Establishing Machine Learning Operations for Continual Learning in Computing Clusters: A Framework for Monitoring and Optimizing Cluster Behavior
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97 MATHEMATICS AND COMPUTING↗
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Results for “operator 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.
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TANTE: Time-adaptive operator learning via neural Taylor expansion
Operator learning for time-dependent partial differential equations (PDEs) has seen rapid progress in recent years, enabling efficient approximation of complex spatiotemporal dynamics. However, most existing methods rely on fixed time step sizes during rollout, which limits their ability to adapt to varying temporal complexity and often leads to error accumulation. In this work, we propose the Time-Adaptive Transformer with Neural Taylor Expansion (TANTE), a novel operator-learning framework that produces continuous-time predictions with adaptive step sizes. TANTE predicts future states by performing a Taylor expansion at the current state, where neural networks learn both the higher-order temporal derivatives and the local radius of convergence. This allows the model to dynamically adjust its rollout based on the local behavior of the solution, thereby reducing cumulative error and improving computational efficiency. We demonstrate the effectiveness of TANTE across a wide range of PDE benchmarks, achieving superior accuracy and adaptability compared to fixed-step baselines, delivering accuracy gains of 60-80 % and speed-ups of 30-40 % at inference time.
97 MATHEMATICS AND COMPUTING↗
Space Operations Learning Center
The Space Operations Learning Center (SOLC) is a tool that provides an online learning environment where students can learn science, technology, engineering, and mathematics (STEM) through a series of training modules. SOLC is also an effective media for NASA to showcase its contributions to the general public. SOLC is a Web-based environment with a learning platform for students to understand STEM through interactive modules in various engineering topics. SOLC is unique in its approach to develop learning materials to teach schoolaged students the basic concepts of space operations. SOLC utilizes the latest Web and software technologies to present this educational content in a fun and engaging way for all grade levels. SOLC uses animations, streaming video, cartoon characters, audio narration, interactive games and more to deliver educational concepts. The Web portal organizes all of these training modules in an easily accessible way for visitors worldwide. SOLC provides multiple training modules on various topics. At the time of this reporting, seven modules have been developed: Space Communication, Flight Dynamics, Information Processing, Mission Operations, Kids Zone 1, Kids Zone 2, and Save The Forest. For the first four modules, each contains three components: Flight Training, Flight License, and Fly It! Kids Zone 1 and 2 include a number of educational videos and games designed specifically for grades K-6. Save The Forest is a space operations mission with four simulations and activities to complete, optimized for new touch screen technology. The Kids Zone 1 module has recently been ported to Facebook to attract wider audience.
Lui, Ben↗
Predicting nonequilibrium Green’s function dynamics and photoemission spectra via nonlinear integral operator learning
Understanding the dynamics of nonequilibrium quantum many-body systems is an important research topic in a wide range of fields across condensed matter physics, quantum optics, and high-energy physics. However, numerical studies of large-scale nonequilibrium phenomena in realistic materials face serious challenges due to intrinsic high-dimensionality of quantum many-body problems and the absence of time-invariance. The nonequilibrium properties of many-body systems can be described by the dynamics of the correlator, or the Green's function of the system, whose time evolution is given by a high-dimensional system of integro-differential equations, known as the Kadanoff–Baym equations (KBEs). The time-convolution term in KBEs, which needs to be recalculated at each time step, makes it difficult to perform long-time numerical simulation. In this paper, we develop an operator-learning framework based on recurrent neural networks (RNNs) to address this challenge. We utilize RNNs to learn the nonlinear mapping between Green's functions and convolution integrals in KBEs. By using the learned operators as a surrogate model in the KBE solver, we obtain a general machine-learning scheme for predicting the dynamics of nonequilibrium Green's functions. Besides significant savings per each time step, the new methodology reduces the temporal computational complexity from $O(N_t^3)$ to $O(N_t)$ where N t is the number of steps taken in a simulation, thereby making it possible to study large many-body problems which are currently infeasible with conventional KBE solvers. Through various numerical examples, we demonstrate the effectiveness of the operator-learning based approach in providing accurate predictions of physical observables such as the reduced density matrix and time-resolved photoemission spectra. Moreover, our framework exhibits clear numerical convergence and can be easily parallelized, thereby facilitating many possible further developments and applications.
97 MATHEMATICS AND COMPUTING↗
Improved Adjoint-Operator Learning For A Neural Network
Improved method of adjoint-operator learning reduces amount of computation and associated computational memory needed to make electronic neural network learn temporally varying pattern (e.g., to recognize moving object in image) in real time. Method extension of method described in "Adjoint-Operator Learning for a Neural Network" (NPO-18352).
Toomarian, Nikzad↗
Generalization error guaranteed auto-encoder-based nonlinear model reduction for operator learning
Many physical processes in science and engineering are naturally represented by operators between infinite-dimensional function spaces. The problem of operator learning, in this context, seeks to extract these physical processes from empirical data, which is challenging due to the infinite or high dimensionality of data. An integral component in addressing this challenge is model reduction, which reduces both the data dimensionality and problem size. In this paper, we utilize low-dimensional nonlinear structures in model reduction by investigating Auto-Encoder-based Neural Network (AENet). AENet first learns the latent variables of the input data and then learns the transformation from these latent variables to corresponding output data. Our numerical experiments validate the ability of AENet to accurately learn the solution operator of nonlinear partial differential equations. Furthermore, we establish a mathematical and statistical estimation theory that analyzes the generalization error of AENet. Finally, our theoretical framework shows that the sample complexity of training AENet is intricately tied to the intrinsic dimension of the modeled process, while also demonstrating the robustness of AENet to noise.
Auto-encoder↗
Space Operations Learning Center Facebook Application
The proposed Space Operations Learning Center (SOLC) Facebook module, initially code-named Spaceville, is intended to be an educational online game utilizing the latest social networking technology to reach a broad audience base and inspire young audiences to be interested in math, science, and engineering. Spaceville will be a Facebook application/ game with the goal of combining learning with a fun game and social environment. The mission of the game is to build a scientific outpost on the Moon or Mars and expand the colony. Game activities include collecting resources, trading resources, completing simple science experiments, and building architectures such as laboratories, habitats, greenhouses, machine shops, etc. The player is awarded with points and achievement levels. The player s ability increases as his/her points and levels increase. A player can interact with other players using multiplayer Facebook functionality. As a result, a player can discover unexpected treasures through scientific missions, engineering, and working with others. The player creates his/her own avatar with his/her selection of its unique appearance, and names the character. The player controls the avatar to perform activities such as collecting oxygen molecules or building a habitat. From observations of other successful social online games such as Farmville and Restaurant City, a common element of these games is having eye-catching and cartoonish characters, and interesting animations for all activities. This will create a fun, educational, and rewarding environment. The player needs to accumulate points in order to be awarded special items needed for advancing to higher levels. Trophies will be awarded to the player when certain goals are reached or tasks are completed. In order to acquire some special items needed for advancement in the game, the player will need to visit his/her neighboring towns to discover the items. This is the social aspect of the game that requires the player to go out of his/her own establishment to explore what is in the neighborhood. Spaceville will take advantage of Facebook s successful architecture to inspire a new audience of scientists and engineers for the future.
Lui, Ben↗
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↗
Transformer-based operator learning framework for self-energy in strongly correlated systems
We introduce Σ-Attention, a transformer-based operator-learning framework for approximating the self-energy operator of strongly correlated electronic systems. By creating a batched dataset that combines results from three complementary approaches, i.e., many-body perturbation theory, strong-coupling expansion, and exact diagonalization, each effective in specific parameter regimes, Σ-Attention is applied to learn an accurate approximation for the self-energy operator that is valid across a wide range of parameter regimes. This hybrid strategy leverages the strengths of existing methods while relying on the transformer's ability to generalize beyond individual limitations. More importantly, the scalability of the transformer architecture allows the learned self-energy to be extended to systems with larger sizes, leading to much improved computational scaling. Using the one-dimensional Hubbard model, we demonstrate that Σ-Attention can accurately predict the Matsubara Green's function of large systems with a wide range of coupling strength. Our framework offers a promising and scalable pathway for studying strongly correlated systems with many possible generalizations.
Zhu, Yuanran↗
Operator learning for energy-efficient building ventilation control with computational fluid dynamics simulation of a real-world classroom
Energy-efficient ventilation control plays an important role in reducing building energy consumption while ensuring occupant health and comfort. While Computational Fluid Dynamics (CFD) simulations provide detailed and physically accurate representations of indoor airflow, their high computational cost limits their use in real-time building control. In this work, we present a neural operator learning framework that combines the physical accuracy of CFD with the computational efficiency of machine learning to enable building ventilation control with the high-fidelity fluid dynamics models. Our method jointly optimizes the airflow supply rates and vent angles to reduce energy use and adhere to air quality constraints. We train an ensemble of neural operator transformer models to learn the mapping from building control actions to airflow fields using high-resolution CFD data. This learned neural operator is then embedded in an optimization-based control framework for building ventilation control. Experimental results show that our approach achieves significant energy savings compared to maximum airflow rate control, rule-based control, as well as data-driven control methods using spatially averaged CO 2 prediction and deep learning–based reduced-order models, while consistently maintaining safe indoor air quality. These results highlight the practicality and scalability of our method in maintaining energy efficiency and indoor air quality in real-world buildings.
32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
Coefficient-to-Basis Network: a fine-tunable operator learning framework for inverse problems with adaptive discretizations and theoretical guarantees
We propose a Coefficient-to-Basis Network (C2BNet), a novel framework for solving inverse problems within the operator learning paradigm. C2BNet efficiently adapts to different discretizations through fine-tuning, using a pre-trained model to significantly reduce computational cost while maintaining high accuracy. Unlike traditional approaches that require retraining from scratch for new discretizations, our method enables seamless adaptation without sacrificing predictive performance. Furthermore, we establish theoretical approximation and generalization error bounds for C2BNet by exploiting low-dimensional structures in the underlying datasets. Our analysis demonstrates that C2BNet adapts to low-dimensional structures without relying on explicit encoding mechanisms, highlighting its robustness and efficiency. To validate our theoretical findings, we conducted extensive numerical experiments that showcase the superior performance of C2BNet on several inverse problems. The results confirm that C2BNet effectively balances computational efficiency and accuracy, making it a promising tool to solve inverse problems in scientific computing and engineering applications.
97 MATHEMATICS AND COMPUTING↗
Active operator learning with predictive uncertainty quantification for partial differential equations
With the increased prevalence of neural operators being used to provide rapid solutions to partial differential equations (PDEs), understanding the accuracy of model predictions and the associated error levels is necessary for deploying reliable surrogate models in scientific applications. Existing uncertainty quantification (UQ) frameworks employ ensembles or Bayesian methods, which can incur substantial computational costs during both training and inference. Here, we propose a lightweight predictive UQ method tailored for Deep operator networks (DeepONets) that also generalizes to other operator networks. Numerical experiments on linear and nonlinear PDEs demonstrate that the framework’s uncertainty estimates are unbiased and provide accurate out-of-distribution uncertainty predictions with a sufficiently large training dataset. Our framework provides fast inference and uncertainty estimates that can efficiently drive outer-loop analyses that would be prohibitively expensive with conventional solvers. We demonstrate how predictive uncertainties can be used in the context of Bayesian optimization and active learning problems to yield improvements in accuracy and data-efficiency for outer-loop optimization procedures. In the active learning setup, we extend the framework to Fourier Neural Operators (FNO) and describe a generalized method for other operator networks. To enable real-time deployment, we introduce an inference strategy based on precomputed trunk outputs and a sparse placement matrix, reducing evaluation time by more than a factor of five. Our method provides a practical route to uncertainty-aware operator learning in time-sensitive settings.
97 MATHEMATICS AND COMPUTING↗
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↗
The Impact of Pictorial Display on Operator Learning and Performance
The effects of pictorially displayed information on human learning and performance of a simple control task were investigated. The controlled system was a harmonic oscillator and the system response was displayed to subjects as either an animated pendulum or a horizontally moving dot. Results indicated that the pendulum display did not effect performance scores but did significantly effect the learning processes of individual operators. The subjects with the pendulum display demonstrated more vertical internal models early in the experiment and the manner in which their internal models were tuned with practice showed increased variability between subjects.
Miller, R. A.↗
Parallel-in-Time Solution of Allen-Cahn Equations by Integrating Operator Learning into the Parareal Method
While recent advances in deep learning have shown promising efficiency gains in solving time-dependent partial differential equations (PDEs), matching the accuracy of conventional numerical solvers still remains a challenge. One strategy to improve the accuracy of deep learning-based solutions for time-dependent PDEs is to use the learned model as the coarse propagator in the Parareal method and a traditional numerical method as the fine solver. However, successful integration of deep learning into the Parareal method requires consistency between the coarse and fine solvers, particularly for PDEs exhibiting rapid changes such as sharp transitions. Here, to ensure this consistency, we propose using convolutional neural networks (CNNs) to learn the fully discrete time-stepping operator defined by the same numerical scheme employed as the fine solver. We demonstrate the effectiveness of the proposed method in solving the classical and mass-conservative Allen–Cahn (AC) equations. Through iterative updates in the Parareal algorithm, our approach achieves a significant computational speedup compared to traditional fine solvers while converging to high-accuracy solutions. Our results highlight that the proposed hybrid Parareal algorithm effectively accelerates simulations, particularly when implemented on multiple GPUs, and converges to the desired accuracy in only a few iterations. Another advantage of our method is that the CNN model is trained on trajectory-based data generated from random initial conditions, such that the trained model can be used to solve the AC equations with various initial conditions without retraining. This work demonstrates the potential of integrating neural network methods into parallel-in-time frameworks for efficient and accurate simulations of time-dependent PDEs.
97 MATHEMATICS AND COMPUTING↗
Adjoint-Operator Learning For A Neural Network
Electronic neural networks made to synthesize initially unknown mathematical models of time-dependent phenomena or to learn temporally evolving patterns by use of algorithms based on adjoint operators. Algorithms less complicated, involve less computation and solve learning equations forward in time possibly simultaneously with equations of evolution of neural network, thereby both increasing computational efficiency and making real-time applications possible.
Barhen, Jacob↗
Human Flight to Lunar and Beyond - Re-Learning Operations Paradigms
For the first time since the Apollo era, NASA is planning on sending astronauts on flights beyond LEO. The Human Space Flight (HSF) program started with a successful initial flight in Earth orbit, in December 2014. The program will continue with two Exploration Missions (EM): EM-1 will be unmanned and EM-2, carrying astronauts, will follow. NASA established a multi-center team to address the communications, and related tacking/navigation needs. This paper will focus on the lessons learned by the team designing the architecture and operations for the missions. Many of these Beyond Earth Orbit lessons had to be re-learned, as the HSF program has operated for many years in Earth orbit. Unlike the Apollo missions that were largely tracked by a dedicated ground network, the HSF planned missions will be tracked (at distances beyond GEO) by the DSN, a network that mostly serves robotic missions. There have been surprising challenges to the DSN as unique modern human spaceflight needs stretch the experience base beyond that of tracking robotic missions in deep space. Close interaction between the DSN and the HSF community to understand the unique needs (e.g. 2-way voice) resulted in a Concept of Operations (ConOps) that leverages both the deep space robotic and the Human LEO experiences. Several examples will be used to highlight the unique challenges the team faced in establishing the communications and tracking capabilities for HSF missions beyond Earth Orbit, including: Navigation. At LEO, HSF missions can rely on GPS devices for orbit determination. For Lunar-and-beyond HSF missions, techniques such as precision 2-way and 3-way Doppler and ranging, Delta-Difference-of-range, and eventually possibly on-board navigation will be used. At the same time, HSF presents a challenge to navigators, beyond those presented by robotic missions - navigating a dynamic/"noisy" spacecraft. Impact of latency - the delay associated with Round-Trip-Light-Time (RTLT). Imagine trying to have a 2-way discussion (audio or video) with an astronaut, with a 2-3 sec or more delay inserted (for lunar distances) or 20 minutes delay (for Mars distances). Balanced communications link. For robotic missions, there has been a heavy emphasis on higher downlink data rates, e.g. bringing back science data. Higher uplink data rates were of secondary importance, as uplink was used only to send commands (and occasionally small files) to the spacecraft. The ratio of downlink-to-uplink data rates was often 10:1 or more. For HSF, a continuous forward link is established and rates for uplink and downlink are more similar.
Kenny, Edward (Ted)↗
Human Flight to Lunar and Beyond - Re-Learning Operations Paradigms
For the first time since the Apollo era, NASA is planning on sending astronauts on flights beyond Low-Earth Orbit (LEO). The Human Space Flight (HSF) program started with a successful initial flight in Earth orbit, in December 2014. The program will continue with two Exploration Missions (EM) to Lunar orbit: EM-1 will be unmanned and EM-2, carrying astronauts, will follow. NASA established a multi-center team to address the communications, and related navigation, needs. This paper will focus on the lessons learned in the team, planning for the missions' parts that are beyond Earth orbit. Many of these lessons had to be re-learned, as the HSF program after operated for many years in Earth orbit. Fortunately, the experience base from tracking robotic missions in deep space by the Deep Space Network (DSN) and close interaction with the HSF community to understand the unique needs (e.g. 2-way voice) resulted in a ConOps that leverages of both the deep space robotic and the Human LEO experiences. Several examples will be used to highlight the unique operational needs for HSF missions beyond Earth Orbit, including: - Navigation. At LEO, HSF missions can rely on Global Positioning System (GPS) devices for orbit determination. For Lunar-and-beyond HSF missions, techniques such as precision 2-way and 3-way Doppler and ranging, Delta-Difference-of-range, and eventually on-board navigation will be used. - Impact of latency - the delay associated with Round-Trip-Light-Time (RTLT). Imagine trying to have a 2-way discussion (audio or video) with an astronaut, with a 2-3 sec delay inserted (for Lunar distances) or 20 minutes delay (for Mars distances). - Balanced communications link. For robotic missions, there has been a heavy emphasis on the downlink data rates, bringing back science data from the instruments on-board the spacecraft. Uplink data rates were of secondary importance, used to send commands to the spacecraft. The ratio of downlink-to-uplink data rates was often 10:1 or more. For HSF, rates for uplink and downlink, at least for high-quality video, need to be similar.
Kenny, Ted↗