Uncorrelated Input Research Articles

Enhancing Immunotherapy Response Prediction in Metastatic Lung Adenocarcinoma: Leveraging Shallow and Deep Learning with CT-Based Radiomics across Single and Multiple Tumor Sites.

This study aimed to evaluate the potential of pre-treatment CT-based radiomics features (RFs) derived from single and multiple tumor sites, and state-of-the-art machine-learning survival algorithms, in predicting progression-free survival (PFS) for patients with metastatic lung adenocarcinoma (MLUAD) receiving first-line treatment including immune checkpoint inhibitors (CPIs). To do so, all adults with newly diagnosed MLUAD, pre-treatment contrast-enhanced CT scan, and performance status ≤ 2 who were treated at our cancer center with first-line CPI between November 2016 and November 2022 were included. RFs were extracted from all measurable lesions with a volume ≥ 1 cm3 on the CT scan. To capture intra- and inter-tumor heterogeneity, RFs from the largest tumor of each patient, as well as lowest, highest, and average RF values over all lesions per patient were collected. Intra-patient inter-tumor heterogeneity metrics were calculated to measure the similarity between each patient lesions. After filtering predictors with univariable Cox p < 0.100 and analyzing their correlations, five survival machine-learning algorithms (stepwise Cox regression [SCR], LASSO Cox regression, random survival forests, gradient boosted machine [GBM], and deep learning [Deepsurv]) were trained in 100-times repeated 5-fold cross-validation (rCV) to predict PFS on three inputs: (i) clinicopathological variables, (ii) all radiomics-based and clinicopathological (full input), and (iii) uncorrelated radiomics-based and clinicopathological variables (uncorrelated input). The Models' performances were evaluated using the concordance index (c-index). Overall, 140 patients were included (median age: 62.5 years, 36.4% women). In rCV, the highest c-index was reached with Deepsurv (c-index = 0.631, 95%CI = 0.625-0.647), followed by GBM (c-index = 0.603, 95%CI = 0.557-0.646), significantly outperforming standard SCR whatever its input (c-index range: 0.560-0.570, all p < 0.0001). Thus, single- and multi-site pre-treatment radiomics data provide valuable prognostic information for predicting PFS in MLUAD patients undergoing first-line CPI treatment when analyzed with advanced machine-learning survival algorithms.

Study of the kurtoses transmission of linear structures under stationary non-Gaussian random loadings

Recent research has discovered that the kurtoses of non-Gaussian stress loadings can significantly impact the fatigue life of in-service structures. To predict potential damage, a fatigue life estimation method based on Gaussian damage and a kurtosis-related corrective coefficient has been developed. However, the transmission of kurtoses from excitations to responses has not been well studied, particularly in multi-input cases. This hinders efforts to accelerate fatigue damage by controlling input kurtoses. Therefore, this paper aims to establish a kurtoses transmission model for a linear structure under multiple uncorrelated stationary non-Gaussian excitations generated by zero-memory non-linear method. Firstly, a single-input kurtosis transmission equation is proposed to establish the relationship between excitation kurtosis and response kurtosis. Using this equation, the response kurtosis is formulated with input kurtosis and the system parameters of a linear structure. Secondly, the response kurtosis of multi-input cases is deduced, and the study shows that the response kurtosis is the weighted sum of the kurtosis induced by each excitation. Finally, numerical validation and experimental studies are conducted to verify the accuracy of both the single-input and multi-input kurtoses transmission models. The validations demonstrate that the proposed models can predict the response kurtoses with satisfactory precision, regardless of the input spectral shape and the number of uncorrelated input forces.

A Low-Cost and Fault-Tolerant Stochastic Architecture for the Bernsen Algorithm Using Bitstream Correlation

Many algorithms for image processing do not require particularly high precision, but they rely on complicated arithmetic operations for every pixel in an image. The Bernsen algorithm is a typical local thresholding algorithm for solving the problem of uneven lighting. However, this algorithm requires a significant computing overhead and is extremely sensitive to noise. In this work, two stochastic computing architectures are proposed for implementing the Bernsen algorithm by using, respectively, uncorrelated and correlated input bitstreams. Experimental results show that both designs, especially the one using correlated bitstreams, present high fault tolerance of soft errors and low hardware cost in comparison with its conventional binary implementation. However, SC logic with uncorrelated inputs is not always superior to its corresponding binary circuit in energy consumption, especially the circuit that needs long input bitstreams. That means that a reasonable use of correlation can further optimize the SC circuit design.

Alteration assemblage characterization using machine learning applied to high-resolution drill-core images, hyperspectral data and geochemistry

Integration of multiple data types is beneficial for prediction of geological characteristics. From the perspective that geochemistry characterizes the composition of a rock mass, hyperspectral data characterizes alteration mineralogy and image feature extraction characterizes texture, most geological classifications would be well-informed by the combination of these three features. The process of meaningfully integrating distinctly sourced datasets and producing scale-relevant predictions for geological classifications involves several steps. We demonstrate a workflow to comprehensively structure and integrate these three feature families, refine training data, predict alteration classes and mitigate noise derived from scale mismatch in output predictions. The dataset, compiled from the Josemaria porphyry copper–gold deposit in Argentina, is comprised of more than 14 000 intervals of approximately 2 m, taken from 36 drillholes, where geochemistry was merged with hyperspectral mineralogy represented as tabular pixel abundances, and textural metrics extracted from core imagery, structured into the geochemical interval. Feature engineering and principal component analysis provided insights into the behaviour of the ore system during intermediate steps, as well as providing uncorrelated feature inputs for a random forest predictor. Training data were refined by producing an initial prediction, thresholding the predictions to &gt;70% dominant class probability and using those (high-probability) samples to produce a final model encoding better constrained separation between alteration assemblages. Prediction using the final model returned an accuracy of 82.5%, as a function of model discrepancy combined with logging ambiguity and a scale mismatch between generalized logged intervals and much more granular (2 m) feature inputs. Noise reduction and generalization to the desired resolution of output was achieved by applying the multiscale multivariate continuous wavelet transform tessellation method to class membership probabilities. Ultimately, a large database of logged drill core was homogenized using empirical methodologies. The described workflow is adaptable to distinct scenarios with some modification and is apt for integrating multiple input feature types and using them to systematically define geological classifications in drillhole data.

Simulation of induced flyrock due to open-pit blasting using the PCA-CART hybrid modeling

The primary objective of open-pit blasting is to achieve optimal rock fragmentation while minimizing undesirable side effects such as flyrock, backbreak, and vibrations. Accurate prediction of these blasting side effects plays a pivotal role in preventing accidents and enhancing overall mine safety and productivity. This paper introduces a simulation model that integrates Classification and Regression Tree (CART) analysis and Principal Component Analysis (PCA) to simulate and predict flyrock occurrences during mine blasting operations. The research focuses on the Sangan iron ore mine as a case study, collecting a dataset of 96 observations from various mine locations during blasting operations. The dataset includes controllable parameters such as burden (B), spacing (S), hole length (L), sub-drilling (U), specific drilling (SD), stemming (T), specific charge (P), and average charge per blast hole (Q). PCA is employed to assess correlations among these input variables. Notably, strong correlations are identified between variables S and B, T and L, and SD and P, leading to the selection of S, L, and P as representative variables. Consequently, S, L, U, P, and Q are chosen as uncorrelated inputs for the CART-based flyrock modeling. Of the 96 datasets, 81 are employed to train the CART model, while the remaining datasets are reserved for testing its performance. The determination coefficients achieved during the training and testing phases are 0.924 and 0.945, respectively, indicating a high level of accuracy in the CART modeling. Additionally, performance indices, including performance index (PI), variance accounted for (VAF), and normalized root mean square error (NRMSE), are calculated, yielding values of 1.423, 92.351 %, and 0.095, respectively. Based on the CART modeling, 21 key guidelines for designing blasting patterns are proposed. Furthermore, sensitivity analysis conducted to highlight the varying importance of input variables in flyrock prediction, wherein variables P and S identified as having the highest and lowest impact, respectively. Finally, the study's findings are applied to recommend and implement a modified blasting pattern in the Sangan iron mine, resulting in a substantial reduction in both vertical and horizontal flyrock incidents.

Uncorrelated bilateral cortical input becomes timed across hippocampal subfields for long waves whereas gamma waves are largely ipsilateral.

The role of interhemispheric connections along successive segments of cortico-hippocampal circuits is poorly understood. We aimed to obtain a global picture of spontaneous transfer of activity during non-theta states across several nodes of the bilateral circuit in anesthetized rats. Spatial discrimination techniques applied to bilateral laminar field potentials (FP) across the CA1/Dentate Gyrus provided simultaneous left and right readouts in five FP generators that reflect activity in specific hippocampal afferents and associative pathways. We used a battery of correlation and coherence analyses to extract complementary aspects at different time scales and frequency bands. FP generators exhibited varying bilateral correlation that was high in CA1 and low in the Dentate Gyrus. The submillisecond delays indicate coordination but not support for synaptic dependence of one side on another. The time and frequency characteristics of bilateral coupling were specific to each generator. The Schaffer generator was strongly bilaterally coherent for both sharp waves and gamma waves, although the latter maintained poor amplitude co-variation. The lacunosum-moleculare generator was composed of up to three spatially overlapping activities, and globally maintained high bilateral coherence for long but not short (gamma) waves. These two CA1 generators showed no ipsilateral relationship in any frequency band. In the Dentate Gyrus, strong bilateral coherence was observed only for input from the medial entorhinal areas, while those from the lateral entorhinal areas were largely asymmetric, for both alpha and gamma waves. Granger causality testing showed strong bidirectional relationships between all homonymous bilateral generators except the lateral entorhinal input and a local generator in the Dentate Gyrus. It also revealed few significant relationships between ipsilateral generators, most notably the anticipation of lateral entorhinal cortex toward all others. Thus, with the notable exception of the lateral entorhinal areas, there is a marked interhemispheric coherence primarily for slow envelopes of activity, but not for pulse-like gamma waves, except in the Schafer segment. The results are consistent with essentially different streams of activity entering from and returning to the cortex on each side, with slow waves reflecting times of increased activity exchange between hemispheres and fast waves generally reflecting ipsilateral processing.

Flight-Test System Identification Techniques and Applications for Small, Low-Cost, Fixed-Wing Aircraft

This paper provides an overview of flight-test system identification methods applied in the Virginia Tech Nonlinear Systems Laboratory that focus on modeling small, inexpensive, fixed-wing aircraft controlled by a ground-based pilot. The general aircraft system identification approach is outlined with details provided on the flight-test facilities, experiment design methods, instrumentation systems, flight-test operations, data processing techniques, and model identification methods enabling small aircraft flight dynamics model development. Specific small aircraft system identification challenges are overcome, including low-cost control surface servo-actuators and instrumentation systems, as well as a greater sensitivity to atmospheric disturbances and limited piloting cues. Four recent system identification research advancements using the general system identification process are featured, including application of uncorrelated pilot inputs for remotely piloted aircraft, aero-propulsive model development for propeller aircraft, spin aerodynamic model development, and nonlinear dynamic modeling without mass properties information. Although this paper provides a summary of several research efforts, the core system identification approach is presented with sufficient detail to allow the methods to be readily adapted to other research efforts leveraging small, low-cost aircraft.

Medium-long-term prediction of water level based on an improved spatio-temporal attention mechanism for long short-term memory networks

River water level usually given by nonlinear and nonstationary time series and affected by numerous complex spatial and temporal factors. But not all input factors are positively correlated with flood forecasting, and uncorrelated inputs tend to bring a lot of noise, resulting in lower prediction accuracy. Attention mechanism is an effective solution to feature filter. This study presents a spatio-temporal attention LSTM (STA-LSTM). Meanwhile, attention visualization makes models interpretation providing the basis. In this work, the attention-weight matrix is generated independently by the spatial attention module and the temporal attention module to focus on the more valuable feature factors in time and space, respectively. Spatio-temporal attention is a combination of spatial attention and temporal attention to assign scores to input sequences. Forecasting the water level of Hanchuan in the next 6 h based on the observations in the past 6 h in the middle and lower reaches of the Hanjiang River, China. The experimental results indicate that the mean absolute error (MAE) of STA-LSTM is 0.74835, root mean square error (RMSE) is 0.90197, mean absolute percent error (MAPE) is 3.29924, and coefficient of determination (R2) is 0.94138. This study improves the accuracy of the water level prediction method, which makes it easier for decision-makers to plan evacuations in advance.

An Encrypted On-Chip Power Supply With Random Parallel Power Injection and Charge Recycling Against Power/EM Side-Channel Attacks

As hardware security becomes a crucial challenge for modern electronic devices to protect information privacy, this article presents an encrypted on-chip power supply to combat power and electromagnetic (EM) side-channel attacks (SCAs) for crypto cores. By splitting power and security into separate paths, random parallel power injection and charge recycling are realized to encrypt supply power activities for high SCA immunity while largely minimizing power and performance overheads. Despite of crypto core power variations, the proposed power supply can maintain uncorrelated input profile by adaptively modulating parallel noisy power injection. Moreover, its EM leakage is highly attenuated by spreading the spectrum energy to a wide frequency range and increasing noise floor. To achieve such, a recycled masking power stage with an encryption interface is designed to randomly inject power noise and recycle system charge with random <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">on</small> -time modulation while still retaining nominal power delivery. A silicon IC prototype of this design is fabricated using a 65 nm CMOS process with an active die area of 0.19 mm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . Measured input power profiles are fully encrypted to improve power SCA immunity. Operating at a nominal switching frequency of 10 MHz, random parallel power encryption reduces peak EM interference noise from 64.57 dBμV to 43.12 dBμV to meet EN55032 Class B standards and verify EM spectrum profile encryption. In response to a step-up load change from zero to full load current of 200 mA, the power supply achieves 1% settling time of 0.78 μs, with measured voltage droop of 54 mV with no other performance overhead. It achieves a peak efficiency of 90.5%, only suffering the maximum power overhead of 4.9%.

ANOVA Assisted Variable Selection in High-dimensional Multicategory Response Data

Multinomial logistic regression is preferred in the classification of multicategory response data for its ease of interpretation and the ability to identify the associated input variables for each category. However, identifying important input variables in high-dimensional data poses several challenges as the majority of variables are unnecessary in discriminating the categories. Frequently used techniques in identifying important input variables in high-dimensional data include regularisation techniques such as Least Absolute Selection Shrinkage Operator (LASSO) and sure independent screening (SIS) or combinations of both. In this paper, we propose to use ANOVA, to assist the SIS in variable screening for high-dimensional data when the response variable is multicategorical. The new approach is straightforward and computationally effective. Simulated data without and with correlation are generated for numerical studies to illustrate the methodology, and the results of applying the methods on real data are presented. In conclusion, ANOVA performance is comparable with SIS in variable selection for uncorrelated input variables and performs better when used in combination with both ANOVA and SIS for correlated input variables.

Prediction of Compressive Strength of Concrete Using the Spearman and PCA-Based BP Neural Network

Using real-time production data of concrete to predict its 28-day compressive strength has great significance for improving the engineering structure quality and overcoming the shortage of the traditional tests long period of concrete compressive strength. The current research has the shortcomings such as insufficient prediction accuracy, inadequate matching between data characteristics and model characteristics, and redundant input parameter information. This paper proposes a BP neural network prediction model optimized by Spearman and PCA. The model first uses Spearman method to reduce the number of the input variables by eliminating variables that have a low correlation with the compressive strength and then uses PCA to eliminate the correlation between material-related variables. Following this, the new uncorrelated input variables optimized by Spearman and PCA are input to the BP neural network-based model to predict the compressive strength of concrete. The results showed that it yielded the mean absolute percentage error (MAPE) of 2.78% and the root mean-squared error (RMSE) of 1.66 MPa, far lower than the error of 4.82% and 2.92 MPa obtained by the nonoptimized BP neural network, respectively. The proposed model fully exploits real-time monitoring data from the concrete mixing station, and its results are close to those of traditional tests. It has great practical significance to guide the concrete production and construction, shorten the production cycle, and reduce the project cost.

Input correlations impede suppression of chaos and learning in balanced firing-rate networks.

Neural circuits exhibit complex activity patterns, both spontaneously and evoked by external stimuli. Information encoding and learning in neural circuits depend on how well time-varying stimuli can control spontaneous network activity. We show that in firing-rate networks in the balanced state, external control of recurrent dynamics, i.e., the suppression of internally-generated chaotic variability, strongly depends on correlations in the input. A distinctive feature of balanced networks is that, because common external input is dynamically canceled by recurrent feedback, it is far more difficult to suppress chaos with common input into each neuron than through independent input. To study this phenomenon, we develop a non-stationary dynamic mean-field theory for driven networks. The theory explains how the activity statistics and the largest Lyapunov exponent depend on the frequency and amplitude of the input, recurrent coupling strength, and network size, for both common and independent input. We further show that uncorrelated inputs facilitate learning in balanced networks.

A self-consistent analytical theory for rotator networks under stochastic forcing: Effects of intrinsic noise and common input.

Despite the incredible complexity of our brains' neural networks, theoretical descriptions of neural dynamics have led to profound insights into possible network states and dynamics. It remains challenging to develop theories that apply to spiking networks and thus allow one to characterize the dynamic properties of biologically more realistic networks. Here, we build on recent work by van Meegen and Lindner who have shown that "rotator networks," while considerably simpler than real spiking networks and, therefore, more amenable to mathematical analysis, still allow one to capture dynamical properties of networks of spiking neurons. This framework can be easily extended to the case where individual units receive uncorrelated stochastic input, which can be interpreted as intrinsic noise. However, the assumptions of the theory do not apply anymore when the input received by the single rotators is strongly correlated among units. As we show, in this case, the network fluctuations become significantly non-Gaussian, which calls for reworking of the theory. Using a cumulant expansion, we develop a self-consistent analytical theory that accounts for the observed non-Gaussian statistics. Our theory provides a starting point for further studies of more general network setups and information transmission properties of these networks.

Biophysical parameter estimation of crops from polarimetric synthetic aperture radar imagery with data-driven polynomial chaos expansion and global sensitivity analysis

• Polynomial Chaos Expansion regression is suitable for estimation of crop biophysical parameters. • Correlated input vectors prevent reliable Global Sensitivity Analysis with Sobol’s method. • Relationship between polynomial coefficients and Sobol sensitivity indices is an advantage. • Interactions between features affect regression of biophysical parameters in different level. Data-driven machine learning regression methods are easy to implement and applicable to a wide-range of data in biophysical parameter estimation and have become a common approach in the remote sensing field. Among the regression methods, polynomial chaos expansion (PCE) is one of the reliable and interesting ones due to its tight relationship with uncertainty quantification. One of the advantages of PCE is that global sensitivity analysis (GSA) with Sobol’s method can be analytically computed from polynomial coefficients if the input space is statistically independent. However, most of the phenomena include dependent features either statistically or physically. Though the physical independence is provided between inputs, they must be statistically uncorrelated. Therefore, an independent and uncorrelated input space must be created before the regression analysis. In this paper, we performed PCE-based regression analysis for the estimation of biophysical parameters of crops. The study was conducted in the experimental fields of field pea, barley, canola, and oat of the AgriSAR2009 campaign. The input parameters of the regression model were formed by creating polarimetric features derived from RADARSAT-2 imagery. The estimated biophysical parameters were based on the discrete in situ measurements of leaf area index (LAI) and normalized difference vegetation index (NDVI), scattered semi-randomly in each crop field. We implemented neighbourhood component analysis (NCA) to create an independent and uncorrelated input space by eliminating correlations. Finally, we investigated the importance of features, which drive the PCE-based regression models applying GSA with Sobol’s method. Besides the individual effects of each feature, their interactions were found significant.

Real‐Time Correlation Detection via Online Learning of a Spiking Neural Network with a Conductive‐Bridge Neuron

AbstractThe neuronal density of complementary metal‐oxide‐semiconductor field‐effect transistor‐based neurons is limited because of the use of capacitors. Therefore, a novel neuron is fabricated using a conductive‐bridge‐neuron device, current‐mirror‐type sense amplifier, latch, micro‐controller‐unit, and digital‐analog‐converters. This neuron exhibits a typical integrate‐and‐fire function; in particular, the generation frequency of the fire spikes at the neuron exponentially increases with the input‐voltage‐spike amplitude. Using the proposed designed neuron in combination with an input spike generation and spike‐timing‐dependent‐plasticity algorithm, a real‐time correlation detection based on online learning is realized. With the increase in the number of learning iterations, the weight of synapses for 100 correlated input neurons gradually increase, whereas that for 900 uncorrelated input neurons steadily reduce. In addition, after 700 learning iterations, the output neuron is almost synchronized with the 100 correlated input neurons, thereby achieving correlation detection for cognitive functions in neuromorphic architectures and demonstrating the possibility of development of a neuromorphic chip based on the conductive‐bridge neurons and synapses.

Uncorrelated stochastic bitstream generation and arithmetic computations using Cu:ZnO memristors

One popular approach under approximate computing is stochastic computing, wherein values are encoded in bitstreams to perform arithmetic operations in a low-power and computationally inexpensive manner. The stochastic computing paradigm, despite many advantages, is hindered by the need to generate uncorrelated input bitstream, which results in additional hardware overhead. In this work, Pt/Cu:ZnO/Nb:STO memristive devices were experimentally found to exhibit random switching behavior. The switching time was modeled as a Poisson distribution and the obtained results were utilized to develop a Stochastic Voltage Threshold Adaptive Memristor (SVTAM) model. Based on this circuit model, an innovative method to perform addition, multiplication and sum of product operations has been proposed by employing the memristor crossbar architecture. This implementation is advantageous as the row inputs can be reused across different columns of the crossbar to implement a large number of functions simultaneously in a compact arrangement. The functionality was verified using just 50 input pulses, and the error is just 1% when the number of pulses was increased to 200. In addition, the output is an uncorrelated bitstream, making it compatible with the existing stochastic computing circuits. This work presents a novel solution towards advancement in the field of approximate computing.

Global sensitivity analysis in epidemiological modeling

Operations researchers worldwide rely extensively on quantitative simulations to model alternative aspects of the COVID-19 pandemic. Proper uncertainty quantification and sensitivity analysis are fundamental to enrich the modeling process and communicate correctly informed insights to decision-makers. We develop a methodology to obtain insights on key uncertainty drivers, trend analysis and interaction quantification through an innovative combination of probabilistic sensitivity techniques and machine learning tools. We illustrate the approach by applying it to a representative of the family of susceptible-infectious-recovered (SIR) models recently used in the context of the COVID-19 pandemic. We focus on data of the early pandemic progression in Italy and the United States (the U.S.). We perform the analysis for both cases of correlated and uncorrelated inputs. Results show that quarantine rate and intervention time are the key uncertainty drivers, have opposite effects on the number of total infected individuals and are involved in the most relevant interactions.

Machine learning classification algorithms and anomaly detection in conventional meters and Tunisian electricity consumption large datasets

Although fraud in electricity consumption is easier to detect when consumption is recorded hourly by smart meters, in most developing countries, where the propensity for fraud is higher, conventional meters are not yet affordable. Fraud detection is easier with time series data-logging due to the periodicity and variability of consumption that reveals deviations from a regular consumption pattern. In contrast, fraud detection with conventional meters remains a significant challenge because anomalies in consumption are well hidden within the normal consumption of other consumers. In this paper, large datasets regarding consumers and invoice data from Tunisia are combined and investigated with several Machine Learning (ML) classification algorithms, to detect irregularities in electricity consumption. By performing extensive feature engineering, including multivariate Gaussian distribution, the efficiency of ensemble classifiers such as Light Gradient Boosting (LGB) outperforms other algorithms and achieves realistic performance from challenging, unbalanced and uncorrelated input datasets.

Detecting Gaussian entanglement via local quantities

Entanglement detection and quantification usually require the knowledge of correlations and thus is a global issue. For both theoretical and practical reasons, it is desirable to characterize entanglement via local quantities. Of course, this can only be achieved in special setups with some underlying structure encoding the correlations. For the often encountered bilinear interactions such as beamsplitter experiments, we propose an operational and friendly entanglement criterion for the two-mode Gaussian output states. Specifically, we show that in order to detect and evaluate the entanglement of the output states of a bilinear interaction involving uncorrelated Gaussian input states, we do not need to know the joint covariance matrix of the whole two-mode output states. Instead, we only need the respective covariances of the reduced states of the two input modes and that of the two output modes, which are all local quantities pertaining to a single mode and can be easily measured locally without explicitly considering the correlations between the modes. This is achieved by exploiting the input-output characteristics of bilinear interactions. The result recasts the celebrated entanglement criteria of Simon (Phys. Rev. Lett., 84 (2000) 2726) and Duan et al. (Phys. Rev. Lett., 84 (2000) 2722) into a more convenient and simple form.

Non-intrusive reduced-order modeling for uncertainty quantification of space–time-dependent parameterized problems

We propose a non-intrusive reduced-order modeling method for space–time-dependent parameterized problems in the context of uncertainty quantification. In the offline stage, proper orthogonal decomposition (POD) is used to extract the spatial modes based on a set of high-fidelity time-parameter-dependent snapshots and then to extract the temporal modes of the projection coefficients of the spatial modes. Finally, parameter-dependent combination coefficients are approximated using polynomial chaos expansions (PCEs). Cubic spline interpolation is used to evaluate the temporal modes at any other given time. In the online stage, a fast evaluation is provided at any given time and parameter by simply estimating the values of the polynomials and temporal modes. To validate the numerical performance of the proposed method, three time-dependent parameterized problems are tested: a one-dimensional (1-D) forced Burger’s equation with a random force term, a 1-D diffusion–reaction equation with a random field force term, and a two-dimensional incompressible fluid flow over a cylinder with a random inflow boundary condition. The results indicate that the proposed method is able to approximate the full-order model very inexpensively with a reasonable loss of accuracy for problems with uncorrelated or correlated input parameters. Furthermore, the proposed method is effective in estimating low-order moments, indicating that it has great potential for use in uncertainty quantification analysis of space–time-dependent problems.