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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 109 records · Page 6
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Sparsified Time-dependent PDEs FNO (STFNO) v1.0.0

STFNO (Sparsified Time-dependent PDEs FNO code) is an extension of the popular Fourier Neural Operator (FNO) architecture to the solution of coupled systems of time-dependent partial differential equations. STFNO leverages the sparsified dependencies on the field quantities based on the semi-discretiezed form of the PDEs, enabling significant reduction in the number of model parameters. STFNO has been extensively tested on two fusion simulation codes, NIMROD and GTC, and can be easily tailored to other systems of PDEs.

Rahman, Mustafa [Lawrence Berkeley National Labora↗
DOE OSTI2026

SPUS-Small-PDE-U-net-Solver

Small PDE U-Net Solver (SPUS) is a compact and efficient foundation model (FM) designed as a unified neural operator for solving a wide range of partial differentialequations (PDEs). SPUS leverages a lightweight residual U-Net-based architecture as a foundation model architecture. To enable effective learning in this minimalist framework, SPUS utilizes a simple yet powerful auto-regressive pretraining strategy which closely replicates the behavior of numerical solvers to learn the underlying physics. SPUS is designed to be pretrained on a diverse set of fluid dynamics PDEs from public benchmark datasets.

Siddik, Abu↗
DOE OSTI2024

Enhancing Fluid Flow Pressure and Saturation Prediction Accuracy and Reducing Uncertainty with Committee Machine – Illinois Basin Decatur Project (IBDP) as a Case Study

Presentation at the 17th International Conference on Greenhouse Gas Control Technologies GHGT-17 held in Calgary, Canada, October 20-24, 2024. Carbon capture and storage (CCS) is a way to play a critical role in the global transition to a low-emission economy. Current progress is hampered by a number of factors, among which the lack of risk-informed design tools and decision support frameworks is seen as a major roadblock. Significant interest exists in using artificial intelligence to accelerate CCS site feasibility studies, as well as to facilitate the permit application process. Existing works commonly train a single deep learning model. This work investigates the feasibility of using a conventional ensemble learning (committee machine) technique to further improve prediction accuracy. Ensemble-based algorithms generally improve over individual base learners in terms of robustness and accuracy. Deep ensembles, however, are time-consuming to create and train. A pragmatic question is whether small-sized ensembles may lead to prediction improvement. Here we evaluated the efficacy of an ensemble learning technique using the latent spectral model (LSM), an efficient deep neural operator algorithm, as base learners. Preliminary results, obtained using the Illinois Basin-Decatur Project (IBDP) carbon sequestration data/model, show that small-sized ensembles can improve prediction over the base learners, achieving prediction accuracy of ~1.6 psi root mean square error (RMSE) on pressure (relative the average reservoir pressure of 3150 psi), and less than 1.3% for saturation.

Sun, Alexander↗
DOE OSTI2024

Enhancing Fluid Flow Pressure and Saturation Prediction Accuracy and Reducing Uncertainty with Committee Machine – Illinois Basin Decatur Project (IBDP) as a Case Study

This is the conference paper accompanying an oral presentation at the 17th International Conference on Greenhouse Gas Control Technologies GHGT-17 held in Calgary, Canada, October 20-24, 2024. Carbon capture and storage (CCS) is a way to play a critical role in the global transition to a low-emission economy. Current progress is hampered by a number of factors, among which the lack of risk-informed design tools and decision support frameworks is seen as a major roadblock. Significant interest exists in using artificial intelligence to accelerate CCS site feasibility studies, as well as to facilitate the permit application process. Existing works commonly train a single deep learning model. This work investigates the feasibility of using a conventional ensemble learning (committee machine) technique to further improve prediction accuracy. Ensemble-based algorithms generally improve over individual base learners in terms of robustness and accuracy. Deep ensembles, however, are time-consuming to create and train. A pragmatic question is whether small-sized ensembles may lead to prediction improvement. Here we evaluated the efficacy of an ensemble learning technique using the latent spectral model (LSM), an efficient deep neural operator algorithm, as base learners. Preliminary results, obtained using the Illinois Basin-Decatur Project (IBDP) carbon sequestration data/model, show that small-sized ensembles can improve prediction over the base learners, achieving prediction accuracy of ~1.6 psi root mean square error (RMSE) on pressure (relative the average reservoir pressure of 3150 psi), and less than 1.3% for saturation.

Sun, Alexander↗
DOE OSTI2025

Physics-Informed Recurrent Neural Networks to Predict Reactor Operations of the AGN-201 Nuclear Reactor

4 page paper submitted to ANS Student conference. Summary of paper similar to the following abstract: The ability to predict how a reactor will operate, understand when anomalous conditions arise, and ensure a reactor is being operated as expected is crucial for deploying new nuclear facilities. Digital twins serve as a unique solution to recognizing reactor behavior; however, they require data to be useful. For next-generation reactors, this data may not currently be available. To explore how synthetic physics-informed reactor data can be used to predict reactor operations, a recurrent neural network was implemented for the Idaho State University AGN-201 digital twin. The goal of this work is to determine how synthetic data can be used to train a recurrent neural network model for predicting the reactor power of the AGN-201. The recurrent neural network was validated using both synthetic and real operational data. We envision this approach will help bridge the gap between the virtual and physical sides of a digital twin, where reactor physics models based on as-built data can be corrected for actual operating parameters to ensure the virtual model mirrors reality.

98 NUCLEAR DISARMAMENT, SAFEGUARDS, AND PHYSICAL P↗
DOE OSTI2024

RandONets: Shallow networks with random projections for learning linear and nonlinear operators

Deep neural networks have been extensively used for the solution of both the forward and the inverse problem for dynamical systems. However, their implementation necessitates optimizing a high-dimensional space of parameters and hyperparameters. This fact, along with the requirement of substantial computational resources, pose a barrier to achieving high numerical accuracy, but also interpretability. Here, to address the above challenges, we present Random Projection-based Operator Networks (RandONets): shallow networks with random projections and tailor-made numerical analysis methods that learn accurately and fast linear and nonlinear operators. Building on previous works, we prove that RandOnets are universal approximators of linear and nonlinear operators. Due to their simplicity, RandONets provide a one-step transformation of the input space, facilitating interpretability. For the evaluation of their performance, we focus on operators of PDEs. We show, that RandONets outperform by several orders of magnitude, both in terms of numerical approximation accuracy and computational cost, the “vanilla” DeepONets. Hence, we believe that our method will trigger further developments in the field of scientific machine learning, for the development of new ‘’light”schemes that will provide high accuracy while reducing dramatically the computational cost. A MATLAB toolbox for RandONets, including demos, is available on GitHub at https://github.com/GianlucaFabiani/RandONets.

Interpretable machine learning↗
DOE OSTI2025

Self-adaptive weights based on balanced residual decay rate for physics-informed neural networks and deep operator networks

Physics-informed deep learning has emerged as a promising alternative for solving partial differential equations. However, for complex problems, training these networks can still be challenging, often resulting in unsatisfactory accuracy and efficiency. In this work, we demonstrate that the failure of plain physics-informed neural networks arises from the significant discrepancy in the convergence rate of residuals at different training points, where the slowest convergence rate dominates the overall solution convergence. Based on these observations, we propose a pointwise adaptive weighting method that balances the residual decay rate across different training points. The performance of our proposed adaptive weighting method is compared with current state-of-the-art adaptive weighting methods on benchmark problems for both physics-informed neural networks and physics-informed deep operator networks. In conclusion, through extensive numerical results we demonstrate that our proposed approach of balanced residual decay rates offers several advantages, including bounded weights, high prediction accuracy, fast convergence rate, low training uncertainty, low computational cost, and ease of hyperparameter tuning.

Balanced convergence rate↗
DOE OSTI2026

Neural Scaling Laws of Deep ReLU and Deep Operator Network: A Theoretical Study

Neural scaling laws play a pivotal role in the performance of deep neural networks and have been observed in a wide range of tasks. However, a complete theoretical framework for understanding these scaling laws remains underdeveloped. In this paper, we explore the neural scaling laws for deep operator networks, which involve learning mappings between function spaces, with a focus on the Chen and Chen style architecture. These approaches, which include the popular Deep Operator Network (DeepONet), approximate the output functions using a linear combination of learnable basis functions and coefficients that depend on the input functions. We establish a theoretical framework to quantify the neural scaling laws by analyzing its approximation and generalization errors. We articulate the relationship between the approximation and generalization errors of deep operator networks and key factors such as network model size and training data size. Moreover, we address cases where input functions exhibit low-dimensional structures, allowing us to derive tighter error bounds. These results also hold for deep ReLU networks and other similar structures. Our results offer a partial explanation of the neural scaling laws in operator learning and provide a theoretical foundation for their applications.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

A variational framework for residual-based adaptivity in neural PDE solvers and operator learning

Residual-based adaptive strategies are widely used in scientific machine learning yet remain largely heuristic. We introduce a variational framework that formalizes these methods through convex transformations of the residual, where different transformations correspond to distinct objective functionals. For instance, exponential weights target uniform error minimization, while linear weights recover quadratic error minimization. This perspective reveals adaptive weighting as a means of selecting sampling distributions that optimize a primal objective, directly linking discretization choices to error metrics. This principled approach yields three key benefits: it enables systematic design of adaptive schemes, reduces discretization error by lowering estimator variance, and enhances learning dynamics by improving gradient signal-to-noise ratio. Extending the framework to operator learning, we demonstrate substantial performance gains across diverse optimizers and architectures. Our results provide a theoretical perspective for residual-based adaptivity and establish a foundation for principled discretization and training.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Enhancing Lattice Kinetic Schemes for Fluid Dynamics with Lattice-Equivariant Neural Networks

A new class of equivariant neural networks is presented, hereby dubbed lattice-equivariant neural networks (LENNs), designed to satisfy local symmetries of a lattice structure. The approach develops within a recently introduced framework aimed at learning neural network-based surrogate models’ lattice Boltzmann collision operators. Whenever neural networks are employed to model physical systems, respecting symmetries and equivariance properties has been shown to be key for accuracy, numerical stability, and performance. Here, hinging on ideas from group representation theory, trainable layers are defined whose algebraic structure is equivariant with respect to the symmetries of the lattice cell. In this work, the presented method naturally allows for efficient implementations, in terms of both memory usage and computational costs, supporting scalable training/testing for lattices in two spatial dimensions and higher (in which the size of symmetry group grows). The approach is validated and tested considering 2D and 3D flowing dynamics, both in laminar and turbulent regimes. It is compared with group-averaged-based symmetric networks and with plain, nonsymmetric, networks, showing how the presented approach unlocks the (a posteriori) accuracy and training stability of the former models and the train/inference speed of the latter networks. (LENNs are about one order of magnitude faster than group-averaged networks in 3D.) The work in this paper opens toward practical use of machine learning-augmented lattice Boltzmann CFD in real-world simulations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

ExtremeMETA: High-speed Lightweight Image Segmentation Model by Remodeling Multi-channel Metamaterial Imagers

Deep neural networks (DNNs) have heavily relied on traditional computational units, such as CPUs and GPUs. However, this conventional approach brings significant computational burden, latency issues, and high power consumption, limiting their effectiveness. This has sparked the need for lightweight networks such as ExtremeC3Net. Meanwhile, there have been notable advancements in optical computational units, particularly with metamaterials, offering the exciting prospect of energy-efficient neural networks operating at the speed of light. Yet, the digital design of metamaterial neural networks (MNNs) faces precision, noise, and bandwidth challenges, limiting their application to intuitive tasks and low-resolution images. In this study, we proposed a large kernel lightweight segmentation model, ExtremeMETA. Based on ExtremeC3Net, our proposed model, ExtremeMETA maximized the ability of the first convolution layer by exploring a larger convolution kernel and multiple processing paths. With the large kernel convolution model, we extended the optic neural network application boundary to the segmentation task. To further lighten the computation burden of the digital processing part, a set of model compression methods was applied to improve model efficiency in the inference stage. The experimental results on three publicly available datasets demonstrated that the optimized efficient design improved segmentation performance from 92.45 to 95.97 on mIoU while reducing computational FLOPs from 461.07 MMacs to 166.03 MMacs. The large kernel lightweight model ExtremeMETA showcased the hybrid design’s ability on complex tasks.

large convolution kernel↗
DOE OSTI2024

Separable physics-informed DeepONet: Breaking the curse of dimensionality in physics-informed machine learning

The deep operator network (DeepONet) has shown remarkable potential in solving partial differential equations (PDEs) by mapping between infinite-dimensional function spaces using labeled datasets. However, in scenarios lacking labeled data, the physics-informed DeepONet (PI-DeepONet) approach, which utilizes the residual loss of the governing PDE to optimize the network parameters, faces significant computational challenges, particularly due to the curse of dimensionality. This limitation has hindered its application to high-dimensional problems, making even standard 3D spatial with 1D temporal problems computationally prohibitive. Additionally, the computational requirement increases exponentially with the discretization density of the domain. Here, to address these challenges and enhance scalability for high-dimensional PDEs, we introduce the Separable physics-informed DeepONet (Sep-PI-DeepONet). This framework employs a factorization technique, utilizing sub-networks for individual one-dimensional coordinates, thereby reducing the number of forward passes and the size of the Jacobian matrix required for gradient computations. By incorporating forward-mode automatic differentiation (AD), we further optimize computational efficiency, achieving linear scaling of computational cost with discretization density and dimensionality, making our approach highly suitable for high-dimensional PDEs. We demonstrate the effectiveness of Sep-PI-DeepONet through three benchmark PDE models: the viscous Burgers’ equation, Biot’s consolidation theory, and a parameterized heat equation. Our framework maintains accuracy comparable to the conventional PI-DeepONet while reducing training time by two orders of magnitude. Notably, for the heat equation solved as a 4D problem, the conventional PI-DeepONet was computationally infeasible (estimated 289.35 h), while the Sep-PI-DeepONet completed training in just 2.5 h. These results underscore the potential of Sep-PI-DeepONet in efficiently solving complex, high-dimensional PDEs, marking a significant advancement in physics-informed machine learning.

Neural operator↗
DOE OSTI2025

UQpy Version 4.2: Uncertainty quantification with Python

We introduce a new module for the UQpy software package which extends its capabilities into the field of Scientific Machine Learning. This module builds on PyTorch to create a flexible and robust platform for uncertainty quantification in machine learning. The scientific machine learning module of UQpy introduces custom layers, neural networks, and neural network trainers that are compatible with torch version 2.2.2 and allow for “plug and play” integration into existing torch code.

Neural networks↗
DOE OSTI2025

A Neural Optimizer With Decision-Focused Learning for Optimal Energy Storage Operation

Here, this article introduces a neural optimizer-based framework for optimizing battery energy storage system (BESS) control for grid services, including demand charge and energy cost reduction. By leveraging decision-focused learning (DFL), the proposed framework ensures seamless integration and adaptation, significantly enhancing control performance. A patch time-series transformer is employed for peak load forecasting, incorporating aleatoric uncertainty quantification to account for forecasting uncertainties within the decision-making process. The framework utilizes a solver-in-the-loop approach to generate optimal BESS actions, which are then used to train the neural optimizer-based agent. By co-optimizing both BESS operational modes and output power within the NN, the system achieves improved performance and robustness. After initial training, the forecasting and control models are jointly fine-tuned to account for forecasting errors, further improving decision precision and efficiency through DFL. Case studies are performed to validate the performance of the framework using multiple real-world datasets, demonstrating superior performance in monthly peak load forecasting compared to state-of-the-art models. In addition, the results are compared against existing decision-making approaches. The results demonstrate a reduction in monthly peak forecasting error by approximately 15% across various performance measures and achieve an optimization gap for BESS operation that is about three times smaller compared to existing methods.

Kim, Hyeonjin [Pacific Northwest National Laborato↗
DOE OSTI2026

NN-OpInf

SAND2026-18878O The NN-OpInf tool is a PyTorch-based approach to operator inference that uses composable, structure-preserving neural networks to represent nonlinear operators. Operator inference is a machine learning method for inferring low-dimensional systems from data and polynomial models for system dynamics. However, many systems do not conform to polynomial structures, which NN-OpInf addresses by parameterizing operators with neural networks. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-NA0003525.

SciDAC↗
DOE OSTI2026

Deep nonparametric estimation of operators between infinite dimensional spaces

Learning operators between infinitely dimensional spaces is an important learning task arising in machine learning, imaging science, mathematical modeling and simulations, etc. This paper studies the nonparametric estimation of Lipschitz operators using deep neural networks. Non-asymptotic upper bounds are derived for the generalization error of the empirical risk minimizer over a properly chosen network class. Under the assumption that the target operator exhibits a low dimensional structure, our error bounds decay as the training sample size increases, with an attractive fast rate depending on the intrinsic dimension in our estimation. Our assumptions cover most scenarios in real applications and our results give rise to fast rates by exploiting low dimensional structures of data in operator estimation. We also investigate the influence of network structures (e.g., network width, depth, and sparsity) on the generalization error of the neural network estimator and propose a general suggestion on the choice of network structures to maximize the learning efficiency quantitatively.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

On the Training and Generalization of Deep Operator Networks

Here, we present a novel training method for deep operator networks (DeepONets), one of the most popular neural network models for operators. DeepONets are constructed by two subnetworks, namely the branch and trunk networks. Typically, the two subnetworks are trained simultaneously, which amounts to solving a complex optimization problem in a high dimensional space. In addition, the nonconvex and nonlinear nature makes training very challenging. To tackle such a challenge, we propose a two-step training method that trains the trunk network first and then sequentially trains the branch network. The core mechanism is motivated by the divide-and-conquer paradigm and is the decomposition of the entire complex training task into two subtasks with reduced complexity. Therein the Gram–Schmidt orthonormalization process is introduced which significantly improves stability and generalization ability. On the theoretical side, we establish a generalization error estimate in terms of the number of training data, the width of DeepONets, and the number of input and output sensors. Numerical examples are presented to demonstrate the effectiveness of the two-step training method, including Darcy flow in heterogeneous porous media.

deep operator networks↗