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All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Digital zero noise extrapolation for quantum error mitigation

Zero-noise extrapolation (ZNE) is an increasingly popular technique for mitigating errors in noisy quantum computations without using additional quantum resources. We review the fundamentals of ZNE and propose several improvements to noise scaling and extrapolation, the two key components in the technique. We introduce unitary folding and parameterized noise scaling. Furthermore, these are digital noise scaling frameworks, i.e. one can apply them using only gate-level access common to most quantum instruction sets. We also study different extrapolation methods, including a new adaptive protocol that uses a statistical inference framework. Benchmarks of our techniques show error reductions of 18X to 24X over non-mitigated circuits and demonstrate ZNE effectiveness at larger qubit numbers than have been tested previously. In addition to presenting new results, this work is a self-contained introduction to the practical use of ZNE by quantum programmers.

quantum computing↗
DOE OSTI2020

How Should Machine Learning Be Successfully Used for Wind Speed Vertical Extrapolation?

An accurate characterization of the wind resource available at hub-height is required for an efficient and bankable wind farm project. However, direct measurement of wind speed at the constantly increasing height of the hub of commercial wind turbines is oftentimes challenging and expensive, so that it is common practice to vertically extrapolate the wind resource from lower and more easily accessible levels. Conventional techniques for wind speed vertical extrapolation include the use of a power law and a logarithmic profile. While simple, the limits in accuracy of these methods have been shown in various studies. Recently, machine learning has been proposed as a new method to vertically extrapolate winds. All the published studies on the topic assess the performance of machine learning techniques in vertically extrapolating the wind resource at the same location where the algorithm has been trained. However, in real-world applications, the wind resource is measured at the instrument location, but it then needs to be extrapolated at hub height at the location of the wind turbines within the find farm. To be able to fully recommend the use of machine learning techniques over the simple power law and logarithmic law, the spatial variability of the performance improvements of the machine learning approaches needs to be assessed. Here, we propose a round-robin validation of a machine learning-based method for wind speed extrapolation. We use 20 months of observations at four locations spanning a 100 km wide region at the Southern Great Plains (SGP) atmospheric observatory, in north-central Oklahoma. At each location, we train a random forest to predict 30-min average wind speed at 143 m AGL. We use as input features lidar wind speed at 65 m AGL, time of day, sonic anemometer wind speed at 4 m AGL, turbulent kinetic energy, and Obukhov length. First, we perform a same-site comparison of the performance of the proposed random forest against the conventional techniques for wind speed extrapolation (namely power law and logarithmic profile, with widely accepted stability corrections). We find that the random forest outperforms the power law in vertically extrapolating wind speed in all the considered stability regimes, with a 33% reduction in MAE for stable conditions, and a 31% reduction in unstable conditions. Similar results are found when comparing predictions of extrapolated winds from the logarithmic profile and the random forest with the observed values. Next, we propose a round-robin validation, to use the random forest trained at each site to extrapolate wind speed at the remaining three sites. We find that the performance of the random forest approach degrades when the algorithm is tested at a site different than the training one. However, even under those circumstances, the machine learning-based approach still outperforms the conventional techniques for wind speed extrapolation, with, on average, a reduction in mean absolute error between 15 and 20% over the conventional methods, with the largest benefits obtained under stable conditions.

Monte Carlo↗
DOE OSTI2023

Reliable extrapolation of deep neural operators informed by physics or sparse observations

Deep neural operators can learn nonlinear mappings between infinite-dimensional function spaces via deep neural networks. As promising surrogate solvers of partial differential equations (PDEs) for real-time prediction, deep neural operators such as deep operator networks (DeepONets) provide a new simulation paradigm in science and engineering. Pure data-driven neural operators and deep learning models, in general, are usually limited to interpolation scenarios, where new predictions utilize inputs within the support of the training set. However, in the inference stage of real-world applications, the input may lie outside the support, i.e., extrapolation is required, which may result to large errors and unavoidable failure of deep learning models. Here, we address this challenge of extrapolation for deep neural operators. First, we systematically investigate the extrapolation behavior of DeepONets by quantifying the extrapolation complexity, via the 2-Wasserstein distance between two function spaces and propose a new strategy of bias–variance trade-off for extrapolation with respect to model capacity. Subsequently, we develop a complete workflow, including extrapolation determination, and we propose five reliable learning methods that guarantee a safe prediction under extrapolation by requiring additional information—the governing PDEs of the system or sparse new observations. The proposed methods are based on either fine-tuning a pre-trained DeepONet or multifidelity learning. We demonstrate the effectiveness of the proposed framework for various types of parametric PDEs. Furthermore, our systematic comparisons provide practical guidelines for selecting a proper extrapolation method depending on the available information, desired accuracy, and required inference speed.

42 ENGINEERING↗
DOE OSTI2021

Constrained Extrapolation Problem and Order-Dependent Mappings

In this work, we consider the problem of extrapolating the perturbation series for the dilute Fermi gas in three dimensions to the unitary limit of infinite scattering length and into the BEC region, using the available strong-coupling information to constrain the extrapolation problem. In this constrained extrapolation problem (CEP) the goal is to find classes of approximants that give well converged results already for low perturbative truncation orders. First, we show that standard Padé and Borel methods are too restrictive to give satisfactory results for this CEP. A generalization of Borel extrapolation is given by the so-called Maximum Entropy extrapolation method (MaxEnt). However, we show that MaxEnt requires extensive elaborations to be applicable to the dilute Fermi gas and is thus not practical for the CEP in this case. Instead, we propose order-dependent-mapping extrapolation (ODME) as a simple, practical, and general method for the CEP. Here, we find that the ODME approximants for the ground-state energy of the dilute Fermi gas are robust with respect to changes of the mapping choice and agree with results from quantum Monte Carlo simulations within uncertainties.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

How accurate is a machine learning-based wind speed extrapolation under a round-robin approach?

As the size of commercial wind turbines keeps increasing, having accurate ways to vertically extrapolate wind speed is essential to obtain a precise characterization of the wind resource for wind energy production. Recently, machine learning has been proposed and applied to extrapolate wind speed to hub heights. However, previous studies trained and tested the machine learning methods at the same site, giving them an unfair advantage over the conventional extrapolation techniques, which are instead more universal. Here, we use data from four sites in Oklahoma to test a round-robin validation approach for machine learning, under which we train a random forest at a site, and test it at a different site, where the model has no prior knowledge of the wind resource. We quantify how the accuracy of this technique varies with distance from the training site, and we find that it outperforms conventional techniques for wind extrapolation at all the considered spatial separations. We then assess how the accuracy of the machine-learning based approach varies when it is used to predict wind speed in a wind farm far wake. Finally, we explore as case study the performance of the random forest in extrapolating winds during a low-level jet event.

17 WIND ENERGY↗
DOE OSTI2020

Decreasing wind speed extrapolation error via domain-specific feature extraction and selection

Abstract. Model uncertainty is a significant challenge in the wind energy industry and can lead to mischaracterization of millions of dollars' worth of wind resources. Machine learning methods, notably deep artificial neural networks (ANNs), are capable of modeling turbulent and chaotic systems and offer a promising tool to produce high-accuracy wind speed forecasts and extrapolations. This paper uses data collected by profiling Doppler lidars over three field campaigns to investigate the efficacy of using ANNs for wind speed vertical extrapolation in a variety of terrains, and it quantifies the role of domain knowledge in ANN extrapolation accuracy. A series of 11 meteorological parameters (features) are used as ANN inputs, and the resulting output accuracy is compared with that of both standard log-law and power-law extrapolations. It is found that extracted nondimensional inputs, namely turbulence intensity, current wind speed, and previous wind speed, are the features that most reliably improve the ANN's accuracy, providing up to a 65 % and 52 % increase in extrapolation accuracy over log-law and power-law predictions, respectively. The volume of input data is also deemed important for achieving robust results. One test case is analyzed in depth using dimensional and nondimensional features, showing that the feature nondimensionalization drastically improves network accuracy and robustness for sparsely sampled atmospheric cases.

17 WIND ENERGY↗
NASA NTRS1992

Extrapolation limitations of multilayer feedforward neural networks

The limitations of backpropagation used as a function extrapolator were investigated. Four common functions were used to investigate the network's extrapolation capability. The purpose of the experiment was to determine whether neural networks are capable of extrapolation and, if so, to determine the range for which networks can extrapolate. The authors show that neural networks cannot extrapolate and offer an explanation to support this result.

Haley, Pamela J.↗
NASA NTRS1992

The Extrapolation of Elementary Sequences

We study sequence extrapolation as a stream-learning problem. Input examples are a stream of data elements of the same type (integers, strings, etc.), and the problem is to construct a hypothesis that both explains the observed sequence of examples and extrapolates the rest of the stream. A primary objective -- and one that distinguishes this work from previous extrapolation algorithms -- is that the same algorithm be able to extrapolate sequences over a variety of different types, including integers, strings, and trees. We define a generous family of constructive data types, and define as our learning bias a stream language called elementary stream descriptions. We then give an algorithm that extrapolates elementary descriptions over constructive datatypes and prove that it learns correctly. For freely-generated types, we prove a polynomial time bound on descriptions of bounded complexity. An especially interesting feature of this work is the ability to provide quantitative measures of confidence in competing hypotheses, using a Bayesian model of prediction.

Laird, Philip↗
NASA NTRS1996

Implicit Extrapolation Methods for Variable Coefficient Problems

Implicit extrapolation methods for the solution of partial differential equations are based on applying the extrapolation principle indirectly. Multigrid tau-extrapolation is a special case of this idea. In the context of multilevel finite element methods, an algorithm of this type can be used to raise the approximation order, even when the meshes are nonuniform or locally refined. Here previous results are generalized to the variable coefficient case and thus become applicable for nonlinear problems. The implicit extrapolation multigrid algorithm converges to the solution of a higher order finite element system. This is obtained without explicitly constructing higher order stiffness matrices but by applying extrapolation in a natural form within the algorithm. The algorithm requires only a small change of a basic low order multigrid method.

Jung, M.↗
NASA NTRS2007

The Extrapolation of High Altitude Solar Cell I(V) Characteristics to AM0

The high altitude aircraft method has been used at NASA GRC since the early 1960's to calibrate solar cell short circuit current, ISC, to Air Mass Zero (AMO). This method extrapolates ISC to AM0 via the Langley plot method, a logarithmic extrapolation to 0 air mass, and includes corrections for the varying Earth-Sun distance to 1.0 AU and compensating for the non-uniform ozone distribution in the atmosphere. However, other characteristics of the solar cell I(V) curve do not extrapolate in the same way. Another approach is needed to extrapolate VOC and the maximum power point (PMAX) to AM0 illumination. As part of the high altitude aircraft method, VOC and PMAX can be obtained as ISC changes during the flight. These values can then the extrapolated, sometimes interpolated, to the ISC(AM0) value. This approach should be valid as long as the shape of the solar spectra in the stratosphere does not change too much from AMO. As a feasibility check, the results are compared to AMO I(V) curves obtained using the NASA GRC X25 based multi-source simulator. This paper investigates the approach on both multi-junction solar cells and sub-cells.

Snyder, David B.↗
DOE OSTI2025

Combining physics-based and data-driven models for quantitatively accurate plasma profile prediction that extrapolates well; with application to DIII-D, AUG, and ITER tokamaks

For design, scenario planning, and control, ITER and all other envisioned tokamaks rely on a variety of statistical and physics-based models to extrapolate to unseen regimes; most notably from low plasma current to high. A 'meta-learning' methodology for combining the accuracy of data-driven models with the generalizability of physics-based models is described and tested, yielding a 5–10 percent improvement in performance beyond either alone for the task of extrapolating time-dependent plasma profile prediction from low- to high- plasma current DIII-D tokamak discharges. Meanwhile, it is shown that both machine learning models extrapolated far-distribution and state-of-the-art 'physics-based' profile predictors fare worse than merely assuming plasma profiles do not change from their initial values. Finally, a variety of other mechanisms for helping data-driven models generalize—transfer learning, adding contextual information from physics simulators, and adding data from the ASDEX Upgrade tokamak—are attempted for similar extrapolation tasks but, in the methodology used in this paper, yield no significant improvement beyond simple data-driven models. Results are summarized in figures 15 and 16.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2025

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

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

Deep learning↗
DOE OSTI2026

Machine learning for seismic low-frequency extrapolation

The cycle-skipping problem that plagues full waveform inversion (FWI) can be at least partially mitigated if low frequencies (which encode the kinematics of wave propagation in seismic data) are recorded. However, seismic sources and receivers are band-limited, so seismic data does not generally include signals down to 0 Hz. To improve our ability to solve the seismic inverse problem, one can synthesize this missing low-frequency (LF) content from the recorded high-frequency (HF) data using machine learning (ML) models. Deep learning models such as convolutional neural networks (CNNs) demonstrate impressive ability to perform low frequency extrapolation. However, such models require powerful hardware (GPU machines) and careful training. We assess the extrapolation capabilities of three different ML models that do not require GPU machines, namely, random forest, Gaussian process regression and gradient boosting, on both synthetic and real data. Experimental results on two synthetic data sets (generated from a low velocity lens embedded in a homogeneous medium, and the Marmousi model) demonstrate that FWI applied to the extrapolated data consistently improves inversion accuracy relative to FWI applied to the original data sets that do not contain low frequencies. Application of low-frequency extrapolation to real data from the Northwest Shelf of Australia demonstrates that tree-based ML models such as gradient boosting can outperform CNNs in terms of both accuracy and computational cost on non-GPU architectures.

58 GEOSCIENCES↗
DOE OSTI2026

Large-momentum effective theory’s asymptotic extrapolation vs the inverse problem

Large-momentum effective theory is a physics-guided systematic expansion to calculate light-cone parton distributions, including collinear (PDFs) and transverse-momentum-dependent ones, at any fixed momentum fraction 𝑥 within a range of [𝑥 min , 𝑥 max ]. It theoretically solves the ill-posed inverse problem that afflicts other theoretical approaches to collinear PDFs, such as short-distance factorizations. Recently, Dutrieux et al. raised practical concerns about whether current or even future lattice data will have sufficient precision in the subasymptotic correlation region to support an error-controlled extrapolation—and if not, whether it becomes an inverse problem where the relevant uncertainties cannot be properly quantified. While we agree that not all current lattice data have the desired precision to qualify for an asymptotic extrapolation, some calculations do, and more are expected in the future. We comment on the analysis and results in Dutrieux et al. and argue that a physics-based systematic extrapolation still provides the most reliable error estimates, even when the data quality is not ideal. In contrast, reframing the long-distance asymptotic extrapolation as a data-driven-only inverse problem with ad hoc mathematical conditioning could lead to unnecessarily conservative errors.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2024

Error mitigation, optimization, and extrapolation on a trapped-ion testbed

Current noisy intermediate-scale quantum (NISQ) trapped-ion devices are subject to errors which can significantly impact the accuracy of calculations if left unchecked. A form of error mitigation called zero noise extrapolation (ZNE) can decrease an algorithm’s sensitivity to these errors without increasing the number of required qubits. Here we explore different methods for integrating this error mitigation technique into the Variational Quantum Eigensolver (VQE) algorithm for calculating the ground state of the HeH + molecule at 0.8 Å in the presence of experimental noise. Using the Quantum Scientific Computing Open User Testbed (QSCOUT) trapped-ion device, we test three methods of scaling noise for extrapolation: time stretching the two-qubit gates, scaling the sideband detuning parameter, and inserting two-qubit gate identity operations into the ansatz circuit. We find that time stretching and sideband detuning scaling fail to scale the noise on our particular hardware in a way that can be extrapolated to zero noise. Scaling our noise with global gate identity insertions and extrapolating after variational optimization, we achieve error suppression of 96.8%, resulting in an energy estimate within –0.004 ± 0.04 hartree of the ground state energy. This is an improvement, but still outside the chemical accuracy threshold of 0.0016 hartree. Furthermore, our results show that the efficacy of this error mitigation technique depends on choosing the correct implementation for a given device architecture.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

Quantum error mitigation by layerwise Richardson extrapolation

A widely used method for mitigating errors in noisy quantum computers is Richardson extrapolation, a technique in which the overall effect of noise on the estimation of quantum expectation values is captured by a single parameter that, after being scaled to larger values, is eventually extrapolated to the zero-noise limit. We generalize this approach by introducing layerwise Richardson extrapolation (LRE), an error mitigation protocol in which the noise of different individual layers (or larger chunks of the circuit) is amplified and the associated expectation values are linearly combined to estimate the zero-noise limit. The coefficients of the linear combination are analytically obtained from the theory of multivariate Lagrange interpolation. LRE leverages the flexible configurational space of layerwise unitary folding, allowing for a more nuanced mitigation of errors by treating the noise level of each layer of the quantum circuit as an independent variable. Furthermore, we provide numerical simulations demonstrating scenarios where LRE achieves superior performance compared to traditional (single-variable) Richardson extrapolation.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2023

Physical implications of the extrapolation and statistical bootstrap of nucleon structure function ratios F 2 n F 2 p for mirror nuclei He 3 and H 3

A nuclear physics example of statistical bootstrap is used on the MARATHON nucleon structure function ratio data in the quark momentum fraction regions x B → 0 and x B → 1. The extrapolated F 2 ratio as quark momentum fraction x B → 1 is $\frac{F^n_2}{F^p_2}$ → 0.4 ± 0.05 and this value is compared to theoretical predictions. The extrapolated ratio when x B → 0 favors the simple model of isospin symmetry with the complete dominance of sea quarks at low momentum fraction. At high- x B , the proton quark distribution function ratio d/u is derived from the F 2 ratio and found to be d/u → 1/6. Our extrapolated values for both the $\frac{F^n_2}{F^p_2}$ ratio and the d/u parton distribution function ratio are within uncertainties of perturbative QCD values from quark counting, helicity conservation arguments, and a Dyson-Schwinger equation with a contact interaction model. In addition, it is possible to match the statistical bootstrap value to theoretical predictions by allowing two compatible models to act simultaneously in the nucleon wave function. Finally, one such example is nucleon wave functions composed of a linear combination of a quark-diquark state and a three-valence quark correlated state with coefficients that combine to give the extrapolated F 2 ratio at x B = 1.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗