Model Reduction Error Research Articles

On the Accuracy in Modeling the Statistical Distribution of Random Telegraph Noise Amplitude

The power consumption of digital circuits is proportional to the square of operation voltage and the demand for low power circuits reduces the operation voltage towards the threshold of MOSFETs. A weak voltage signal makes circuits vulnerable to noise and the optimization of circuit design requires modelling noise. Random Telegraph Noise (RTN) is the dominant noise for modern CMOS technologies and Monte Carlo modelling has been used to assess its impact on circuits. This requires statistical distributions of RTN amplitude and three different distributions were proposed by early works: Lognormal, Exponential, and Gumbel distributions. They give substantially different RTN predictions and agreement has not been reached on which distribution should be used, calling the modelling accuracy into questions. The objective of this work is to assess the accuracy of these three distributions and to explore other distributions for better accuracy. A novel criterion has been proposed for selecting distributions, which requires a monotonic reduction of modelling errors with increasing number of traps. The three existing distributions do not meet this criterion and thirteen other distributions are explored. It is found that the Generalized Extreme Value (GEV) distribution has the lowest error and meet the new criterion. Moreover, to reduce modelling errors, early works used bimodal Lognormal and Exponential distributions, which have more fitting parameters. Their errors, however, are still higher than those of the monomodal GEV distribution. GEV has a long distribution tail and predicts substantially worse RTN impact. The work highlights the uncertainty in predicting the RTN distribution tail by different statistical models.

Complex Frequency Domain Model Reduction: A Comparison of Techniques

A mathematical model of the plant to be controlled must first be identified in order to design a controller to improve the process’ performance and stability. This mathematical model usually is in the form of complex frequency domain state-space realization or continuous/ discrete time domain regression algorithm. This paper works on the complex frequency domain state-space realization. Reducing the order of the state-space model of the physical plant is usually desirable since it reduces computational time and hardware implementation for the resulting controller designed based on the model. For a frequency domain state space realization of a linear time-invariant and non-causal plant, there are numerous methods available to reduce its order. These include Balancing Model Reduction, Frequency-Weighted Balancing, Low-Frequency Approximation and Balanced Truncation with Reduced Error Bound. Besides background information and a summary, the model-reduction methodology for each of the above-said techniques is also presented. Using a two-input two-output six-state model for a large turbo-generator and a large industrial boiler with 8 states, 2 inputs and 2 outputs model, this paper provides a comparison between three of the four methodologies mentioned above in terms of the spectral norms of the model reduction error.

Modeling subfilter soot-turbulence interactions in Large Eddy Simulation: An a priori study

A new LES model for subfilter soot-turbulence interactions is developed based on an a priori analysis using large-scale DNS data of temporally evolving non premixed n-heptane jet flames at a jet Reynolds number of 15,000. In this work, soot formation is modeled in LES by solving explicit transport equations for soot moments, and the unclosed filtered soot moment source terms are closed by a presumed PDF approach. Due to the strong intermittency of soot fields, a previous modeling approach assumes the presumed PDF to be bimodal accounting for sooting and non-sooting subfilter regions but neglects any sub-structure of the soot distribution. In this work, the modeling framework is improved by a new presumed PDF model that explicitly accounts for the sub-structure of the sooting mode, which is modeled by a log-normal distribution. The previous and new models are assessed by means of their prediction of the filtered source terms and the filtered intermittency, and the log-normal distribution is found to significantly reduce modeling errors, in particular, for the coagulation source term. Introducing a log-normal distribution for the PDF of the sooting mode involves a large amount of additional model parameters, such as the width of the distribution and correlation coefficients among different soot moments, so model assumptions to reduce the number of model parameters are discussed by means of the DNS data. The conclusions are found to be robust with respect to a variation in the global Damköhler number in the DNS datasets. The final model formulation only requires solving two additional transport equations in LES compared to previous models, while significantly improved model predictions are obtained for the coagulation source term which is import for predicting the number of soot particles.

Can Short and Partial Observations Reduce Model Error and Facilitate Machine Learning Prediction?

Predicting complex nonlinear turbulent dynamical systems is an important and practical topic. However, due to the lack of a complete understanding of nature, the ubiquitous model error may greatly affect the prediction performance. Machine learning algorithms can overcome the model error, but they are often impeded by inadequate and partial observations in predicting nature. In this article, an efficient and dynamically consistent conditional sampling algorithm is developed, which incorporates the conditional path-wise temporal dependence into a two-step forward-backward data assimilation procedure to sample multiple distinct nonlinear time series conditioned on short and partial observations using an imperfect model. The resulting sampled trajectories succeed in reducing the model error and greatly enrich the training data set for machine learning forecasts. For a rich class of nonlinear and non-Gaussian systems, the conditional sampling is carried out by solving a simple stochastic differential equation, which is computationally efficient and accurate. The sampling algorithm is applied to create massive training data of multiscale compressible shallow water flows from highly nonlinear and indirect observations. The resulting machine learning prediction significantly outweighs the imperfect model forecast. The sampling algorithm also facilitates the machine learning forecast of a highly non-Gaussian climate phenomenon using extremely short observations.

Comparing Numerical Accuracy of Icosahedral A-Grid and C-Grid Schemes in Solving the Shallow-Water Model

AbstractA single software framework is introduced to evaluate numerical accuracy of the A-grid (NICAM) versus C-grid (MPAS) shallow-water model solvers on icosahedral grids. The C-grid staggering scheme excels in numerical noise control and total energy conservation, which results in exceptional stability for long time integration. Its weakness lies in the lack of model error reduction with increasing resolution in specific test cases (especially the root-mean-square error). The A-grid method conserves well potential enstrophy and shows a linear reduction of error with increasing resolution. The gridpoint noise manifests itself clearly on A-grid, but much less on C-grid. We show that the Coriolis force term on C-grid has a larger error than on A-grid. To treat the Coriolis term and kinetic energy gradient on an equal footing on C-grid, we propose combining these two quantities into a single tendency term and computing its value by a linear combination operation. This modification alone reduces numerical errors but still fails to converge the maximum error with resolution. The method of Peixoto can solve the maximum-error nonconvergence problem on C-grid but degrades the numerical stability. For the steady-state thin-layer test (0.01 m in depth), the A-grid method is less susceptible than C-grid methods, which are presumably disrupted by the Hollingsworth instability. The effect of horizontal diffusion on model accuracy and energy conservation is shown in detail. Programming experience shows that software implementation and optimization can strongly influence computational performance for models, although memory requirement and computational load of the two schemes are comparable.

Spectroscopy of binary black hole ringdown using overtones and angular modes

The black hole uniqueness and the no-hair theorems imply that the quasinormal spectrum of any astrophysical black hole is determined solely by its mass and spin. The countably infinite number of quasinormal modes of a Kerr black hole are thus related to each other and any deviations from these relations provide a strong hint for physics beyond the general theory of relativity. To test the no-hair theorem using ringdown signals, it is necessary to detect at least two quasinormal modes. In particular, one can detect the fundamental mode along with a subdominant overtone or with another angular mode, depending on the mass ratio and the spins of the progenitor binary. Also in the light of the recent discovery of GW190412, studying how the mass ratio affects the prospect of black hole spectroscopy using overtones or angular modes is pertinent, and this is the major focus of our study. First, we provide ready-to-use fits for the amplitudes and phases of both the angular modes and overtones as a function of mass ratio $q\in[0,10]$. Using these fits we estimate the minimum signal-to-noise ratio for detectability, resolvability, and measurability of subdominant modes/tones. We find that performing black-hole spectroscopy with angular modes is preferable when the binary mass ratio is larger than $q\approx 1.2$ (provided that the source is not located at a particularly disfavoured inclination angle). For nonspinning, equal-mass binary black holes, the overtones seem to be the only viable option to perform a spectroscopy test of the no-hair theorem. However this would require a large ringdown signal-to-noise ratio ($\approx 100$ for a $5\%$ accuracy test with two overtones) and the inclusion of more than one overtone to reduce modelling errors, making black-hole spectroscopy with overtones impractical in the near future.

Improved Transferability of Data-Driven Damage Models Through Sample Selection Bias Correction.

Damage models for natural hazards are used for decision making on reducing and transferring risk. The damage estimates from these models depend on many variables and their complex sometimes nonlinear relationships with the damage. In recent years, data‐driven modeling techniques have been used to capture those relationships. The available data to build such models are often limited. Therefore, in practice it is usually necessary to transfer models to a different context. In this article, we show that this implies the samples used to build the model are often not fully representative for the situation where they need to be applied on, which leads to a “sample selection bias.” In this article, we enhance data‐driven damage models by applying methods, not previously applied to damage modeling, to correct for this bias before the machine learning (ML) models are trained. We demonstrate this with case studies on flooding in Europe, and typhoon wind damage in the Philippines. Two sample selection bias correction methods from the ML literature are applied and one of these methods is also adjusted to our problem. These three methods are combined with stochastic generation of synthetic damage data. We demonstrate that for both case studies, the sample selection bias correction techniques reduce model errors, especially for the mean bias error this reduction can be larger than 30%. The novel combination with stochastic data generation seems to enhance these techniques. This shows that sample selection bias correction methods are beneficial for damage model transfer.

Uncertainty-quantified parametrically homogenized constitutive models (UQ-PHCMs) for dual-phase \u03b1/\u03b2 titanium alloys

This paper develops an uncertainty-quantified parametrically homogenized constitutive model (UQ-PHCM) for dual-phase α/β titanium alloys such as Ti6242S. Their microstructures are characterized by primary α-grains consisting of hcp crystals and transformed β-grains consisting of alternating laths of α (hcp) and β (bcc) phases. The PHCMs bridge length-scales through explicit microstructural representation in structure-scale constitutive models. The forms of equations are chosen to reflect fundamental deformation characteristics such as anisotropy, length-scale dependent flow stresses, tension-compression asymmetry, strain-rate dependency, and cyclic hardening under reversed loading conditions. Constitutive coefficients are functions of representative aggregated microstructural parameters or RAMPs that represent distributions of crystallographic orientation and morphology. The functional forms are determined by machine learning tools operating on a data-set generated by crystal plasticity FE analysis. For the dual phase alloys, an equivalent PHCM is developed from a weighted averaging rule to obtain the equivalent material response from individual PHCM responses of primary α and transformed β phases. The PHCMs are readily incorporated in FE codes like ABAQUS through user-defined material modeling windows such as UMAT. Significantly reduced number of solution variables in the PHCM simulations compared to micromechanical models, make them several orders of magnitude more efficient, but with comparable accuracy. Bayesian inference along with a Taylor-expansion based uncertainty propagation method is employed to quantify and propagate different uncertainties in PHCM such as model reduction error, data sparsity error and microstructural uncertainty. Numerical examples demonstrate the accuracy of PHCM and the relative importance of different sources of uncertainty.

The Badhwar‐O'Neill 2020 GCR Model

AbstractThe Badhwar‐O'Neill (BON) model has been used for some time to describe the galactic cosmic ray (GCR) environment encountered in deep space by astronauts and sensitive electronics. The most recent version of the model, BON2014, was calibrated to available measurements to reduce model errors for particles and energies of significance to astronaut exposure. Although subsequent studies showed the model to be reasonably accurate for such applications, modifications to the sunspot number (SSN) classification system and a large number of new high‐precision measurements suggested the need to develop an improved and more capable model. In this work, the BON2020 model is described. The new model relies on daily integral flux from the Advanced Composition Explorer Cosmic Ray Isotope Spectrometer (ACE/CRIS) to describe solar activity. For time periods not covered by ACE/CRIS, the updated international SSN database is used. Parameters in the new model are calibrated to available data, which include the new Alpha Magnetic Spectrometer (AMS‐02) and Payload for Antimatter Matter Exploration and Light‐nuclei Astrophysics (PAMELA) high‐precision measurements. It is found that the BON2020 model is a significant improvement over BON2014. Systematic bias associated with BON2014 has been removed. The average relative error of the BON2020 model compared to all available measurements is found to be <1%, and BON2020 is found to be within ±15% of a large fraction of the available measurements (26,269 of 27,646 → 95%).

Global evaluation of WBGT and SET indices for outdoor environments using thermal imaging and artificial neural networks

The health and well-being of occupants of outdoor environments are largely affected by thermal stress, and therefore a global assessment is essential. Wet-bulb globe-temperature (WBGT) is used as a heat stress indicator and standard effective temperature (SET) is used as a thermal comfort index for assessment of thermal comfort in indoor and outdoor environments. These indices are usually evaluated point-wise which could be sufficient for relatively small spaces, but not suitable for large outdoor environments. This research proposes using a system combining climate sensors readings and thermal imaging to globally evaluate WBGT and SET values for outdoor environments. The algorithm derives air temperature from surface temperature values obtained using a thermal imaging camera. The obtained results were validated using readings of available sensors. Point-wise validation showed that the proposed methodology yielded results with a maximum average error of 11.4% compared to the average of point-wise local measurement for the WBGT, and an error of 8.5% for the SET. To minimize the error, an error reduction model based on artificial neural networks has been implemented. The error was further reduced to a maximum average error of 1.76% and 1.25% for WBGT and SET respectively.

Adapting a fatigue model for shoulder flexion fatigue: Enhancing recovery rate during intermittent rest intervals

Although the rotator cuff muscles are susceptible to fatigue, shoulder fatigue studies reporting torque decline during intermittent tasks are relatively uncommon in the literature. A previous modification to the three-compartment controller (3CC) fatigue model incorporated a rest recovery multiplier (3CC-r model) to represent augmented blood flow to muscle during rest intervals (Looft et al., 2018). A rest recovery value of r = 15 was optimal for ankle, knee, and elbow joint regions, whereas r = 30 was better for hand/grip muscles. However, shoulder torque decline data was unavailable in the literature for comparison. Thus, the purpose of this study was to collect fatigue data for two different intermittent, isometric shoulder flexion fatiguing tasks and assess the 3CC-r model with r = 15 or 30 compared to the original 3CC model. Twenty healthy participants (9 M) completed two fatigue tasks: 50% maximum voluntary contraction (MVC) with 50% duty cycle (DC) and 70% MVC with 70% DC. MVCs were assessed at discrete time points (1, 3, 5, 10, and 15 min) until endurance time (MET). Mean observed percent torque decline (%TD) for the two tasks were compared to three model estimates: 3CC-r (using r = 15 and r = 30) and 3CC. Using these data, we confirmed that the addition of a rest multiplier (r = 15 somewhat better than r = 30) substantially improved predictions of shoulder fatigue using a previously validated analytical fatigue model (3CC). The relatively large reduction in model errors over the original model suggests the importance of representing augmented recovery during rest periods.

A Gaussian moment method and its augmentation via LSTM recurrent neural networks for the statistics of cavitating bubble populations

Phase-averaged dilute bubbly flow models require high-order statistical moments of the bubble population. The method of classes, which directly evolve bins of bubbles in the probability space, are accurate but computationally expensive. Moment-based methods based upon a Gaussian closure present an opportunity to accelerate this approach, particularly when the bubble size distributions are broad (polydisperse). For linear bubble dynamics a Gaussian closure is exact, but for bubbles undergoing large and nonlinear oscillations, it results in a large error from misrepresented higher-order moments. Long short-term memory recurrent neural networks, trained on Monte Carlo truth data, are proposed to improve these model predictions. The networks are used to correct the low-order moment evolution equations and improve prediction of higher-order moments based upon the low-order ones. Results show that the networks can reduce model errors to less than 1% of their unaugmented values.

A theory-guided deep-learning formulation and optimization of seismic waveform inversion

Deep-learning techniques appear to be poised to play very important roles in our processing flows for inversion and interpretation of seismic data. The most successful seismic applications of these complex pattern-identifying networks will, presumably, be those that also leverage the deterministic physical models on which we normally base our seismic interpretations. If this is true, algorithms belonging to theory-guided data science, whose aim is roughly this, will have particular applicability in our field. We have developed a theory-designed recurrent neural network (RNN) that allows single- and multidimensional scalar acoustic seismic forward-modeling problems to be set up in terms of its forward propagation. We find that training such a network and updating its weights using measured seismic data then amounts to a solution of the seismic inverse problem and is equivalent to gradient-based seismic full-waveform inversion (FWI). By refining these RNNs in terms of optimization method and learning rate, comparisons are made between standard deep-learning optimization and nonlinear conjugate gradient and limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) optimized algorithms. Our numerical analysis indicates that adaptive moment (or Adam) optimization with a learning rate set to match the magnitudes of standard FWI updates appears to produce the most stable and well-behaved waveform inversion results, which is reconfirmed by a multidimensional 2D Marmousi experiment. Future waveform RNNs, with additional degrees of freedom, may allow optimal wave propagation rules to be solved for at the same time as medium properties, reducing modeling errors.

Reducing error in water distribution network simulations with field measurements

Reduction of error in water distribution network (WDN) models leads to simulations that are more representative of actual network conditions and allows for more realistic system responses. Technological improvements have resulted in data collection becoming more prevalent in WDNs. This study quantifies the reduction in model error when considering demand uncertainty by incorporating pressure reducing valve (PRV) monitoring, operational monitoring, and supervisory control and data acquisition (SCADA) system data. Model implementation procedures were developed for each of these data types. For this study, outputs obtained by the modeling software EPANET for a WDN model built with hourly measured demands were treated as actual network observations. Pressures simulated by the network model that incorporated all three types of data had less error than pressures simulated by a base model representative of what water managers would use without access to this data. Model improvement varies both spatially and temporally.

The convolutional interpretation of registration-based plane wave steered pulse-echo local sound speed estimators

Pulse-echo reconstruction of sound speed has long been considered a difficult problem within the domain of quantitative biomedical ultrasound. However, recent results (Jaeger 2015 Ultrasound Med. Biol. 41 235–50; Jaeger and Frenz 2015 Ultrasonics 62 299–304; Jaeger et al 2015 Phys. Med. Biol. 60 4497–515) have demonstrated that pulse-echo reconstructions of sound speed are achievable by exploiting correlations in post-beamformed data from steered, plane-wave excitations in the presence of diffuse scatterers. Despite these recent advances, a coherent theoretical imaging framework for describing the approach and results is lacking in the literature. In this work, the problem of sound speed reconstruction using steered, plane-wave excitations is reformulated as a truncated convolutional problem, and the theoretical implications of this reformulation are explored. Additionally, a matrix-free algorithm is proposed that leverages the computational and storage advantages of the fast Fourier transform (FFT) while simultaneously avoiding FFT wraparound artifacts. In particular, the storage constraints of the approach are reduced down from to over full matrix reconstruction, making this approach a better candidate for large reconstructions on clinical machines. This algorithm was then tested in the open source simulation package k-Wave to assess its robustness to modeling error and resolution reduction was demonstrated under full-wave propagation conditions relative to ideal straight-ray simulations. The method was also validated in a phantom experiment.

Neural network method as means of processing experimental data on grain crop yields

In the work based on agroecological and technological testing of varieties of grain crops of domestic and foreign breeding, winter triticale in particular, conducted on the experimental field of the Smolensk State Agricultural Academy between 2015 and 2019, we present the methodology and results of processing the experimental data used for constructing the neural network model. Neural networks are applicable for solving tasks that are difficult for computers of traditional design and humans alike. Those are processing large volumes of experimental data, automation of image recognition, approximation of functions and prognosis. Neural networks include analyzing subject areas and weight coefficients of neurons, detecting conflict samples and outliers, normalizing data, determining the number of samples required for teaching a neural network and increasing the learning quality when their number is insufficient, as well as selecting the neural network type and decomposition based on the number of input neurons. We consider the technology of initial data processing and selecting the optimal neural network structure that allows to significantly reduce modeling errors in comparison with neural networks created with unprepared source data. Our accumulated experience of working with neural networks has demonstrated encouraging results, which indicates the prospects of this area, especially when describing processes with large amounts of variables. In order to verify the resulting neural network model, we have carried out a computational experiment, which showed the possibility of applying scientific results in practice.

Atmospheric Predictability of the Tropics, Middle Latitudes, and Polar Regions Explored through Global Storm-Resolving Simulations

AbstractThe predictability of the atmosphere has important implications for weather prediction, because it determines what forecast problems are potentially tractable. Even though our general understanding of error growth and predictability has been increasing, relatively little is known about the detailed structure of atmospheric predictability, such as how it varies between climate regions. The present study addresses this issue by exploring error growth and predictability in three latitude zones, using model output from a previous global storm-resolving predictability experiment by Judt published in 2018. It was determined that the tropics have longer predictability than the middle latitudes and polar regions (tropics, >20 days; middle latitudes and polar regions, a little over 2 weeks). Each latitude zone had distinct error growth characteristics, and error growth was broadly consistent with the underlying dynamics of each zone. Evidence suggests that equatorial waves play a role in the comparatively long predictability of the tropics; specifically, equatorial waves seem to be less prone to error growth than middle-latitude baroclinic disturbances. Even though the generality of the findings needs to be assessed in future studies, the overall conclusions agree with previous work in that current numerical weather prediction procedures have not reached the limits of atmospheric predictability, especially in the tropics. One way to exploit tropical predictability is to reduce model error, for example, by using global storm-resolving models instead of conventional models that parameterize convection.

An emergent constraint on future Arctic sea-ice albedo feedback

Arctic sea ice has decreased substantially over recent decades, a trend projected to continue. Shrinking ice reduces surface albedo, leading to greater surface solar absorption, thus amplifying warming and driving further melt. This sea-ice albedo feedback (SIAF) is a key driver of Arctic climate change and an important uncertainty source in climate model projections. Using an ensemble of models, we demonstrate an emergent relationship between future SIAF and an observable version of SIAF in the current climate’s seasonal cycle. This relationship is robust in constraining SIAF over the coming decades (Pearson’s r = 0.76), and then it degrades. The degradation occurs because some models begin producing ice-free conditions, signalling a transition to a new ice regime. The relationship is strengthened when models with unrealistically thin historical ice are excluded. Because of this tight relationship, reducing model errors in the current climate’s seasonal SIAF and ice thickness can narrow SIAF spread under climate change. The reflectivity of the Arctic Ocean decreases as sea ice decreases, creating a feedback of more heat absorption, warming and further melt. An ensemble of models is used to gain understanding of this in the current climate to constrain the intermodel spread in predictions of sea-ice albedo changes.

Using a Nonlinear Forcing Singular Vector Approach to Reduce Model Error Effects in ENSO Forecasting

Abstract Nonlinear forcing singular vector (NFSV)-based assimilation is adopted to determine the model tendency errors that represent the combined effect of different kinds of model errors; then, an NFSV-tendency error forecast model is formulated. This error forecast model is coupled with an intermediate complex model (ICM) and makes the ICM output closer to the observations; finally, an NFSV-ICM forecast model for ENSO is constructed. The competing aspect of the NFSV-ICM is to consider not only model errors but also the interaction between model errors and initial errors because of the mathematical nature of the NFSV-tendency errors. Based on the prediction experiments for tropical SSTAs during either the training period (1960–96; i.e., when the NFSV-ICM is formulated) or the cross-validation period (1997–2016), the NFSV-ICM is determined to have a much higher forecast skill in predicting ENSO that, specifically, extends the skillful predictions of ENSO from a lead time of 6 months in the original ICM to a lead time of 12 months. The higher skill of the NFSV-ICM is especially reflected in the predictions of SSTAs in the central and western Pacific. For the well-known spring predictability barrier (SPB) phenomenon that greatly limits ENSO forecasting skill, the NFSV-ICM also shows great abilities in suppressing its negative effect on ENSO predictions. Although the NFSV-ICM is presently only involved with the NFSV-related assimilation of SSTs, it has shown its usefulness in predicting ENSO. It is clear that the NFSV-based assimilation approach is effective in dealing with the effect of model errors on ENSO forecasts.

A Kriging-Based Approach to Autonomous Experimentation with Applications to X-Ray Scattering

Modern scientific instruments are acquiring data at ever-increasing rates, leading to an exponential increase in the size of data sets. Taking full advantage of these acquisition rates will require corresponding advancements in the speed and efficiency of data analytics and experimental control. A significant step forward would come from automatic decision-making methods that enable scientific instruments to autonomously explore scientific problems—that is, to intelligently explore parameter spaces without human intervention, selecting high-value measurements to perform based on the continually growing experimental data set. Here, we develop such an autonomous decision-making algorithm that is physics-agnostic, generalizable, and operates in an abstract multi-dimensional parameter space. Our approach relies on constructing a surrogate model that fits and interpolates the available experimental data, and is continuously refined as more data is gathered. The distribution and correlation of the data is used to generate a corresponding uncertainty across the surrogate model. By suggesting follow-up measurements in regions of greatest uncertainty, the algorithm maximally increases knowledge with each added measurement. This procedure is applied repeatedly, with the algorithm iteratively reducing model error and thus efficiently sampling the parameter space with each new measurement that it requests. We validate the method using synthetic data, demonstrating that it converges to faithful replica of test functions more rapidly than competing methods, and demonstrate the viability of the approach in an experimental context by using it to direct autonomous small-angle (SAXS) and grazing-incidence small-angle (GISAXS) x-ray scattering experiments.