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NASA NTRS1993

On the dynamics of approximating schemes for dissipative nonlinear equations

Since one can rarely write down the analytical solutions to nonlinear dissipative partial differential equations (PDE's), it is important to understand whether, and in what sense, the behavior of approximating schemes to these equations reflects the true dynamics of the original equations. Further, because standard error estimates between approximations of the true solutions coming from spectral methods - finite difference or finite element schemes, for example - and the exact solutions grow exponentially in time, this analysis provides little value in understanding the infinite time behavior of a given approximating scheme. The notion of the global attractor has been useful in quantifying the infinite time behavior of dissipative PDEs, such as the Navier-Stokes equations. Loosely speaking, the global attractor is all that remains of a sufficiently large bounded set in phase space mapped infinitely forward in time under the evolution of the PDE. Though the attractor has been shown to have some nice properties - it is compact, connected, and finite dimensional, for example - it is in general quite complicated. Nevertheless, the global attractor gives a way to understand how the infinite time behavior of approximating schemes such as the ones coming from a finite difference, finite element, or spectral method relates to that of the original PDE. Indeed, one can often show that such approximations also have a global attractor. We therefore only need to understand how the structure of the attractor for the PDE behaves under approximation. This is by no means a trivial task. Several interesting results have been obtained in this direction. However, we will not go into the details. We mention here that approximations generally lose information about the system no matter how accurate they are. There are examples that show certain parts of the attractor may be lost by arbitrary small perturbations of the original equations.

Jones, Donald A.↗

NASA NTRS2016

Novel Photon-Counting Detectors for Free-Space Communication

We present performance data for novel photon counting detectors for free space optical communication. NASA GSFC is testing the performance of three novel photon counting detectors 1) a 2x8 mercury cadmium telluride avalanche array made by DRS Inc. 2) a commercial 2880 silicon avalanche photodiode array and 3) a prototype resonant cavity silicon avalanche photodiode array. We will present and compare dark count, photon detection efficiency, wavelength response and communication performance data for these detectors. We discuss system wavelength trades and architectures for optimizing overall communication link sensitivity, data rate and cost performance. The HgCdTe APD array has photon detection efficiencies of greater than 50 were routinely demonstrated across 5 arrays, with one array reaching a maximum PDE of 70. High resolution pixel-surface spot scans were performed and the junction diameters of the diodes were measured. The junction diameter was decreased from 31 m to 25 m resulting in a 2x increase in e-APD gain from 470 on the 2010 array to 1100 on the array delivered to NASA GSFC. Mean single photon SNRs of over 12 were demonstrated at excess noise factors of 1.2-1.3.The commercial silicon APD array has a fast output with rise times of 300ps and pulse widths of 600ps. Received and filtered signals from the entire array are multiplexed onto this single fast output. The prototype resonant cavity silicon APD array is being developed for use at 1 micron wavelength.

Photons↗

NASA NTRS1998

Current and Future Critical Issues in Rocket Propulsion Systems

The objective of this research was to tackle several problems that are currently of great importance to NASA. In a liquid rocket engine several complex processes take place that are not thoroughly understood. Droplet evaporation, turbulence, finite rate chemistry, instability, and injection/atomization phenomena are some of the critical issues being encountered in a liquid rocket engine environment. Pulse Detonation Engines (PDE) performance, combustion chamber instability analysis, 60K motor flowfield pattern from hydrocarbon fuel combustion, and 3D flowfield analysis for the Combined Cycle engine were of special interest to NASA. During the summer of 1997, we made an attempt to generate computational results for all of the above problems and shed some light on understanding some of the complex physical phenomena. For this purpose, the Liquid Thrust Chamber Performance (LTCP) code, mainly designed for liquid rocket engine applications, was utilized. The following test cases were considered: (1) Characterization of a detonation wave in a Pulse Detonation Tube; (2) 60K Motor wall temperature studies; (3) Propagation of a pressure pulse in a combustion chamber (under single and two-phase flow conditions); (4) Transonic region flowfield analysis affected by viscous effects; (5) Exploring the viscous differences between a smooth and a corrugated wall; and (6) 3D thrust chamber flowfield analysis of the Combined Cycle engine. It was shown that the LTCP-2D and LTCP-3D codes are capable of solving complex and stiff conservation equations for gaseous and droplet phases in a very robust and efficient manner. These codes can be run on a workstation and personal computers (PC's).

Navaz, Homayun K.↗

NASA NTRS2000

JANNAF 25th Airbreathing Propulsion Subcommittee, 37th Combustion Subcommittee and 1st Modeling and Simulation Subcommittee Joint Meeting

Volume I, the first of three volumes, is a compilation of 24 unclassified/unlimited-distribution technical papers presented at the Joint Army-Navy-NASA-Air Force (JANNAF) 25th Airbreathing Propulsion Subcommittee, 37th Combustion Subcommittee and 1st Modeling and Simulation Subcommittee (MSS) meeting held jointly with the 19th Propulsion Systems Hazards Subcommittee. The meeting was held 13-17 November 2000 at the Naval Postgraduate School and Hyatt Regency Hotel, Monterey, California. Topics covered include: a Keynote Address on Future Combat Systems, a review of the new JANNAF Modeling and Simulation Subcommittee, and technical papers on Hyper-X propulsion development and verification; GTX airbreathing launch vehicles; Hypersonic technology development, including program overviews, fuels for advanced propulsion, ramjet and scramjet research, hypersonic test medium effects; and RBCC engine design and performance, and PDE and UCAV advanced and combined cycle engine technologies.

Fry, Ronald S.↗

DOE OSTI2025

Physics-informed latent neural operator for real-time predictions of time-dependent parametric PDEs

Deep operator network (DeepONet) has shown significant promise as surrogate models for systems governed by partial differential equations (PDEs), enabling accurate mappings between infinite-dimensional function spaces. However, when applied to systems with high-dimensional input-output mappings arising from large numbers of spatial and temporal collocation points, these models often require heavily overparameterized networks, leading to long training times. Latent DeepONet addresses some of these challenges by introducing a two-step approach: first learning a reduced latent space using a separate model, followed by operator learning within this latent space. While efficient, this method is inherently data-driven and lacks mechanisms for incorporating physical laws, limiting its robustness and generalizability in data-scarce settings. Here, in this work, we propose PI-Latent-NO, a physics-informed latent neural operator framework that integrates governing physics directly into the learning process. Our architecture features two coupled DeepONets trained end-to-end: a Latent-DeepONet that learns a low-dimensional representation of the solution, and a Reconstruction-DeepONet that maps this latent representation back to the physical space. By embedding PDE constraints into the training via automatic differentiation, our method eliminates the need for labeled training data and ensures physics-consistent predictions. The proposed framework is both memory and compute-efficient, exhibiting near-constant scaling with problem size and demonstrating significant speedups over traditional physics-informed operator models. We validate our approach on a range of parametric PDEs, showcasing its accuracy, scalability, and suitability for real-time prediction in complex physical systems.

Latent representations↗

DOE OSTI2024

AEOLUS: Advances in Experimental Design, Optimal Control, and Learning for Uncertain Complex Systems

Sustained advances in the mathematics of modeling and simulation have resulted in the capability today for routine simulation of a number of large scale complex DOE-relevant systems. As remarkable as this capability for solving the so-called forward problem is, it is typically only the first step-an inner loop within an outer loop that explores the simulation model's parameter space and decision space to characterize uncertainty in the model's predictions, learn unknown model parameters from data, design the most informative experiments, determine optimal control strategies, and create optimal designs. Broadly, what unifies all of these outer loop problems is that they are, in one form or another, optimization problems over parameter/control/design space that are constrained by complex uncertain models. To fully realize the power of scientific simulation as a basis for scientific discovery, technological innovation, and rational decision-making, it is imperative to move beyond simulation to tackle the outer loop of optimization for learning from data, experimental design, and control with complex uncertain models. When the models under consideration are large-scale and complex, and when the optimization variable and uncertain parameter spaces are high (or infinite) dimensional, this constitutes a grand challenge of the highest order, and is intractable with conventional methods. To overcome these challenges, the AEOLUS Center was established to develop a unified mathematical, computational, and statistical framework for (1) Learning predictive models from complex data via Bayesian inference and optimization, and (2) Optimizing experiments, processes, and designs using the resulting uncertain models. These problems are intractable with conventional methods, for several reasons: (1) The simulation problems that govern the inner loops of the optimization problems are expensive to execute (due to severe nonlinearity, heterogeneity, multiphysics/multiscale coupling); (2) The optimization variable and uncertain parameter spaces are high dimensional, often stemming from discretizations of infinite dimensional fields such as initial conditions, sources, or material properties. We argue that the key to overcoming these challenges is to develop new mathematical, computational, and statistical methods that exploit the structure of the Bayesian inference and optimization problems mediated by their underlying complex uncertain models. This structure includes the regularity, sparsity, geometry, low intrinsic dimensionality, and multifidelity nature of the maps from uncertain parameter/optimization variable spaces to the specific objectives targeted: Bayesian inference, optimal experimental design, and optimal control design. Black box methods developed as generic tools are incapable of exploiting this structure. To be successful, we must create, integrate, and cross-fertilize ideas across multiple areas of applied math--including approximation theory, Bayesian inference, data science, experimental design, information theory, machine learning, model reduction, optimal control theory, parallel algorithms, PDE-constrained optimization, randomized algorithms, stochastic optimization, and uncertainty quantification--all while exploiting the structure of the problems at hand. With this goal in mind, we have marshaled a team of leading authorities in these areas. While the methods we develop will be broadly applicable across a wide spectrum of DOE problems in which experiments inform models and the systems those models describe must be optimized under uncertainty, we have chosen a specific area, advanced manufacturing and materials, to drive our work. AMM is characterized by complex models across multiple scales, and is a rich source of challenging problems in inference, experimental design, and optimal control, requiring multifaceted and integrated advances in applied mathematics. As such, AMM serves as an excellent vehicle to motivate and demonstrate the advances in applied mathematics developed by our center.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS2002

26th JANNAF Airbreathing Propulsion Subcommittee Meeting

This volume, the first of four volumes, is a collection of 28 unclassified/unlimited-distribution papers which were presented at the Joint Army-Navy-NASA-Air Force (JANNAF) 26th Airbreathing Propulsion Subcommittee (APS) was held jointly with the 38th Combustion Subcommittee (CS), 20th Propulsion Systems Hazards Subcommittee (PSHS), and 2nd Modeling and Simulation Subcommittee. The meeting was held 8-12 April 2002 at the Bayside Inn at The Sandestin Golf & Beach Resort and Eglin Air Force Base, Destin, Florida. Topics covered include: scramjet and ramjet R&D program overviews; tactical propulsion; space access; NASA GTX status; PDE technology; actively cooled engine structures; modeling and simulation of complex hydrocarbon fuels and unsteady processes; and component modeling and simulation.

Fry, Ronald S.↗

NASA NTRS2008

Thermal Barrier and Protective Coatings to Improve the Durability of a Combustor Under a Pulse Detonation Engine Environment

Pulse detonation engine (PDE) concepts are receiving increasing attention for future aeronautic propulsion applications, due to their potential thermodynamic cycle efficiency and higher thrust to density ratio that lead to the decrease in fuel consumption. But the resulting high gas temperature and pressure fluctuation distributions at high frequency generated with every detonation are viewed to be detrimental to the combustor liner material. Experimental studies on a typical metal combustion material exposed to a laser simulated pulse heating showed extensive surface cracking. Coating of the combustor materials with low thermal conductivity ceramics is shown to protect the metal substrate, reduce the thermal stresses, and hence increase the durability of the PDE combustor liner material. Furthermore, the temperature fluctuation and depth of penetration is observed to decrease with increasing the detonation frequency. A crack propagation rate in the coating is deduced by monitoring the variation of the coating apparent thermal conductivity with time that can be utilized as a health monitoring technique for the coating system under a rapid fluctuating heat flux.

Ghosn, Louis J.↗

DOE OSTI2025

Leveraging operator learning to accelerate convergence of the preconditioned conjugate gradient method

We propose a new deflation strategy to accelerate the convergence of the preconditioned conjugate gradient (PCG) method for solving parametric large-scale linear systems of equations. Unlike traditional deflation techniques that rely on eigenvector approximations or recycled Krylov subspaces, we generate the deflation subspaces using operator learning, specifically the Deep Operator Network (DeepONet). To this aim, we introduce two complementary approaches for assembling the deflation operators. The first approach approximates near-null space vectors of the discrete PDE operator using the basis functions learned by the DeepONet. The second approach directly leverages solutions predicted by the DeepONet. To further enhance convergence, we also propose several strategies for prescribing the sparsity pattern of the deflation operator. Here, a comprehensive set of numerical experiments encompassing steady-state, time-dependent, scalar, and vector-valued problems posed on both structured and unstructured geometries is presented and demonstrates the effectiveness of the proposed DeepONet-based deflated PCG method, as well as its generalization across a wide range of model parameters and problem resolutions.

Deflation↗

DOE OSTI2026

Latent Twins

Over the past decade, scientific machine learning has transformed the development of mathematical and computational frameworks for analyzing, modeling, and predicting complex systems. From inverse problems to numerical partial differential equations (PDEs), dynamical systems, and model reduction, these advances have pushed the boundaries of what can be simulated. Yet they have often progressed in parallel, with representation learning and algorithmic solution methods evolving largely as separate pipelines. With Latent Twins, we propose a unifying mathematical framework that creates a hidden surrogate in latent space for the underlying equations. Whereas digital twins mirror physical systems in the digital world, Latent Twins mirror mathematical systems in a learned latent space governed by operators. Through this lens, classical modeling, inversion, model reduction, and operator approximation all emerge as special cases of a single principle. We establish the fundamental approximation properties of Latent Twins for both ordinary differential equations (ODEs) and PDEs and demonstrate the framework across three representative settings: (i) canonical ODEs, capturing diverse dynamical regimes; (ii) a PDE benchmark using the shallow-water equations, contrasting Latent Twin simulations with deep operator network and forecasts with a four-dimensional variational method baseline; and (iii) a challenging real-data geopotential reanalysis dataset, reconstructing and forecasting from sparse, noisy observations. Latent Twins provide a compact, interpretable surrogate for solution operators that evaluate across arbitrary time gaps in a single-shot, while remaining compatible with scientific pipelines such as assimilation, control, and uncertainty quantification. Looking forward, this framework offers scalable, theory-grounded surrogates that bridge data-driven representation learning and classical scientific modeling across disciplines.

Latent Twins↗

DOE OSTI2024

Bayesian reduced-order deep learning surrogate model for dynamic systems described by partial differential equations

We propose a reduced-order deep-learning surrogate model for dynamic systems described by time-dependent partial differential equations. This method employs space–time Karhunen–Loève expansions (KLEs) of the state variables and space-dependent KLEs of space-varying parameters to identify the reduced (latent) dimensions. Subsequently, a deep neural network (DNN) is used to map the parameter latent space to the state variable latent space. An approximate Bayesian method is developed for uncertainty quantification (UQ) in the proposed KL-DNN surrogate model. The KL-DNN method is tested for the linear advection–diffusion and nonlinear diffusion equations, and the Bayesian approach for UQ is compared with the deep ensembling (DE) approach, commonly used for quantifying uncertainty in DNN models. It was found that the approximate Bayesian method provides a more informative distribution of the PDE solutions in terms of the coverage of the reference PDE solutions (the percentage of nodes where the reference solution is within the confidence interval predicted by the UQ methods) and log predictive probability. The DE method is found to underestimate uncertainty and introduce bias. For the nonlinear diffusion equation, we compare the KL-DNN method with the Fourier Neural Operator (FNO) method and find that KL-DNN is 10% more accurate and needs less training time than the FNO method.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1991

Spillover, nonlinearity, and flexible structures

Many systems whose evolution in time is governed by Partial Differential Equations (PDEs) are linearized around a known equilibrium before Computer Aided Control Engineering (CACE) is considered. In this case, there are infinitely many independent vibrational modes, and it is intuitively evident on physical grounds that infinitely many actuators would be needed in order to control all modes. A more precise, general formulation of this grave difficulty (spillover problem) is due to A.V. Balakrishnan. A possible route to circumvention of this difficulty lies in leaving the PDE in its original nonlinear form, and adding the essentially finite dimensional control action prior to linearization. One possibly applicable technique is the Liapunov Schmidt rigorous reduction of singular infinite dimensional implicit function problems to finite dimensional implicit function problems. Omitting details of Banach space rigor, the formalities of this approach are given.

Bass, Robert W.↗

NASA NTRS1995

The Use of Steady and Unsteady Detonation Waves for Propulsion Systems

Detonation wave enhanced supersonic combustors such as the Oblique Detonation Wave Engine (ODWE) are attractive propulsion concepts for hypersonic flight. These engines utilize detonation waves to enhance fuel-air mixing and combustion. The benefits of wave combustion systems include shorter and lighter engines which require less cooling and generate lower internal drag. These features allow air-breathing operation at higher Mach numbers than the diffusive burning scramjet delaying the need for rocket engine augmentation. A comprehensive vehicle synthesis code has predicted the aerodynamic characteristics and structural size and weight of a typical single-stage-to-orbit vehicle using an ODWE. Other studies have focused on the use of unsteady or pulsed detonation waves. For low speed applications, pulsed detonation engines (PDE) have advantages in low weight and higher efficiency than turbojets. At hypersonic speeds, the pulsed detonations can be used in conjunction with a scramjet type engine to enhance mixing and provide thrust augmentation.

Adelman, Henry G.↗

NASA NTRS1995

A General Purpose Sparse Matrix Parallel Solvers Package

A general purpose solver package for constructing and solving a range of sparse linear systems arising from discretization of PDEs on unstructured meshes is developed. Once the sparse symmetric complex matrix is constructed, it can be solved by either a preconditioned bi-conjugate gradient solver, a two-stage Cholesky LDLT factorization solver, or a hybrid solver combining the above two methods.

solver sparse matrix solvers package PDE↗

NASA NTRS1995

A General Purpose Sparse Matrix Parallel Solvers Package

solver sparse matrix solvers package PDE↗

DOE OSTI2025

Constrained or unconstrained? Neural-network-based equation discovery from data

Throughout many fields, practitioners often rely on differential equations to model systems. Yet, for many applications, the theoretical derivation of such equations and/or the accurate resolution of their solutions may be intractable. Instead, recently developed methods, including those based on parameter estimation, operator subset selection, and neural networks, allow for the data-driven discovery of both ordinary and partial differential equations (PDEs), on a spectrum of interpretability. The success of these strategies is often contingent upon the correct identification of representative equations from noisy observations of state variables and, as importantly and intertwined with that, the mathematical strategies utilized to enforce those equations. Specifically, the latter has been commonly addressed via unconstrained optimization strategies. Representing the PDE as a neural network, we propose to discover the PDE (or the associated operator) by solving a constrained optimization problem and using an intermediate state representation similar to a physics-informed neural network (PINN). The objective function of this constrained optimization problem promotes matching the data, while the constraints require that the discovered PDE is satisfied at a number of spatial collocation points. We present a penalty method and a widely used trust-region barrier method to solve this constrained optimization problem, and we compare these methods on numerical examples. Our results on several example problems demonstrate that the latter constrained method outperforms the penalty method, particularly for higher noise levels or fewer collocation points. This work motivates further exploration into using sophisticated constrained optimization methods in scientific machine learning, as opposed to their commonly used, penalty-method or unconstrained counterparts. For both of these methods, we solve these discovered neural network PDEs with classical methods, such as finite difference methods, as opposed to PINNs-type methods relying on automatic differentiation. Here, we briefly highlight how simultaneously fitting the data while discovering the PDE improves the robustness to noise and other small, yet crucial, implementation details.

Data-driven discovery↗

DOE OSTI2025

Optimization problems governed by systems of PDEs with uncertainties

This paper reviews current theoretical and numerical approaches to optimization problems governed by partial differential equations (PDEs) that depend on random variables or random fields. Such problems arise in many engineering, science, economics and societal decision-making tasks. This paper focuses on problems in which the governing PDEs are parametrized by the random variables/fields, and the decisions are made at the beginning and are not revised once uncertainty is revealed. Examples of such problems are presented to motivate the topic of this paper, and to illustrate the impact of different ways to model uncertainty in the formulations of the optimization problem and their impact on the solution. A linear–quadratic elliptic optimal control problem is used to provide a detailed discussion of the set-up for the risk-neutral optimization problem formulation, study the existence and characterization of its solution, and survey numerical methods for computing it. Different ways to model uncertainty in the PDE-constrained optimization problem are surveyed in an abstract setting, including risk measures, distributionally robust optimization formulations, probabilistic functions and chance constraints, and stochastic orders. Furthermore, approximation-based optimization approaches and stochastic methods for the solution of the large-scale PDE-constrained optimization problems under uncertainty are described. Some possible future research directions are outlined.

Heinkenschloss, Matthias [Rice Univ., Houston, TX ↗

NASA NTRS1997

Improving Discrete-Sensitivity-Based Approach for Practical Design Optimization

In developing the automated methodologies for simulation-based optimal shape designs, their accuracy, efficiency and practicality are the defining factors to their success. To that end, four recent improvements to the building blocks of such a methodology, intended for more practical design optimization, have been reported. First, in addition to a polynomial-based parameterization, a partial differential equation (PDE) based parameterization was shown to be a practical tool for a number of reasons. Second, an alternative has been incorporated to one of the tedious phases of developing such a methodology, namely, the automatic differentiation of the computer code for the flow analysis in order to generate the sensitivities. Third, by extending the methodology for the thin-layer Navier-Stokes (TLNS) based flow simulations, the more accurate flow physics was made available. However, the computer storage requirement for a shape optimization of a practical configuration with the -fidelity simulations (TLNS and dense-grid based simulations), required substantial computational resources. Therefore, the final improvement reported herein responded to this point by including the alternating-direct-implicit (ADI) based system solver as an alternative to the preconditioned biconjugate (PbCG) and other direct solvers.

Baysal, Oktay↗