Search NASA⌕ Search

SEARCH · Search NASA

Results for “MATHEMATICS - CALCULUS, OPERATIONAL”

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 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Towards Automated Reasoning Chains for Verification of LLM-Generated Scientific Code

With the rise of Large Language Model (LLM) generated code, including in domains like scientific computing, ensuring not only syntactical, but also mathematical correctness, has become a critical task. Traditional formal methods approaches often struggle with the ambiguity of floating-point code, and full symbolic execution is extremely costly and limited. We propose a chain-of-reasoning approach that iteratively lifts basic semantics from code into the SPIRAL system and then establishes numerical equivalency to the desired mathematical operation. Here, we leverage the ample mathematical knowledge already formalized in SPIRAL to enable the system to recognize not just different implementations of the same algorithm but fully separate approaches to solving the given problem. The chain establishes tight error bounds on the output of given code with respect to the true continuous solution it approximates, quantifying all sources of error. We demonstrate this approach by establishing the correctness of a pseudospectral solver for a simple 1-dimensional Poisson problem.

Oschatz, Quentin [Carnegie Mellon University,Pitts↗
DOE OSTI2026

torch-einshard v1.0

torch-einshard is a Python library for describing local and distributed PyTorch tensor computations with compact, einsum-like notation. Its expressions name logical axes, specify how they are sharded across a PyTorch DeviceMesh, and represent partial reductions. The library automatically performs contractions, permutations, reshaping, splitting, gathering, reduction, reduce-scatter, and repartitioning while preserving autograd. Additional features include sharding-aware FFTs, tensor rolls, halo exchange, sliding windows, 1D–3D convolutions, uneven-shard handling, parameter initialization and gradient management, and cost-based execution planning. It is designed for scientific machine learning and large-model workloads, including tensor-, sequence-, and spatial-parallel MLPs, attention, convolutions, and spectral operations. Compared with manually combining torch.einsum and distributed collectives, torch-einshard expresses both the mathematical operation and data placement in one readable formula. This reduces boilerplate and synchronization errors, keeps forward and backward communication consistent, and allows the library to select optimized collective strategies without changing model code.

Morozov, Dmitriy [Lawrence Berkeley National Labor↗
DOE OSTI2025

Employing a Hardware-in-the-Loop Approach to Realize a Fully Homomorphic Controller for a Small Modular Advanced High Temperature Reactor

This paper addresses the cybersecurity challenges of advanced nuclear reactors by integrating fully homomorphic encryption (FHE) into their control systems, enabling encrypted processing of control signals without compromising functionality. Advanced nuclear reactors, including Small Modular Reactors (SMRs) and microreactors, aim to achieve autonomous and remote operations, reducing costs and enhancing competitiveness. However, these advancements expand the attack surface for cyberattacks, particularly in autonomous and remote operation scenarios. Cyberattacks can exploit vulnerabilities to manipulate physical processes, causing shutdowns, asset damage, or public harm. Such attacks begin with passive reconnaissance, where adversaries intercept communications or observe behaviors to gather information, which is then leveraged to execute cyber-physical attacks by injecting malicious commands. Nuclear power must adopt cybersecurity protection measures to secure the integrity and availability of their digital control systems. This paper demonstrates the application of FHE to secure operations by enabling encrypted processing of sensitive signals and parameters -- ensuring privacy without exposing data. FHE supports secure mathematical operations on encrypted data without requiring decryption. Using a hardware-in-the-loop (HIL) approach, this paper implements an FHE-integrated controller on a BeagleBone Black (BBB) controlling a simulation of the Small Modular Advanced High Temperature Reactor (SmAHTR). By doing so, the encrypted controller protects the integrity of critical set points and control signals during transmission and processing. Thus, FHE-integrated controllers enhance secure operations of advanced nuclear reactors while maintaining functionality.

control systems↗
DOE OSTI2025

AutoUncertainties: A Python Package for Uncertainty Propagation

Propagation of uncertainties is of great utility in the experimental sciences. While the rules of (linear) uncertainty propagation are straightforward, managing many variables with uncertainty information can quickly become complicated in large scientific software stacks. Often, this requires programmers to keep track of many variables and implement custom error propagation rules for each mathematical operator and function. The Python package AutoUncertainties, described here, provides a solution to this problem.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Practical Probabilistic Programming

Recent advances in probabilistic programming languages (PPLs) have provided the capability for exact inference: computing a closed-form probability distribution for a given probabilistic program. In particular, the new language Roulette uses a language oriented programming (LOP) approach, wherein analysts build new programming languages on top of a set of primitives provided by Roulette, which then translates these structures into a weighted model counting problem which can be solved by automated reasoning tools. However, because Roulette provides few convenience features, developing these new languages is challenging even for expert users. We developed a standard library of common probability functions for Roulette with the goal of improved usability. This included approximation of continuous probability density functions using discrete probability mass functions. We demonstrated this approach by modeling a cosmic ray striking a RAM controller. We found that Roulette provides a powerful interface for highly expressive probabilistic programs to be generated. In collaboration with the NNSA Advanced Simulation and Computing program, which resulted in development of a tool called Circulette, we were able to model complex circuits expressed in Verilog using probabilistic programs with an expressivity not previously possible. Our research question that motivated the development of a Roulette standard library was to determine whether non-experts could use a PPL to model relevant problems regarding radiation effects on microelectronics. This standard library improved the expressivity of Roulette by implementing common probability density functions, mathematical operators on distributions, and support for empirical distributions. While Roulette is a powerful modeling language, the untyped, LOP approach makes error messages difficult to understand and requires expert aid. We recommend further research on Roulette, especially with its error messages, to enable improved usability. At the same time, this project demonstrated that for users familiar with Roulette and the LOP approach, Roulette provides powerful new capabilities that can be integrated with other Sandia modeling capabilities.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

SymbolNet: neural symbolic regression with adaptive dynamic pruning for compression

Abstract Compact symbolic expressions have been shown to be more efficient than neural network (NN) models in terms of resource consumption and inference speed when implemented on custom hardware such as field-programmable gate arrays (FPGAs), while maintaining comparable accuracy (Tsoi et al 2024 EPJ Web Conf. 295 09036). These capabilities are highly valuable in environments with stringent computational resource constraints, such as high-energy physics experiments at the CERN Large Hadron Collider. However, finding compact expressions for high-dimensional datasets remains challenging due to the inherent limitations of genetic programming (GP), the search algorithm of most symbolic regression (SR) methods. Contrary to GP, the NN approach to SR offers scalability to high-dimensional inputs and leverages gradient methods for faster equation searching. Common ways of constraining expression complexity often involve multistage pruning with fine-tuning, which can result in significant performance loss. In this work, we propose S y m b o l N e t , a NN approach to SR specifically designed as a model compression technique, aimed at enabling low-latency inference for high-dimensional inputs on custom hardware such as FPGAs. This framework allows dynamic pruning of model weights, input features, and mathematical operators in a single training process, where both training loss and expression complexity are optimized simultaneously. We introduce a sparsity regularization term for each pruning type, which can adaptively adjust its strength, leading to convergence at a target sparsity ratio. Unlike most existing SR methods that struggle with datasets containing more than O ( 10 ) inputs, we demonstrate the effectiveness of our model on the LHC jet tagging task (16 inputs), MNIST (784 inputs), and SVHN (3072 inputs).

Tsoi, Ho Fung (ORCID:0000000225502184)↗
DOE OSTI2026

LibraryX: A Framework for Cross-Library-Call Optimization

Scientific applications utilize performance libraries as a software engineering concept: these libraries encapsulate important and well-understood (mathematical) operations, allow for reuse, and are implemented and tuned by experts. Domain scientists then implement complex algorithms based on these domainspecific libraries. While individual library calls are optimized, larger performance gains across sequences of calls—sometimes spanning multiple libraries—are often unrealized, forcing a trade-off between performance and implementation complexity.To overcome this issue, we propose LibraryX, an approach and a system that allows for cross-library-call optimization even when library calls stem from multiple performance libraries. LibraryX annotates library calls with semantic information and optimizes entire directed acyclic graphs (DAGs) of calls dynamically using the SPIRAL code generation system. We demonstrate its effectiveness across a range of memory bound workloads, achieving significant speedups on Nvidia, AMD, and Intel accelerators compared to code using native libraries without cross-call optimization.

Rao, Sanil [Carnegie Mellon University,Department ↗
DOE OSTI2025

QCUncertainty/sigma

Sigma is a header-only C++ library for uncertainty propagation throughout mathematical operations on floating point values.

Waldrop, Jonathan M.↗
DOE OSTI2024

Homomorphic Encryption for Electrical Metering Aggregation: Protecting the Privacy of Building Tenants

Electrical meters are devices that measure consumer electricity usage. The data collected by these meters is necessary for utility billing and electrical grid management but can also be used to assess the environmental impact of buildings. Prior research has found that unprotected metering data could potentially be used to infer some information about the behaviors of building tenants by detecting changes in electricity usage. For example, a period of low electricity usage could suggest that the tenants are not in the building. As smart metering becomes more common, there is a growing need for data privacy protections for metering data that do not negatively impact the quality and availability of data used for energy management and billing applications. To identify potential solutions, we developed a Python-based data aggregation platform to analyze the potential efficacy of privacy-enhancing technologies for energy metering applications. This platform aggregates groups of metering sites into virtual buildings, which could potentially detach changes in electrical activity from individual tenants, making it more difficult to track the activity of a specific tenant. To further protect data during analysis, this project utilizes homomorphic encryption as part of its initial approach. Homomorphic encryption offers a means of protecting energy consumption data while permitting mathematical operations to be performed without the need to know the data contents. This allows for data to be processed into usable statistics without revealing energy consumption information. A series of homomorphic encryption libraries were evaluated to determine their applicability and limitations in the context of metering data. The use of these techniques may help to reassure consumers and encourage further adoption of smart grid infrastructure.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
DOE OSTI2025

Even Higher-Level Synthesis: An Exploration of AI Hardware Accelerators using HLS4ML

With the rise of artificial intelligence, the popularization of deep learning, and a constantly evolving industry, the demand for flexible and efficient tools has never been greater. As algorithms grow more complex, their runtime and energy consumption increase exponentially. Customized hardware accelerators, long used for specific mathematical operations, remain essential for managing modern applications' computational and power demands. Hardware accelerators can speed up complex computations by orders of magnitude, but their manual design and verification processes are often challenging and time-consuming. High-Level Synthesis (HLS) provides a solution by transforming high-level algorithm descriptions, typically written in C++ or SystemC, into synthesizable RTL suitable for hardware implementation. This approach reduces development time for RTL engineers while offering flexibility beyond what traditional handwritten RTL can provide. We extended this capability to the machine-learning domain with the open-source framework hls4ml, which allows neural networks trained in Python frameworks like Tensorflow or PyTorch to be synthesized into efficient hardware representations for the traditional FPGA and ASIC flows. This breakthrough addresses the growing need for reduced design turnaround and easy verification of ML hardware accelerators with low latency and power efficiency constraints. During this tutorial, we will demonstrate how Python complements HLS by simplifying the ML design process, bridging the gap between software and hardware development. Attendees will explore how we translate neural networks modeled in Python into fixed-point C++ models suitable for HLS workflows. We will dive into strategies like Value-Range Analysis and Quantization-Aware Training, which optimize these designs for deployment and evaluate their accuracy, power consumption, and energy efficiency. To exemplify these concepts, experts from Fermilab will share their experiences applying this technology to high-energy physics experiments, where real-time, low-latency processing is critical. Over the years, Fermilab engineers have demonstrated how deep neural networks, optimized for hardware using hls4ml, can meet the stringent requirements of trigger systems at the CERN Large Hadron Collider. These systems rely on rapid decision-making to process immense data volumes while retaining only the most relevant events for further analysis. The application of hls4ml has also been extended to innovative technologies like smart pixel arrays. These smart pixels integrate ML inference capabilities directly into sensor devices, enabling localized data processing at the pixel level. This approach drastically reduces the need to transmit raw data to external processing units, significantly decreasing power consumption and latency. By embedding neural networks within the pixel architecture, the smart pixels can identify and prioritize relevant data in real time, providing a highly efficient solution for edge computing in scenarios such as particle detectors and imaging systems. Fermilab's work highlights the potential of hardware-accelerated ML in scenarios where both speed and power efficiency are mission-critical. Through this tutorial, attendees will gain valuable insights into the challenges and solutions of deploying ML in hardware. Understanding how HLS and hls4ml streamline the development of neural network-based hardware accelerators is fundamental for the industry's future. Participants will learn how these technologies are shaping the future of AI and scientific computing.

Di Guglielmo, Giuseppe [Fermilab]↗
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 OSTI2025

Radiological Safety Analysis Computer (RSAC) Program Version 7.4 Users’ Manual

The Radiological Safety Analysis Computer (RSAC) Program Version 7.4 (RSAC-7) is the newest version of the RSAC legacy code. RSAC-7 calculates the consequences of a release of radionuclides to the atmosphere. Users generates a fission product inventory from either reactor operating history or a nuclear criticality event. RSAC-7 models the effects of high-efficiency particulate air filters or other cleanup systems and calculates the decay and ingrowth during transport through processes, facilities, and the environment. Doses are calculated for inhalation, air immersion, ground surface, ingestion, and cloud gamma pathways. RSAC-7 is used as a tool to evaluate accident conditions in emergency response scenarios, radiological sabotage events, and safety basis accident consequences. This users’ manual contains the mathematical models and operating instructions for RSAC-7. Instructions, screens, and examples are provided to guide the user through the functions provided by RSAC-7. This program is designed for users who are familiar with radiological dose assessment methods.

73 - NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2025

Specificity and tunability of efflux pumps: A new role for the proton gradient?

Efflux pumps that transport antibacterial drugs out of bacterial cells have broad specificity, commonly leading to broad spectrum resistance and limiting treatment strategies for infections. It remains unclear how efflux pumps can maintain this broad spectrum specificity to diverse drug molecules while limiting the efflux of other cytoplasmic content. We have investigated the origins of this broad specificity using theoretical models informed by the experimentally determined structural and kinetic properties of efflux pumps. We developed a set of mathematical models describing operation of efflux pumps as a discrete cyclic stochastic process across a network of states characterizing pump conformations and the presence/absence of bound ligands and protons. These include a minimal three-state model that lends itself to clear analytic calculations as well as a five-state model that relaxes some of the simpler model’s most strict assumptions. We found that the pump specificity is determined not solely by the drug affinity to the pump–as is commonly assumed–but it is also directly affected by the periplasmic pH and the transmembrane potential. Therefore, changes to the proton concentration gradient and voltage drop across the membrane can influence how effective the pump is at extruding a particular drug molecule. Furthermore, we found that while both the proton concentration gradient across the membrane and the transmembrane potential contribute to the thermodynamic force driving the pump, their effects on the efflux enter not strictly in a combined proton motive force. Rather, they have two distinguishable effects on the overall throughput. These results highlight the unexpected effects of thermodynamic driving forces out of equilibrium and illustrate how efflux pump structure and function are conducive to the emergence of multidrug resistance.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2025

What exactly does Bekenstein bound?

The Bekenstein bound posits a maximum entropy for matter with finite energy confined to a spatial region. It is often interpreted as a fundamental limit on the information that can be stored by physical objects. In this work, we test this interpretation by asking whether the Bekenstein bound imposes constraints on a channel's communication capacity, a context in which information can be given a mathematically rigorous and operationally meaningful definition. We study specifically the Unruh channel that describes a stationary Alice exciting different species of free scalar fields to send information to an accelerating Bob, who is confined to a Rindler wedge and exposed to the noise of Unruh radiation. We show that the classical and quantum capacities of the Unruh channel obey the Bekenstein bound that pertains to the decoder Bob. In contrast, even at high temperatures, the Unruh channel can transmit a significant number of zero-bits , which are quantum communication resources that can be used for quantum identification and many other primitive protocols. Therefore, unlike classical bits and qubits, zero-bits and their associated information processing capability are generally not constrained by the Bekenstein bound. However, we further show that when both the encoder and the decoder are restricted, the Bekenstein bound does constrain the channel capacities, including the zero-bit capacity.

Hayden, Patrick [Stanford University, CA (United S↗
DOE OSTI2025

Scattering theory in noncanonical phase space: A Drift-Kinetic collision operator for weakly collisional plasmas

After developing a scattering theory for grazing collisions in general noncanonical phase spaces, we introduce a guiding center collision operator in five-dimensional phase space designed for plasma regimes characterized by long wavelengths (relative to the Larmor radius), low frequencies (relative to the cyclotron frequency), and weak collisionality (where repeated Coulomb collisions induce cumulatively small changes in particle magnetic moment). The collision operator is fully determined by the noncanonical Hamiltonian structure of guiding center dynamics and exhibits a metriplectic structure, ensuring the conservation of particle number, momentum, energy, and interior Casimir invariants. It also satisfies an H-theorem, allowing for deviations from an equilibrium Maxwellian distribution due to the nontrivial kernel of the noncanonical guiding center Poisson tensor, spanned by the magnetic moment. We propose that this collision operator and its underlying mathematical structure may offer valuable insight into the study of turbulence, transport, and self-organizing phenomena in both laboratory and astrophysical plasmas.

Hamiltonian mechanics↗
DOE OSTI2025

Closed-form expressions for unitaries of spin-adapted fermionic operators

One of the open challenges in quantum computing simulations of problems of chemical interest is the proper enforcement of spin symmetry. Efficient quantum circuits implementing unitaries generated by spin-adapted operators remain elusive, while naïve Trotterization schemes break spin symmetry. Here, in this work, we analyze the mathematical structure of spin-adapted operators and derive closed-form expressions for unitaries generated by singlet spin-adapted generalised single and double excitations. These results represent significant progress toward the economical enforcement of spin symmetry in quantum simulations.

fermionic algebra↗
DOE OSTI2026

A mathematical framework for thermodynamic computing with applications to chemical reaction networks

The widespread adoption of energy-intensive computing applications has led to a growing need for energy-efficient computing approaches. Thermodynamic computing offers a promising approach for low-energy computation by leveraging the intrinsic computational capabilities of physical, chemical, or biological systems. However, the mathematical foundations of thermodynamic computing require further development to fully realize the potential energy efficiencies, as well as to assess factors like noise and operational speed. In this paper, we establish a mathematical framework for utilizing thermodynamic processes to perform fundamental operations, including addition, subtraction, multiplication, and division. We highlight the use of chemical reactions as potential computational units and explore synthetic chemical and biochemical systems as practical implementations. Additionally, we demonstrate how these principles can be applied to solving complex mathematical problems, such as ordinary differential equations (ODEs) and suggest the necessary components to implement the thermodynamic computing framework using chemical reactions based in a microfluidic device. This work enhances our understanding of thermodynamic processes for natural computing as a basis for scalable, energy-efficient computation in paradigm disruptive next-generation systems.

Cannon, William R. [Pacific Northwest National Lab↗
DOE OSTI2025

Analog Systems for Edge Optimization

Over the past decade, analog computing has the subject of substantial research interest providing a path toward improved computational efficiency in the post-Dennard era. Analog matrix vector multiplication (MVM) accelerators provide a popular approach given the ubiquity of MVM operations in numerous applications. However, historically analog computing systems can struggle with applications requiring high precision due to the inherent susceptibility of these systems to analog non-idealities. Therefore, prior work on analog systems has focused either on applications known to be tolerant of limited precision (e.g., neural network inference), or using expensive techniques to emulate high-precision using many analog MVM operations. In this work, we propose an alternative approach. Motivated by recent advances in inexact nonlinear solvers and optimizers, we explore the potential of co-designing optimization algorithms which can take full advantage of the fundamentally inexact analog MVM operations. To enable these co-designed algorithms we also develop a general mathematical theory of the precision and energy efficiency of analog operations, and a new system architecture for tightly-coupled analog and digital computation. Finally, we examine the applicability of analog computing to a wider class of symmetric positive definite systems and find potential in using analog operations as a sparse approximate inverse preconditioner. With these core innovations, this project provides a path toward effectively implementing optimization algorithms on power-constrained autonomous and semi-autonomous systems.

97 MATHEMATICS AND COMPUTING↗