Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “Equation learning”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 289 records · Page 16
All sourcesNASA NTRSDOE OSTI
DOE OSTI2026

Scientific Discovery with Physics-Informed System Identification (Abbreviated Report)

My fellowship research focused on making physics-based simulations faster and more useful through machine learning. Many problems in science and engineering are governed by partial differential equations, but high-fidelity simulations are often too expensive to run repeatedly. I worked on improving Latent Space Dynamics Identification (LaSDI), a reduced-order modeling framework that compresses large simulation data sets into a smaller representation and then learns how that representation evolves over time. The motivation was to develop reduced models that remain accurate for more challenging systems, especially when predictions must remain reliable over long time intervals or when the underlying dynamics are more complicated than standard methods can easily handle. I also contributed to related work on Quandary, a high-performance software effort for simulation and control of open quantum systems, before focusing primarily on Latent Space Dynamics Identification methods. The main outcomes of the fellowship were two new algorithms (both of which were published), Rollout-LaSDI and Higher-Order LaSDI, together with supporting work on multi-stage Latent Space Dynamics Identification. Rollout-LaSDI improved long-term prediction by training the model to stay accurate over extended time horizons, and Higher-Order LaSDI broadened the method so it could model systems with higher-order time dynamics. My contributions to multistage Latent Space Dynamics Identification also helped show that its later training stages could be simplified without losing effectiveness, and that this behavior held across different model architectures and training strategies. Taken together, these advances improved the accuracy, flexibility, and practical value of reduced-order modeling tools for computational science.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Equation-based and data-driven modeling: Open-source software current state and future directions

Here, a review of current trends in scientific computing reveals a broad shift to open-source and higher-level programming languages such as Python and growing career opportunities over the next decade. Open-source modeling tools accelerate innovation in equation-based and data-driven applications. Significant resources have been deployed to develop data-driven tools (PyTorch, TensorFlow, Scikit-learn) from tech companies that rely on machine learning services to meet business needs while keeping the foundational tools open. Open-source equation-based tools such as Pyomo, CasADi, Gekko, and JuMP are also gaining momentum according to user community and development pace metrics. Integration of data-driven and principles-based tools is emerging. New compute hardware, productivity software, and training resources have the potential to radically accelerate progress. However, long-term support mechanisms are still necessary to sustain the momentum and maintenance of critical foundational packages.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Nonintrusive projection-based reduced order modeling using stable learned differential operators

Nonintrusive projection-based reduced order models (ROMs) are essential for dynamics prediction in multi-query applications where underlying governing equations are known but the access to the source of the underlying full order model (FOM) is unavailable; that is, FOM is a glass-box. This article proposes a learn-then-project approach for nonintrusive model reduction. In the first step of this approach, high-dimensional stable sparse learned differential operators (S-LDOs) are determined using the generated data. In the second step, the ordinary differential equations, comprising these S-LDOs, are used with suitable dimensionality reduction and low-dimensional subspace projection methods to provide equations for the evolution of reduced states. This approach allows easy integration into the existing intrusive ROM framework to enable nonintrusive model reduction while allowing the use of Petrov–Galerkin projections. The applicability of the proposed approach is demonstrated for Galerkin and LSPG projection-based ROMs through four numerical experiments: 1-D scalar advection, 1-D Burgers, 2-D scalar advection and 1-D scalar advection–diffusion–reaction equations. In conclusion, the results indicate that the proposed nonintrusive ROM strategy provides accurate and stable dynamics prediction.

42 ENGINEERING↗
DOE OSTI2020

Non-autoregressive time-series methods for stable parametric reduced-order models

Advection-dominated dynamical systems, characterized by partial differential equations, are found in applications ranging from weather forecasting to engineering design where accuracy and robustness are crucial. There has been significant interest in the use of techniques borrowed from machine learning to reduce the computational expense and/or improve the accuracy of predictions for these systems. These rely on the identification of a basis that reduces the dimensionality of the problem and the subsequent use of time series and sequential learning methods to forecast the evolution of the reduced state. Often, however, machine-learned predictions after reduced-basis projection are plagued by issues of stability stemming from incomplete capture of multiscale processes as well as due to error growth for long forecast durations. To address these issues, we have developed a non-autoregressive time series approach for predicting linear reduced-basis time histories of forward models. In particular, we demonstrate that non-autoregressive counterparts of sequential learning methods such as long short-term memory (LSTM) considerably improve the stability of machine-learned reduced-order models. Further, we evaluate our approach on the inviscid shallow water equations and show that a non-autoregressive variant of the standard LSTM approach that is bidirectional in the principal component directions obtains the best accuracy for recreating the nonlinear dynamics of partial observations. Moreover-and critical for many applications of these surrogates-inference times are reduced by three orders of magnitude using our approach, compared with both the equation-based Galerkin projection method and the standard LSTM approach.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Adaptive Activation Functions Accelerate Convergence in Deep and Physics-informed Neural Networks

We employ adaptive activation functions for regression in deep and physics-informed neural networks (PINNs) to approximate smooth and discontinuous functions as well as solutions of linear and nonlinear partial differential equations. In particular, we solve the nonlinear Klein-Gordon equation, which has smooth solutions, the nonlinear Burgers equation, which can admit high gradient solutions, and the Helmholtz equation. We introduce a scalable hyper-parameter in the activation function, which can be optimized to achieve best performance of the network as it changes dynamically the topology of the loss function involved in the optimization process. The adaptive activation function has better learning capabilities than the traditional one (fixed activation) as it improves greatly the convergence rate, especially at early training, as well as the solution accuracy. To better understand the learning process, we plot the neural network solution in the frequency domain to examine how the network captures successively different frequency bands present in the solution. We consider both forward problems, where the approximate solutions are obtained, as well as inverse problems, where parameters involved in the governing equation are identified. Our simulation results show that the proposed method is a very simple and effective approach to increase the efficiency, robustness and accuracy of the neural network approximation of nonlinear functions as well as solutions of partial differential equations, especially for forward problems. We theoretically prove that in the proposed method, gradient descent algorithms are not attracted to suboptimal critical points or local minima.

machine leaning, Bad minima, Inverse problems, Phy↗
DOE OSTI2022

Extension of the PINN diffusion model to k-eigenvalue problems

This paper extends our recent work on the Physics-Informed Neural Networks (PINN) approach for the fixed source diffusion models and applies it to the diffusion theory based k-eigenvalue problems. To make the PINN equitable for the eigenvalue problems, we introduce a novel integral regularization term to the loss function in the framework, and allow the direct inference of the principal eigenvalue and the associated eigenfunction. The regularization term enforces a pre-defined value on the integration of the model predictions, and this value can be directly related to a physical property of the system. We also introduce an additional learnable parameter to approximate the principal eigenvalue. As a proof of principle, we solve the one-group two-dimensional k-eigenvalue neutron diffusion equation in this work. We then provide two numerical examples to demonstrate the applicability of the PINN approach. In each example, we solve the k-eigenvalue diffusion equation in a multi-region configuration constrained with a set of Robin boundary conditions for generality. We use a FEM solution based on the power-iteration method to verify the results of the PINN solution. The results showed relative percentage error in the predicted eigenvalue of about 0.77% and about 1.2% for example 1 and example 2, respectively. The mean absolute error in the predicted flux for example 1 is ∼ 0.002 and for example 2 is ∼ 0.0024. These results indicate some preliminary successes of the PINN application to k-eigenvalue problems. (authors)

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
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 OSTI2021

Physics-informed machine learning

Despite great progress in simulating multiphysics problems using the numerical discretization of partial differential equations (PDEs), one still cannot seamlessly incorporate noisy data into existing algorithms, mesh generation remains complex, and high-dimensional problems governed by parameterized PDEs cannot be tackled. Moreover, solving inverse problems with hidden physics is often prohibitively expensive and requires different formulations and elaborate computer codes. Machine learning has emerged as a promising alternative, but training deep neural networks requires big data, not always available for scientific problems. Instead, such networks can be trained from additional information obtained by enforcing the physical laws (for example, at random points in the continuous space-time domain). Such physics-informed learning integrates (noisy) data and mathematical models, and implements them through neural networks or other kernel-based regression networks. Moreover, it may be possible to design specialized network architectures that automatically satisfy some of the physical invariants for better accuracy, faster training and improved generalization. Furthermore, we review some of the prevailing trends in embedding physics into machine learning, present some of the current capabilities and limitations and discuss diverse applications of physics-informed learning both for forward and inverse problems, including discovering hidden physics and tackling high-dimensional problems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

An end-to-end deep learning method for solving nonlocal Allen–Cahn and Cahn–Hilliard phase-field models

Here, we propose an efficient end-to-end deep learning method for solving nonlocal Allen–Cahn (AC) and Cahn–Hilliard (CH) phase-field models. One motivation for this effort emanates from the fact that discretized partial differential equation-based AC or CH phase-field models result in diffuse interfaces between phases, with the only recourse for remediation is to severely refine the spatial grids in the vicinity of the true moving sharp interface whose width is determined by a grid-independent parameter that is substantially larger than the local grid size. In this work, we introduce non-mass conserving nonlocal AC or CH phase-field models with regular, logarithmic, or obstacle double-well potentials. Because of non-locality, some of these models feature totally sharp interfaces separating phases. The discretization of such models can lead to a transition between phases whose width is only a single grid cell wide. Another motivation is to use deep learning approaches to ameliorate the otherwise high cost of solving discretized nonlocal phase-field models. To this end, loss functions of the customized neural networks are defined using the residual of the fully discrete approximations of the AC or CH models, which results from applying a Fourier collocation method and a temporal semi-implicit approximation. To address the long-range interactions in the models, we tailor the architecture of the neural network by incorporating a nonlocal kernel as an input channel to the neural network model. We then provide the results of extensive computational experiments to illustrate the accuracy, predictive capabilities, and cost reductions of the proposed method.

42 ENGINEERING↗
DOE OSTI2023

How machine learning can extend electroanalytical measurements beyond analytical interpretation

Electroanalytical measurements are routinely used to estimate material properties exhibiting current and voltage signatures. Analysis of such measurements relies on analytical expressions of material properties to describe the experiments. The need for analytical expressions limits the experiments that can be used to measure properties as well as the properties that can be estimated from a given experiment. Such analytical relations are essentially solutions of the physics-based differential equations (with properties as coefficients) describing the material behavior under certain specific conditions. In recent years, a new machine learning-based approach has been gaining popularity wherein the differential equations are numerically solved to interpret the electroanalytical experiments in terms of corresponding material properties. Since the physics-based differential equations are solved, one can additionally estimate underlying fields, e.g., concentration profile, using such an approach. To exemplify the characteristics of such a machine learning assisted interpretation of electroanalytical measurements, we use data from the Hebb–Wagner test on a magnesium spinel intercalation host. In conclusion, as compared to the traditional analytical expression-based interpretation, the emerging approach decreases experimental efforts to characterize relevant material properties as well as provides field information that was previously inaccessible.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

Data-driven recovery of hidden physics in reduced order modeling of fluid flows

In this article, we introduce a modular hybrid analysis and modeling (HAM) approach to account for hidden physics in reduced order modeling (ROM) of parameterized systems relevant to fluid dynamics. The hybrid ROM framework is based on using first principles to model the known physics in conjunction with utilizing the data-driven machine learning tools to model the remaining residual that is hidden in data. This framework employs proper orthogonal decomposition as a compression tool to construct orthonormal bases and Galerkin projection (GP) as a model to build the dynamical core of the system. Our proposed methodology hence compensates structural or epistemic uncertainties in models and utilizes the observed data snapshots to compute true modal coefficients spanned by these bases. The GP model is then corrected at every time step with a data-driven rectification using a long short-term memory (LSTM) neural network architecture to incorporate hidden physics. A Grassmannian manifold approach is also adopted for interpolating basis functions to unseen parametric conditions. The control parameter governing the system's behavior is thus implicitly considered through true modal coefficients as input features to the LSTM network. The effectiveness of the HAM approach is then discussed through illustrative examples that are generated synthetically to take hidden physics into account. Furthermore, our approach thus provides insights addressing a fundamental limitation of the physics-based models when the governing equations are incomplete to represent underlying physical processes.

42 ENGINEERING↗
DOE OSTI2024

Machine learning surrogates for surface complexation model of uranium sorption to oxides

Abstract The safety assessments of the geological storage of spent nuclear fuel require understanding the underground radionuclide mobility in case of a leakage from multi-barrier canisters. Uranium, the most common radionuclide in non-reprocessed spent nuclear fuels, is immobile in reduced form (U(IV) and highly mobile in an oxidized state (U(VI)). The latter form is considered one of the most dangerous environmental threats in the safety assessments of spent nuclear fuel repositories. The sorption of uranium to mineral surfaces surrounding the repository limits their mobility. We quantify uranium sorption using surface complexation models (SCMs). Unfortunately, numerical SCM solvers often encounter convergence problems due to the complex nature of convoluted equations and correlations between model parameters. This study explored two machine learning surrogates for the 2-pK Triple Layer Model of uranium retention by oxide surfaces if released as U(IV) in the oxidizing conditions: random forest regressor and deep neural networks. Our surrogate models, particularly DNN, accurately reproduce SCM model predictions at a fraction of the computational cost without any convergence issues. The safety assessment of spent fuel repositories, specifically the migration of leaked radioactive waste, will benefit from having ultrafast AI/ML surrogates for the computationally expensive sorption models that can be easily incorporated into larger-scale contaminant migration models. One such model is presented here.

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗
DOE OSTI2022

MIONet: Learning Multiple-Input Operators via Tensor Product

As an emerging paradigm in scientific machine learning, neural operators aim to learn operators, via neural networks, that map between infinite-dimensional function spaces. Several neural operators have been recently developed. However, all the existing neural operators are only designed to learn operators defined on a single Banach space; i.e., the input of the operator is a single function. Here, for the first time, we study the operator regression via neural networks for multiple-input operators defined on the product of Banach spaces. We first prove a universal approximation theorem of continuous multiple-input operators. We also provide a detailed theoretical analysis including the approximation error, which provides guidance for the design of the network architecture. Based on our theory and a low-rank approximation, we propose a novel neural operator, MIONet, to learn multiple-input operators. MIONet consists of several branch nets for encoding the input functions and a trunk net for encoding the domain of the output function. Here, we demonstrate that MIONet can learn solution operators involving systems governed by ordinary and partial differential equations. In our computational examples, we also show that we can endow MIONet with prior knowledge of the underlying system, such as linearity and periodicity, to further improve accuracy.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Statistical learning for predicting density–matrix-based electron dynamics

In this study, we consider the problem of learning density-dependent molecular Hamiltonian matrices from time series of electron density matrices, all in the context of Hartree–Fock theory. Prior work developed a solution to this problem for small molecular systems with density and Hamiltonian matrices of size at most 6 × 6. Here, using a battery of techniques, we scale prior methods to larger molecular systems with, for example, 29 × 29 matrices. This includes systems that either have more electrons or are expressed in large basis sets such as 6-311++G**. Scaling the method to larger systems enhances its relevance for realistic applications in chemistry and physics. To achieve this scaling, we apply dimensionality reduction, ridge regression and analytic computation of Hessians. Through the combination of these techniques, we are able to learn Hamiltonians by minimizing an objective function that encodes local propagation error. Importantly, these learned Hamiltonians can then be used to predict electron dynamics for thousands of steps: When we use our learned Hamiltonians to numerically solve the time-dependent Hartree–Fock equation, we obtain predicted dynamics that are in close quantitative agreement with ground truth dynamics. This includes field-off trajectories similar to the training data and field-on trajectories outside of the training data.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2022

Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems

Deep learning has been shown to be an effective tool in solving partial differential equations (PDEs) through physics-informed neural networks (PINNs). PINNs embed the PDE residual into the loss function of the neural network, and have been successfully employed to solve diverse forward and inverse PDE problems. However, one disadvantage of the first generation of PINNs is that they usually have limited accuracy even with many training points. Here, we propose a new method, gradient-enhanced physics-informed neural networks (gPINNs), for improving the accuracy of PINNs. gPINNs leverage gradient information of the PDE residual and embed the gradient into the loss function. We tested gPINNs extensively and demonstrated the effectiveness of gPINNs in both forward and inverse PDE problems. Our numerical results show that gPINN performs better than PINN with fewer training points. Additionally, we combined gPINN with the method of residual-based adaptive refinement (RAR), a method for improving the distribution of training points adaptively during training, to further improve the performance of gPINN, especially in PDEs with solutions that have steep gradients.

42 ENGINEERING↗
DOE OSTI2024

Underground hydrogen storage leakage detection and characterization based on machine learning of sparse seismic data

Underground hydrogen storage (UHS) is considered as a scalable approach for massive storage and seasonal extraction of hydrogen (H 2 ). Although conventional leakage detection and characterization methods based on time-lapse seismic imaging and inversion generally apply to H 2 leakage detection problem, a high-fidelity yet cost effective geophysics approach is still missing to reliably inform leakage location and properties based on very sparse data. In response, we develop a novel supervised machine learning method to detect and characterize H 2 leakage from UHS. The input to our neural network are sparse time-lapse seismic waveforms, while the output from the neural network includes the spatial location and physical properties of a H 2 leakage. Here, we generate high-quality time-lapse waveforms using the elastic-wave equations to train the neural network. We train and validate our machine learning model and find that it attains high accuracy in using extremely sparse time-lapse seismic data to detect and characterize H 2 leakage. Our investigation is the first systematic study that focuses on applying machine learning to subsurface H 2 leakage detection and characterization and could potentially serve as a cost-effective geophysical tool for underground hydrogen leakage detection and characterization with high fidelity.

08 HYDROGEN↗
DOE OSTI2023

Anticipating gelation and vitrification with medium amplitude parallel superposition (MAPS) rheology and artificial neural networks

Abstract Anticipating qualitative changes in the rheological response of complex fluids (e.g., a gelation or vitrification transition) is an important capability for processing operations that utilize such materials in real-world environments. One class of complex fluids that exhibits distinct rheological states are soft glassy materials such as colloidal gels and clay dispersions, which can be well characterized by the soft glassy rheology (SGR) model. We first solve the model equations for the time-dependent, weakly nonlinear response of the SGR model. With this analytical solution, we show that the weak nonlinearities measured via medium amplitude parallel superposition (MAPS) rheology can be used to anticipate the rheological aging transitions in the linear response of soft glassy materials. This is a rheological version of a technique called structural health monitoring used widely in civil and aerospace engineering. We design and train artificial neural networks (ANNs) that are capable of quickly inferring the parameters of the SGR model from the results of sequential MAPS experiments. The combination of these data-rich experiments and machine learning tools to provide a surrogate for computationally expensive viscoelastic constitutive equations allows for rapid experimental characterization of the rheological state of soft glassy materials. We apply this technique to an aging dispersion of Laponite ® clay particles approaching the gel point and demonstrate that a trained ANN can provide real-time detection of transitions in the nonlinear response well in advance of incipient changes in the linear viscoelastic response of the system.

Lennon, Kyle R. (ORCID:0000000212515461)↗
DOE OSTI2025

Real-time inference and extrapolation with Time-Conditioned UNet: Applications in hypersonic flows, incompressible flows, and global temperature forecasting

Neural Operators are fast and accurate surrogates for nonlinear mappings between functional spaces within training domains. Extrapolation beyond the training domain remains a grand challenge across all application areas. We present Time-Conditioned UNet (TC-UNet) as an operator learning method to solve time-dependent PDEs continuously in time without any temporal discretization, including in extrapolation scenarios. TC-UNet incorporates the temporal evolution of the PDE into its architecture by combining a parameter conditioning approach with the attention mechanism from the Transformer architecture. After training, TC-UNet makes real-time inferences on an arbitrary temporal grid. We demonstrate its extrapolation capability on a climate problem by estimating the global temperature for several years and also for inviscid hypersonic flow around a double cone. We propose different training strategies involving temporal bundling and sub-sampling. We demonstrate performance improvements for several benchmarks, performing extrapolation for long time intervals and zero-shot super-resolution time.

Deep learning↗