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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.

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At least 181 records · Page 10
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

A gradient-based deep neural network model for simulating multiphase flow in porous media

We report simulation of multiphase flow in porous media is crucial for the effective management of subsurface energy and environment-related activities. The numerical simulators used for modeling such processes rely on spatial and temporal discretization of the governing mass and energy balance partial-differential equations (PDEs) into algebraic systems via finite-difference/volume/element methods. These simulators usually require dedicated software development and maintenance, and suffer low efficiency from a runtime and memory standpoint for problems with multi-scale heterogeneity, coupled-physics processes or fluids with complex phase behavior. Therefore, developing cost-effective, data-driven models can become a practical choice, and in this work, we choose deep learning approaches as they can handle high dimensional data and accurately predict state variables with strong nonlinearity. In this paper, we describe a gradient-based deep neural network (GDNN) constrained by the physics related to multiphase flow in porous media. We tackle the nonlinearity of flow in porous media induced by rock heterogeneity, fluid properties, and fluid-rock interactions by decomposing the nonlinear PDEs into a dictionary of elementary differential operators. We use a combination of operators to handle rock spatial heterogeneity and fluid flow by advection. Since the augmented differential operators are inherently related to the physics of fluid flow, we treat them as first principles prior knowledge to regularize the GDNN training. We use the example of pressure management at geologic CO 2 storage sites, where CO 2 is injected in saline aquifers and brine is produced, and apply GDNN to construct a predictive model that is trained with physics-based simulation data and emulates the physics process. We demonstrate that GDNN can effectively predict the nonlinear patterns of subsurface responses, including the temporal and spatial evolution of the pressure and saturation plumes. We also successfully extend the GDNN to convolutional neural network (CNN), namely gradient-based CNN (GCNN), and validate its capability to improve the prediction accuracy. GDNN has great potential to tackle challenging problems that are governed by highly nonlinear physics and enable the development of data-driven models with higher fidelity.

42 ENGINEERING↗
DOE OSTI2026

Higher-order LaSDI: Reduced order modeling with multiple time derivatives

Solving complex partial differential equations (PDEs) is essential across scientific disciplines but often requires numerical models that can be prohibitively expensive in time-sensitive applications. Reduced-order models (ROMs) address this challenge by exploiting low-dimensional structure to create fast approximations. The Latent Space Dynamics Identification (LaSDI) framework has demonstrated success in learning ROMs for parameterized PDE families, but remains limited to first-order systems. Here, in this paper, we propose Higher-Order LaSDI (HLaSDI), which extends the LaSDI framework to PDEs with arbitrary order of time derivatives. This generalization significantly expands the applicability of LaSDI-based methods to systems previously outside their scope, including hyperbolic PDEs. We demonstrate HLaSDI’s accuracy and efficiency on several linear and nonlinear benchmark problems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Gradient flow based phase-field modeling using separable neural networks

Allen–Cahn equation is a reaction–diffusion equation and is widely used for modeling phase separation. Machine learning methods for solving the Allen–Cahn equation in its strong form suffer from inaccuracies in collocation techniques, errors in computing higher-order spatial derivatives, and the large system size required by the space–time approach. To overcome these challenges, we propose solving the gradient flow of the Ginzburg–Landau free energy functional, which is equivalent to the Allen–Cahn equation, thereby avoiding the second-order spatial derivatives associated with the Allen–Cahn equation. A minimizing movement scheme is employed to solve the gradient flow problem, eliminating the complexities of a space–time approach. We utilize a separable neural network that efficiently represents the phase field through low-rank tensor decomposition. As we use the minimizing movement scheme to numerically solve the gradient flow problem, we thus, refer to the proposed method as the Separable Deep Minimizing Movement (SDMM) method. The evaluation of the functional in the minimizing movement scheme using the Gauss quadrature technique bypasses the inaccuracies associated with collocation techniques traditionally used to solve partial differential equations. A hyperbolic tangent transformation is introduced on the phase field prior to the evaluation of the functional to ensure that it remains strictly bounded within the values of the two phases. For this transformation, theoretical guarantee for energy stability of the minimizing movement scheme is established. Our results suggest that this transformation helps to improve the accuracy and efficiency significantly. The proposed method resolves the challenges faced by state-of-the-art machine learning techniques, outperforming them in both accuracy and efficiency. It is also the first machine learning method to achieve an order of magnitude speed improvement over the finite element method. In addition to its formulation and computational implementation, several case studies illustrate the applicability of the proposed method.

42 ENGINEERING↗
DOE OSTI2022

Context-Aware Learning for Inverse Design in Photovoltaics

This document describes progress in the ARPA-E DIFFERENTIATE project titled “Context-Aware Learning for Inverse Design in Photovoltaics” during the period of May 2019 to May 2022. This project is being performed at Iowa State University, New York University, Stanford University, and National Renewable Energy Laboratory. The project aims to develop a new machine learning (ML) framework to significantly accelerate the design of organic microstructures for improved organic photovoltaic performance. In this project, we had developed an inverse design framework using Deep Learning called InvNets for generating microstructures with desired physics-driven properties. As a preliminary product, in Milestone 3, we demonstrated how InvNets show 20% improvement in the performance of the microstructures and over 100X speedup in the performance compared to traditional processes for physics-driven inverse design. Later, in Milestone 6, we demonstrated that InvNets work for more complex physics properties, specifically, generating microstructures for organic photovoltaic cells with desired current-voltage characteristics. Further, in Milestone 4, we explored the idea of using physics-aware surrogates for obtaining solutions of partial differential equations(PDE) called as DiffNets(now called as NeuFENets to avoid ambiguity of names). The connection between both frameworks is that DiffNet surrogates form the physics-aware surrogate in the InvNet framework. Finally in Milestone 8, we extend our framework for other physics domains. Specifically, we explore building geometry-aware NeuFENets by developing physics surrogates that exploit ideas from traditional immersed boundary finite element methods. With these updates, we are able to achieve all the Milestones.

36 MATERIALS SCIENCE↗
DOE OSTI2022

LaSDI: Parametric Latent Space Dynamics Identification

Enabling fast and accurate physical simulations with data has become an important area of computational physics to aid in inverse problems, design-optimization, uncertainty quantification, and other various decision-making applications. This paper presents a data-driven framework for parametric latent space dynamics identification procedure that enables fast and accurate simulations. The parametric model is achieved by building a set of local latent space model and designing an interaction among them. An individual local latent space dynamics model achieves accurate solution in a trust region. By letting the set of trust region to cover the whole parameter space, our model shows an increase in accuracy with an increase in training data. Herein we introduce two different types of interaction mechanisms, i.e., point-wise and region-based approach. Both linear and nonlinear data compression techniques are used. We illustrate the framework of Latent Space Dynamics Identification (LaSDI) enable a fast and accurate solution process on various partial differential equations, i.e., Burgers’ equations, radial advection problem, and nonlinear heat conduction problem, achieving 0 (100)x speed-up and 0 (1)% relative error with respect to the corresponding full order models.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1997

A Comparison of PETSC Library and HPF Implementations of an Archetypal PDE Computation

Two paradigms for distributed-memory parallel computation that free the application programmer from the details of message passing are compared for an archetypal structured scientific computation a nonlinear, structured-grid partial differential equation boundary value problem using the same algorithm on the same hardware. Both paradigms, parallel libraries represented by Argonne's PETSC, and parallel languages represented by the Portland Group's HPF, are found to be easy to use for this problem class, and both are reasonably effective in exploiting concurrency after a short learning curve. The level of involvement required by the application programmer under either paradigm includes specification of the data partitioning (corresponding to a geometrically simple decomposition of the domain of the PDE). Programming in SPAM style for the PETSC library requires writing the routines that discretize the PDE and its Jacobian, managing subdomain-to-processor mappings (affine global- to-local index mappings), and interfacing to library solver routines. Programming for HPF requires a complete sequential implementation of the same algorithm, introducing concurrency through subdomain blocking (an effort similar to the index mapping), and modest experimentation with rewriting loops to elucidate to the compiler the latent concurrency. Correctness and scalability are cross-validated on up to 32 nodes of an IBM SP2.

Hayder, M. Ehtesham↗
DOE OSTI2023

Applications of physics informed neural operators

Abstract We present a critical analysis of physics-informed neural operators (PINOs) to solve partial differential equations (PDEs) that are ubiquitous in the study and modeling of physics phenomena using carefully curated datasets. Further, we provide a benchmarking suite which can be used to evaluate PINOs in solving such problems. We first demonstrate that our methods reproduce the accuracy and performance of other neural operators published elsewhere in the literature to learn the 1D wave equation and the 1D Burgers equation. Thereafter, we apply our PINOs to learn new types of equations, including the 2D Burgers equation in the scalar, inviscid and vector types. Finally, we show that our approach is also applicable to learn the physics of the 2D linear and nonlinear shallow water equations, which involve three coupled PDEs. We release our artificial intelligence surrogates and scientific software to produce initial data and boundary conditions to study a broad range of physically motivated scenarios. We provide the source code , an interactive website to visualize the predictions of our PINOs, and a tutorial for their use at the Data and Learning Hub for Science .

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

SPIKANs: separable physics-informed Kolmogorov–Arnold networks

Physics-Informed Neural Networks (PINNs) have emerged as a promising method for solving partial differential equations (PDEs) in scientific computing. While PINNs typically use multilayer perceptrons (MLPs) as their underlying architecture, recent advancements have explored alternative neural network structures. One such innovation is the Kolmogorov–Arnold Network (KAN), which has demonstrated benefits over traditional MLPs, including faster neural scaling and better interpretability. The application of KANs to physics-informed learning has led to the development of Physics-Informed KANs (PIKANs), enabling the use of KANs to solve PDEs. However, despite their advantages, KANs often suffer from slower training speeds, particularly in higher-dimensional problems where the number of collocation points grows exponentially with the dimensionality of the system. To address this challenge, we introduce Separable Physics-Informed Kolmogorov–Arnold Networks (SPIKANs). This novel architecture applies the principle of separation of variables to PIKANs, decomposing the problem such that each dimension is handled by an individual KAN. This approach drastically reduces the computational complexity of training without sacrificing accuracy, facilitating their application to higher-dimensional PDEs. Through a series of benchmark problems, we demonstrate the effectiveness of SPIKANs, showcasing their superior scalability and performance compared to PIKANs and highlighting their potential for solving complex, high-dimensional PDEs in scientific computing.

Kolmogorov-Arnold networks↗
DOE OSTI2022

Time-series forecasting using manifold learning, radial basis function interpolation, and geometric harmonics

We address a three-tier numerical framework based on nonlinear manifold learning for the forecasting of high-dimensional time series, relaxing the “curse of dimensionality” related to the training phase of surrogate/machine learning models. At the first step, we embed the high-dimensional time series into a reduced low-dimensional space using nonlinear manifold learning (local linear embedding and parsimonious diffusion maps). Then, we construct reduced-order surrogate models on the manifold (here, for our illustrations, we used multivariate autoregressive and Gaussian process regression models) to forecast the embedded dynamics. Finally, we solve the pre-image problem, thus lifting the embedded time series back to the original high-dimensional space using radial basis function interpolation and geometric harmonics. The proposed numerical data-driven scheme can also be applied as a reduced-order model procedure for the numerical solution/propagation of the (transient) dynamics of partial differential equations (PDEs). In conclusion, we assess the performance of the proposed scheme via three different families of problems: (a) the forecasting of synthetic time series generated by three simplistic linear and weakly nonlinear stochastic models resembling electroencephalography signals, (b) the prediction/propagation of the solution profiles of a linear parabolic PDE and the Brusselator model (a set of two nonlinear parabolic PDEs), and (c) the forecasting of a real-world data set containing daily time series of ten key foreign exchange rates spanning the time period 3 September 2001–29 October 2020.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2023

Solving multiphysics-based inverse problems with learned surrogates and constraints

Abstract Solving multiphysics-based inverse problems for geological carbon storage monitoring can be challenging when multimodal time-lapse data are expensive to collect and costly to simulate numerically. We overcome these challenges by combining computationally cheap learned surrogates with learned constraints. Not only does this combination lead to vastly improved inversions for the important fluid-flow property, permeability, it also provides a natural platform for inverting multimodal data including well measurements and active-source time-lapse seismic data. By adding a learned constraint, we arrive at a computationally feasible inversion approach that remains accurate. This is accomplished by including a trained deep neural network, known as a normalizing flow, which forces the model iterates to remain in-distribution, thereby safeguarding the accuracy of trained Fourier neural operators that act as surrogates for the computationally expensive multiphase flow simulations involving partial differential equation solves. By means of carefully selected experiments, centered around the problem of geological carbon storage, we demonstrate the efficacy of the proposed constrained optimization method on two different data modalities, namely time-lapse well and time-lapse seismic data. While permeability inversions from both these two modalities have their pluses and minuses, their joint inversion benefits from either, yielding valuable superior permeability inversions and CO 2 plume predictions near, and far away, from the monitoring wells.

Yin, Ziyi (ORCID:0000000250248771)↗
NASA NTRS1981

Parameter estimation of large flexible aerospace structures with application to the control of the Maypole Deployable Reflector

Systems such as the Maypole deployable reflector have a distributed parameter nature. The flexible column and hoop structure and the circular antenna of 30-100 meter diameter which it supports are described by partial, rather than ordinary, differential equations. Progress completed in reduced order modelling andd controller design and digital parameter estimation and control is summarized. Topics covered include depolyment and on-orbit operation; quasi-static (steady state) operation; dynamic distributed parameter system; autoregressive moving average identification; frequency domain procedures; direct or implicit active control; adaptive observers; parameter estimation using a linear reinforcement learning factor; feedback control; and reduced order modeling for nonlinear systems.

Balas, M. J.↗
DOE OSTI2024

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

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

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2025

Bayesian learning with Gaussian processes for low-dimensional representations of time-dependent nonlinear systems

This work presents a data-driven method for learning low-dimensional time-dependent physics-based surrogate models whose predictions are endowed with uncertainty estimates. We use the operator inference approach to model reduction that poses the problem of learning low-dimensional model terms as a regression of state space data and corresponding time derivatives by minimizing the residual of reduced system equations. Standard operator inference models perform well with accurate training data that are dense in time, but producing stable and accurate models when the state data are noisy and/or sparse in time remains a challenge. Another challenge is the lack of uncertainty estimation for the predictions from the operator inference models. Our approach addresses these challenges by incorporating Gaussian process surrogates into the operator inference framework to (1) probabilistically describe uncertainties in the state predictions and (2) procure analytical time derivative estimates with quantified uncertainties. The formulation leads to a generalized least-squares regression and, ultimately, reduced-order models that are described probabilistically with a closed-form expression for the posterior distribution of the operators. The resulting probabilistic surrogate model propagates uncertainties from the observed state data to reduced-order predictions. Furthermore, we demonstrate the method is effective for constructing low-dimensional models of two nonlinear partial differential equations representing a compressible flow and a nonlinear diffusion–reaction process, as well as for estimating the parameters of a low-dimensional system of nonlinear ordinary differential equations representing compartmental models in epidemiology.

Data-driven model reduction↗
DOE OSTI2022

Physics-informed neural networks for identification of material properties using standing waves

A metallic structure in its initial stage of failure involves plastic deformation or environmental degradation that changes the elastic modulus and density. This work presents the detection of change in wave velocity (a function of elastic modulus and density) as a system identification problem. A physics-informed neural network (PINN) is proposed to solve the system identification problem. The PINN takes the spatial coordinates of scanning locations and time as inputs and provides the displacement and wave velocity as outputs. The governing partial differential equation of standing waves in a rod is incorporated into the neural network as physics in the form of a loss function. The wave velocity vector is randomly initiated. During the training of the network, physics is used to determine and update the wave velocity target vector from the network’s displacement predictions. The measured data, comprising sparse displacement response on the rod structure, are used to train the PINN. The wave velocity at the sparse locations on the rod is learned from the predicted displacements during the training. Using the predictions of the trained network, the response of free vibration or material property variation can be reconstructed at unscanned locations on the structure to obtain high-resolution maps for full-field imaging to detect and localize the changes caused by plastic deformation. The PINN’s sparse scanning and simultaneous prediction capability during training can lead to high scanning and data-processing speeds. This capability yields a nondestructive evaluation system that can predict the presence of degraded material locations as the structural vibrations are scanned and processed in real time.

Rathod, Vivek↗
NASA NTRS1993

Development of programmable artificial neural networks

Conventionally programmed digital computers can process numbers with great speed and precision, but do not easily recognize patterns or imprecise or contradictory data. Instead of being programmed in the conventional sense, artificial neural networks are capable of self-learning through exposure to repeated examples. However, the training of an ANN can be a time consuming and unpredictable process. A general method is being developed to mate the adaptability of the ANN with the speed and precision of the digital computer. This method was successful in building feedforward networks that can approximate functions and their partial derivatives from examples in a single iteration. The general method also allows the formation of feedforward networks that can approximate the solution to nonlinear ordinary and partial differential equations to desired accuracy without the need of examples. It is believed that continued research will produce artificial neural networks that can be used with confidence in practical scientific computing and engineering applications.

Meade, Andrew J.↗
DOE OSTI2025

Accelerating phase field simulations through a hybrid adaptive Fourier neural operator with U-net backbone

Prolonged contact between a corrosive liquid and metal alloys can cause progressive dealloying. For one such process as liquid-metal dealloying (LMD), phase field models have been developed to understand the mechanisms leading to complex morphologies. However, the LMD governing equations in these models often involve coupled non-linear partial differential equations (PDE), which are challenging to solve numerically. In particular, numerical stiffness in the PDEs requires an extremely refined time step size (on the order of 10 -12 s or smaller). This computational bottleneck is especially problematic when running LMD simulation until a late time horizon is required. This motivates the development of surrogate models capable of leaping forward in time, by skipping several consecutive time steps at-once. In this paper, we propose a U-shaped adaptive Fourier neural operator (U-AFNO), a machine learning (ML) based model inspired by recent advances in neural operator learning. U-AFNO employs U-Nets for extracting and reconstructing local features within the physical fields, and passes the latent space through a vision transformer (ViT) implemented in the Fourier space (AFNO). We use U-AFNOs to learn the dynamics of mapping the field at a current time step into a later time step. We also identify global quantities of interest (QoI) describing the corrosion process (e.g., the deformation of the liquid-metal interface, lost metal, etc.) and show that our proposed U-AFNO model is able to accurately predict the field dynamics, in spite of the chaotic nature of LMD. Most notably, our model reproduces the key microstructure statistics and QoIs with a level of accuracy on par with the high-fidelity numerical solver, while achieving a significant 11, 200 × speed-up on a high-resolution grid when comparing the computational expense per time step. Finally, we also investigate the opportunity of using hybrid simulations, in which we alternate forward leaps in time using the U-AFNO with high-fidelity time stepping. We demonstrate that while advantageous for some surrogate model design choices, our proposed U-AFNO model in fully auto-regressive settings consistently outperforms hybrid schemes.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Machine Learning with Gradient-Based Optimization of Nuclear Waste Vitrification with Uncertainties and Constraints

Gekko is an optimization suite in Python that solves optimization problems involving mixed-integer, nonlinear, and differential equations. The purpose of this study is to integrate common Machine Learning (ML) algorithms such as Gaussian Process Regression (GPR), support vector regression (SVR), and artificial neural network (ANN) models into Gekko to solve data based optimization problems. Uncertainty quantification (UQ) is used alongside ML for better decision making. These methods include ensemble methods, model-specific methods, conformal predictions, and the delta method. An optimization problem involving nuclear waste vitrification is presented to demonstrate the benefit of ML in this field. ML models are compared against the current partial quadratic mixture (PQM) model in an optimization problem in Gekko. GPR with conformal uncertainty was chosen as the best substitute model as it had a lower mean squared error of 0.0025 compared to 0.018 and more confidently predicted a higher waste loading of 37.5 wt% compared to 34 wt%. The example problem shows that these tools can be used in similar industry settings where easier use and better performance is needed over classical approaches. Future works with these tools include expanding them with other regression models and UQ methods, and exploration into other optimization problems or dynamic control.

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗
DOE OSTI2025

Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker–Planck Equations

The Fokker-Planck (FP) equation is a foundational partial differential equation (PDE) in stochastic processes involving Brownian motions. However, the curse of dimensionality (CoD) poses a formidable challenge when dealing with high-dimensional FP equations. Although Monte Carlo simulation and (vanilla) Physics-Informed Neural Networks (PINNs) have shown the potential to tackle CoD, both methods exhibit significant numerical errors in high dimensions when dealing with the probability density function (PDF) associated with Brownian motion. The point-wise PDF values tend to decrease exponentially as dimensionality increases, surpassing the precision of numerical simulations and resulting in substantial errors. In addition, due to its massive sampling, Monte Carlo fails to offer fast sampling. Modeling the logarithm likelihood (LL) via vanilla PINNs transforms the FP equation into a notoriously difficult Hamilton-Jacobi-Bellman (HJB) equation, which is impractical for PINN learning, whose error grows rapidly with dimension. To this end, we propose a novel approach utilizing a score-based solver to fit the score function in stochastic differential equations (SDEs). The score function, defined as the gradient of the LL, plays a fundamental role in inferring LL and PDF and enables fast SDE sampling, offering an effective means to overcome the CoD. Three fitting methods, Score Matching (SM), Sliced Score Matching (SSM), and Score-PINN, are introduced, each contributing unique advantages in computational complexity, accuracy, and generality. The proposed score-based SDE solver operates in two stages: first, employing score matching or Score-PINN to acquire the score function; and second, solving the LL via an ordinary differential equation (ODE) using the obtained score function. Comparative evaluations across these methods showcase varying trade-offs. The proposed methodology is evaluated across diverse SDEs, including anisotropic Ornstein-Uhlenbeck processes, geometric Brownian motion, and Brownian motion with varying eigenspace. We also test various distributions, including Gaussian, Log-normal, Laplace, and Cauchy distributions. The numerical results demonstrate the score-based SDE solver’s stability, speed, and performance across different experimental settings, solidifying its potential as a solution to CoD for high-dimensional FP equations.

97 MATHEMATICS AND COMPUTING↗