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DOE OSTI2025

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↗

DOE OSTI2026

SODAs: sparse optimization for the discovery of differential and algebraic equations

Differential-algebraic equations (DAEs) integrate ordinary differential equations (ODEs) with algebraic constraints, providing a fundamental framework for developing models of dynamical systems characterized by time-scale separation, conservation laws and physical constraints. While sparse optimization has revolutionized model development by allowing data-driven discovery of parsimonious models from a library of possible equations, existing approaches for dynamical systems assume DAEs can be reduced to ODEs by eliminating variables before model discovery. This assumption limits the applicability of such methods for DAE systems with unknown constraints and time scales. We introduce sparse optimization for differential-algebraic systems (SODAs), a data-driven method for the identification of DAEs in their explicit form. By discovering the algebraic and dynamic components sequentially without prior identification of the algebraic variables, this approach leads to a sequence of convex optimization problems. It has the advantage of discovering interpretable models that preserve the structure of the underlying physical system. To this end, SODAs improves since SODAs is singular numerical stability when handling high correlations between library terms, caused by near-perfect algebraic relationships, by iteratively refining the conditioning of the candidate library. We demonstrate the performance of our method on biological, mechanical and electrical systems, showcasing its robustness to noise in both simulated time series and real-time experimental data.

DAE↗

DOE OSTI2024

A Quantum Approach for Implementing Fixed-Point Arithmetic in Solving Ordinary Differential Equations

Differential equations (DEs) serve as fundamental tools in mathematical modeling across scientific disciplines, yet classical numerical solvers face limitations with large-scale or computationally intensive problems. This study explores a quantum-inspired approach to solving DEs, combining quantum-inspired techniques with classical methods. It focuses on fixed-point arithmetic on quantum circuits, utilizing basic quantum gates to manipulate DE solutions. We expand upon the techniques introduced by Zanger et al. [Quantum, 5, 502 (2021)] by offering a precise computation for a fixed-point signed multiplication scheme, while also presenting a quantum circuit capable of executing the fixed-point division algorithm. We demonstrate the feasibility of our approach through the simulation of a linear Ordinary Differential Equation (ODE), where initial conditions and parameters are encoded into quantum circuits using fixed-point representation. By executing sequences of quantum gates mimicking numerical integration steps, we obtain approximate solutions to the ODE with specified fixed-point precision.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

DOE OSTI2026

Dual Representations and H ∞ -Optimal Control of Partial Differential Equations

We consider H ∞ -optimal state-feedback control of the class of linear Partial Differential Equations (PDEs) which admit a Partial Integral Equation (PIE) representation. While linear matrix inequalities are commonly used for optimal control of Ordinary Differential Equations (ODEs), the absence of a universal state-space representation and suitable dual form prevents such methods from being applied to optimal control of PDEs. Specifically, for ODEs, the controller synthesis problem is defined in state-space, and duality is used to resolve the bilinearity of that synthesis problem. Recently, the PIE representation was proposed as a universal state-space representation for linear PDE systems. In this paper, we show that any PDE system represented by a PIE admits a dual PIE with identical stability and I/O properties. This result allows us to reformulate the stabilizing and optimal state-feedback control problems as convex optimization over the cone of positive Partial Integral (PI) operators. Operator inversion formulae then allow us to construct feedback gains for the original PDE system. The results are verified through application to several canonical problems in optimal control of PDEs and indicate the resulting bounds on H ∞ norm are not conservative.

42 ENGINEERING↗

DOE OSTI2024

scikit-SUNDAE ((SUN)DIALS Differential Algebraic Equations) [SWR-24-137]

Scikit-SUNDAE provides Python bindings to SUNDIALS integrators. The implicit differential algebraic (IDA) solver and C-based variable-coefficient ordinary differential equations (CVODE) solver are both included. The name SUNDAE combines (SUN)DIALS and DAE, which stands for differential algebraic equations. Solvers specific to DAE problems are not frequently available in Python. An ordinary differential equation (ODE) solver is also included for completeness. ODEs can be categorized as a subset of DAEs (i.e., DAEs with no algebraic constraints). https://pypi.org/project/scikit-sundae

Randall, Corey↗

DOE OSTI2024

Model-form Error Correction using Universal Differential Equations for an Agent-Based Model of Infectious Disease

This report demonstrates universal differential equations (UDEs) as an approach to bridge the gap between ordinary differential equations (ODE) models and agent-based models (ABMs). Using UDE models as surrogates for ABMs allows us to preserve the foundational ODE that represents global disease dynamics while coupling it with a neural network model to approximate functions for the local behaviors of the ABM.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2025

A Quantum Approach for Implementing Fixed-Point Arithmetic in Solving Ordinary Differential Equations

Differential equations (DEs) serve as fundamental tools in mathematical modeling across scientific disciplines, yet classical numerical solvers face limitations with large-scale or computationally intensive problems. This study explores a quantum-inspired approach to solving DEs, combining quantum- inspired techniques with classical methods. It focuses on fixed- point arithmetic on quantum circuits, utilizing basic quantum gates to manipulate DE solutions. We expand upon the techniques introduced by Zanger et al. [Quantum, 5, 502 (2021)] by offering a precise computation for a fixed-point signed multiplication scheme, while also presenting a quantum circuit capable of executing the fixed-point division algorithm. We demonstrate the feasibility of our approach through the simulation of a linear Ordinary Differential Equation (ODE), where initial conditions and parameters are encoded into quantum circuits using fixed- point representation. By executing sequences of quantum gates mimicking numerical integration steps, we obtain approximate solutions to the ODE with specified fixed-point precision.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

NASA NTRS1984

A new turbulence closure model for boundary layer flows with strong adverse pressure gradients and separation

A new turbulence closure model designed specifically to treat two-dimensional, turbulent boundary layers with strong adverse pressure gradients and attendant separation, is presented. The influence of history effects are modeled by using an ordinary differential equation (ODE) derived from the turbulence kinetic-energy equation, to describe the streamwise development of the maximum Reynolds shear stress in conjunction with an assumed eddy-viscosity distribution which has as its velocity scale the maximum Reynolds shear stress. In the outer part of the boundary layer, the eddy viscosity is treated as a free parameter which is adjusted in order to satisfy the ODE for the maximum shear stress. Because of this, the model s not simply an eddy-viscosity model, but contains features of a Reynolds-stress model. Comparisons with experiments are presented which clearly show the proposed model to be superior to the Cebeci-Smith model in treating strongly retarded and separated flows. In contrast to two-equation, eddy-viscosity models, it requires only slightly more computational effort than simple models like the Cebeci-Smith model.

Johnson, D. A.↗

NASA NTRS1986

Physical and numerical sources of computational inefficiency in integration of chemical kinetic rate equations: Etiology, treatment and prognosis

The design of a very fast, automatic black-box code for homogeneous, gas-phase chemical kinetics problems requires an understanding of the physical and numerical sources of computational inefficiency. Some major sources reviewed in this report are stiffness of the governing ordinary differential equations (ODE's) and its detection, choice of appropriate method (i.e., integration algorithm plus step-size control strategy), nonphysical initial conditions, and too frequent evaluation of thermochemical and kinetic properties. Specific techniques are recommended (and some advised against) for improving or overcoming the identified problem areas. It is argued that, because reactive species increase exponentially with time during induction, and all species exhibit asymptotic, exponential decay with time during equilibration, exponential-fitted integration algorithms are inherently more accurate for kinetics modeling than classical, polynomial-interpolant methods for the same computational work. But current codes using the exponential-fitted method lack the sophisticated stepsize-control logic of existing black-box ODE solver codes, such as EPISODE and LSODE. The ultimate chemical kinetics code does not exist yet, but the general characteristics of such a code are becoming apparent.

Pratt, D. T.↗

NASA NTRS1986

Stability with large step sizes for multistep discretizations of stiff ordinary differential equations

One-leg and multistep discretizations of variable-coefficient linear systems of ODEs having both slow and fast time scales are investigated analytically. The stability properties of these discretizations are obtained independent of ODE stiffness and compared. The results of numerical computations are presented in tables, and it is shown that for large step sizes the stability of one-leg methods is better than that of the corresponding linear multistep methods.

Majda, George↗

NASA NTRS1989

Aspects of efficient and reliable multibody system simulation

Multibody system equations can be generated in various forms. All of these may be interpreted as results of two basic approaches, the augmentation- and the elimination-method. The former method yields the descriptor form of the system motion, a set of differential-algebraic equations (DAE), and the latter the state space representation, a minimal set of ordinary differential equations (ODE). Both of these methods are surveyed. Particular emphasis is on the discussion of recursive computational schemes, generating the equations of motion with a number of operations, which is proportional to the number N of system bodies (O(N)-formulations). For simulation purposes one would like to create that set of system equations, which can be generated most efficiently and for which the most efficient and reliable solution techniques are available. Numerical solution techniques for ODE have been studied in great detail and they are well-developed. By contrast, DAE have not been investigated for such a long time. In view of new developments in the latter field the generation of all the equations required for an efficient and reliable solution of DAE describing multibody system motion is discussed. These methods, i.e., an O(N)-formulation and new techniques for solving DAE, are implemented in the SIMPACK code. Its capabilities are illustrated by simulation of multibody robot models.

Schwertassek, R.↗

NASA NTRS1992

Near-resonant excitation and propagation of eccentric density waves by external forcing

An overview is presented of the astronomical evidence that relatively massive, distended, gaseous disks form as a natural by-product of the process of star formation, and also the numerical evidence that SLING-amplified eccentric modes in the outer parts of such disks can drive one-armed spiral density waves in the inner parts by near-resonant excitation and propagation. An ordinary differential equation (ODE) of the second order that approximately governs the nonlocalized forcing of waves in a disk satisfying Lindblad resonance almost everywhere is derived. When transformed and appended with an extra model term, this ODE implies, for free waves, the usual asymptotic results of the WKBJ dispersion relationship and the propagation Goldreich-Tremaine (1978) formula for the resonant torque exerted on a localized Lindblad resonance. An analytical solution is given for the rate of energy and angular momentum transfer by nonlocalized near-resonant forcing in the case when the disk has power-law dependences on the radius of the surface density and temperature.

Ostriker, Eve C.↗

NASA NTRS1993

FD-TD modeling of 2-D dielectric waveguides for propagation and scattering of femtosecond optical solitons

Experimentalists have produced all-optical switches capable of 100-fs responses. To adequately model such switches, nonlinear effects in optical materials (both instantaneous and dispersive) must be included. In principle, the behavior of electromagnetic fields in nonlinear dielectrics can be determined by solving Maxwell's equations subject to the assumption that the electric polarization has a nonlinear relation to the electric field. However, until our previous work, the resulting nonlinear Maxwell's equations have not been solved directly. Rather, approximations have been made that result in a class of generalized nonlinear Schrodinger equations (GNLSE) that solve only for the envelope of the optical pulses. In this paper, we present first-time calculations from the vector nonlinear Maxwell's equations of femtosecond soliton propagation and scattering, including carrier waves, in two-dimensional systems of dielectric waveguides exhibiting the Kerr and Raman quantum effects. We use the finite-difference time-domain (FD-TD) method in an extension of our 1-D work. There, in a fundamental innovation, we treated the linear and nonlinear convolutions for the electric polarization as new dependent variables. By differentiating these convolutions in the time domain, we derived an equivalent system of coupled, nonlinear second-order ODE's. These equations together with Maxwell's equations form the system that is solved to determine the electromagnetic fields in inhomogeneous nonlinear dispersive media. Backstorage in time is limited to only that needed by the time-integration algorithm for the ODE's, rather than that needed to store the time-history of the kernel functions of the convolutions (1000-10,000 time steps). Thus, a 2-D nonlinear optics model from Maxwell's equations is now feasible.

Joseph, Rose↗

NASA NTRS1994

Global Asymptotic Behavior of Iterative Implicit Schemes

The global asymptotic nonlinear behavior of some standard iterative procedures in solving nonlinear systems of algebraic equations arising from four implicit linear multistep methods (LMMs) in discretizing three models of 2 x 2 systems of first-order autonomous nonlinear ordinary differential equations (ODEs) is analyzed using the theory of dynamical systems. The iterative procedures include simple iteration and full and modified Newton iterations. The results are compared with standard Runge-Kutta explicit methods, a noniterative implicit procedure, and the Newton method of solving the steady part of the ODEs. Studies showed that aside from exhibiting spurious asymptotes, all of the four implicit LMMs can change the type and stability of the steady states of the differential equations (DEs). They also exhibit a drastic distortion but less shrinkage of the basin of attraction of the true solution than standard nonLMM explicit methods. The simple iteration procedure exhibits behavior which is similar to standard nonLMM explicit methods except that spurious steady-state numerical solutions cannot occur. The numerical basins of attraction of the noniterative implicit procedure mimic more closely the basins of attraction of the DEs and are more efficient than the three iterative implicit procedures for the four implicit LMMs. Contrary to popular belief, the initial data using the Newton method of solving the steady part of the DEs may not have to be close to the exact steady state for convergence. These results can be used as an explanation for possible causes and cures of slow convergence and nonconvergence of steady-state numerical solutions when using an implicit LMM time-dependent approach in computational fluid dynamics.

Yee, H. C.↗

NASA NTRS1995

Dynamical Approach Study of Spurious Steady-State Numerical Solutions of Nonlinear Differential Equations: Global Asymptotic Behavior of Time Discretizations - 2

The global asymptotic nonlinear behavior of 1 1 explicit and implicit time discretizations for four 2 x 2 systems of first-order autonomous nonlinear ordinary differential equations (ODES) is analyzed. The objectives are to gain a basic understanding of the difference in the dynamics of numerics between the scalars and systems of nonlinear autonomous ODEs and to set a baseline global asymptotic solution behavior of these schemes for practical computations in computational fluid dynamics. We show how 'numerical' basins of attraction can complement the bifurcation diagrams in gaining more detailed global asymptotic behavior of time discretizations for nonlinear differential equations (DEs). We show how in the presence of spurious asymptotes the basins of the true stable steady states can be segmented by the basins of the spurious stable and unstable asymptotes. One major consequence of this phenomenon which is not commonly known is that this spurious behavior can result in a dramatic distortion and, in most cases, a dramatic shrinkage and segmentation of the basin of attraction of the true solution for finite time steps. Such distortion, shrinkage and segmentation of the numerical basins of attraction will occur regardless of the stability of the spurious asymptotes, and will occur for unconditionally stable implicit linear multistep methods. In other words, for the same (common) steady-state solution the associated basin of attraction of the DE might be very different from the discretized counterparts and the numerical basin of attraction can be very different from numerical method to numerical method. The results can be used as an explanation for possible causes of error, and slow convergence and nonconvergence of steady-state numerical solutions when using the time-dependent approach for nonlinear hyperbolic or parabolic PDES.

Yee, H. C.↗

NASA NTRS1995

Dynamical Approach Study of Spurious Steady-State Numerical Solutions of Nonlinear Differential Equations: Global Asymptotic Behavior of Time Discretizations - Part 2

The global asymptotic nonlinear behavior of 11 explicit and implicit time discretizations for four 2 x 2 systems of first-order autonomous nonlinear ordinary differential equations (ODEs) is analyzed. The objectives are to gain a basic understanding of the difference in the dynamics of numerics between the scalars and systems of nonlinear autonomous ODEs and to set a baseline global asymptotic solution behavior of these schemes for practical computations in computational fluid dynamics. We show how 'numerical' basins of attraction can complement the bifurcation diagrams in gaining more detailed global asymptotic behavior of time discretizations for nonlinear differential equations (DEs). We show how in the presence of spurious asymptotes the basins of the true stable steady states can be segmented by the basins of the spurious stable and unstable asymptotes. One major consequence of this phenomenon which is not commonly known is that this spurious behavior can result in a dramatic distortion and, in most cases, a dramatic shrinkage and segmentation of the basin of attraction of the true solution for finite time steps. Such distortion, shrinkage and segmentation of the numerical basins of attraction will occur regardless of the stability of the spurious asymptotes, and will occur for unconditionally stable implicit linear multistep methods. In other words, for the same (common) steady-state solution the associated basin of attraction of the DE might be very different from the discretized counterparts and the numerical basin of attraction can be very different from numerical method to numerical method. The results can be used as an explanation for possible causes of error, and slow convergence and nonconvergence of steady-state numerical solutions when using the time-dependent approach for nonlinear hyperbolic or parabolic PDEs.

Yee, H. C.↗

NASA NTRS1999

A Hysteresis Model for Piezoceramic Materials

This paper addresses the modeling of nonlinear constitutive relations and hysteresis inherent to piezoceramic materials at moderate to high drive levels. Such models are, necessary to realize the, full potential of the materials in high performance control applications, and a necessary prerequisite is the development of techniques which permit control implementation. The approach employed here is based on the qualification of reversible and irreversible domain wall motion in response to applied electric fields. A comparison with experimental data illustrates that because the resulting ODE model is physics-based, it can be employed for both characterization and prediction of polarization levels throughout the range of actuator operation. Finally, the ODE formulation is amenable to inversion which facilitates the development of an inverse compensator for linear control design.

Smith, Ralph C.↗

NASA NTRS1995

On the Importance of the Dynamics of Discretizations

It has been realized recently that the discrete maps resulting from numerical discretizations of differential equations can possess asymptotic dynamical behavior quite different from that of the original systems. This is the case not only for systems of Ordinary Differential Equations (ODEs) but in a more complicated manner for Partial Differential Equations (PDEs) used to model complex physics. The impact of the modified dynamics may be mild and even not observed for some numerical methods. For other classes of discretizations the impact may be pronounced, but not always obvious depending on the nonlinear model equations, the time steps, the grid spacings and the initial conditions. Non-convergence or convergence to periodic solutions might be easily recognizable but convergence to incorrect but plausible solutions may not be so obvious - even for discretized parameters within the linearized stability constraint. Based on our past four years of research, we will illustrate some of the pathology of the dynamics of discretizations, its possible impact and the usage of these schemes for model nonlinear ODEs, convection-diffusion equations and grid adaptations.

Sweby, Peter K.↗