine the effects of stiffness on the ability of a ResNet to generalize, via computational studies on example problems coming from climate and chemistry models. We find that penalizing stiffness does have a unique regularizing effect, but we see no benefit to penalizing stiffness over L 2 regularization (penalization of network parameter norms) in terms of predictive performance.
97 MATHEMATICS AND COMPUTING↗
Data-Driven Closures and Assimilation for Stiff Multiscale Random Dynamics
Here, we introduce a data-driven and physics-informed framework for propagating uncertainty in stiff, multiscale random ordinary differential equations (RODEs) driven by correlated (colored) noise. Unlike systems subjected to Gaussian white noise, a deterministic equation for the joint probability density function (PDF) of RODE state variables does not exist in closed form. Moreover, such an equation would require as many phase-space variables as there are states in the RODE system. To alleviate this curse of dimensionality, we instead derive exact, albeit unclosed, reduced-order PDF (RoPDF) equations for low-dimensional observables/quantities of interest. The unclosed terms take the form of state-dependent conditional expectations, which are directly estimated from data at sparse observation times. However, for systems exhibiting stiff, multiscale dynamics, data sparsity introduces regression discrepancies that compound during RoPDF evolution. This is overcome by introducing a kinetic-like defect term to the RoPDF equation, which is learned by assimilating in sparse, low-fidelity RoPDF estimates. Two assimilation methods are considered, namely nudging and deep neural networks, which are successfully tested against Monte Carlo simulations.
97 MATHEMATICS AND COMPUTING↗
Implicit Multirate GARK Methods
This article considers multirate generalized-structure additively partitioned Runge–Kutta methods for solving stiff systems of ordinary differential equations with multiple time scales. These methods treat different partitions of the system with different timesteps for a more targeted and efficient solution compared to monolithic single rate approaches. With implicit methods used across all partitions, methods must find a balance between stability and the cost of solving nonlinear equations for the stages. In order to characterize this important trade-off, we explore multirate coupling strategies, problems for assessing linear stability, and techniques to efficiently implement Newton iterations for stage equations. Unlike much of the existing multirate stability analysis which is limited in scope to particular methods, we present general statements on stability and describe fundamental limitations for certain types of multirate schemes. New implicit multirate methods up to fourth order are derived, and their accuracy and efficiency properties are verified with numerical tests.
97 MATHEMATICS AND COMPUTING↗
ORMATEX
The Oak Ridge Matrix Exponential (ORMATEX) software library contains methods to compute the matrix exponential and the action of the matrix exponential on a vector. Additionally, this package contains the related methods for the phi-functions which commonly appear in a wide class of exponential time integration methods. Krylov methods are provided to evaluate the matrix exponential-vector and phi-vector products for cases where the matrix is large and sparse. Utilizing these methods, ORMATEX implements performant exponential integrators for large systems of coupled ordinary differential equations (ODEs). The exponential time integration routines in ORMATEX are particularly suitable to large, stiff systems of equations. These routines may be utilized as a competitive alternative to classical implicit and explicit time integration schemes for certain classes of differential equations where the problem stiffness can be predominately explained by the linear terms.
Gurecky, William [Oak Ridge National Laboratory (O↗