Hybrid Monte Carlo Research Articles

Probabilistic geophysical inversion of complex resistivity measurements using the Hamiltonian Monte Carlo method

Summary In this work, we introduce the probabilistic inversion of tomographic complex resistivity (CR) measurements using the Hamiltonian Monte Carlo (HMC) method. The posterior model distribution on which our approach operates accounts for the underlying complex-valued nature of the CR imaging problem accurately by including the individual errors of the measured impedance magnitude and phase, allowing for the application of independent regularization on the inferred subsurface conductivity magnitude and phase, and incorporating the effects of cross-sensitivities. As the tomographic CR inverse problem is non-linear, of high dimension and features strong correlations between model parameters, efficiently sampling from the posterior model distribution is challenging. To meet this challenge we use HMC, a Markov-chain Monte Carlo method that incorporates gradient information to achieve efficient model updates. To maximize the benefit of a given number of forward calculations, we use the No-U-Turn sampler (NUTS) as a variant of HMC. We demonstrate the probabilistic inversion approach on a synthetic CR tomography measurement. The NUTS succeeds in creating a sample of the posterior model distribution that provides us with the ability to analyze correlations between model parameters and to calculate statistical estimators of interest, such as the mean model and the covariance matrix. Our results provide a strong basis for the characterization of the posterior model distribution and uncertainty quantification in the context of the tomographic CR inverse problem.

A vaccination-based COVID-19 model: Analysis and prediction using Hamiltonian Monte Carlo

Compartmental models have emerged as robust computational frameworks and have yielded remarkable success in the fight against COVID-19. This study proposes a vaccination-based compartmental model for COVID-19 transmission dynamics. The model reflects the specific stages of COVID-19 infection and integrates a vaccination strategy, allowing for a comprehensive analysis of how vaccination rates influence the disease spread. We fit this model to daily confirmed COVID-19 cases in Tennessee, United States of America (USA), from June 4 to November 26, 2021, in a Bayesian inference approach using the Hamiltonian Monte Carlo (HMC) algorithm. First, excluding vaccination dynamics from the model, we estimated key epidemiological parameters like infection, recovery, and disease-induced death rates. This analysis yielded a basic reproduction number (R0) of 1.5. Second, we incorporated vaccination dynamics and estimated the vaccination rate for three vaccines: 0.0051 per day for both Pfizer and Moderna and 0.0059 per day for Janssen. The fitted curves show reductions in the epidemic peak for all three vaccines. Pfizer and Moderna vaccines bring the peak down from 8,029 infected cases to 5,616 infected cases, while the Janssen vaccine reduces it, to 6,493 infected cases. Simulations of the model by varying the vaccination rate and vaccine efficacy were performed. A highly effective vaccine (95% efficacy) with a daily vaccination rate of 0.006 halved COVID-19 infections, reducing cases from 8,029 to around 4,000. The results also show that the model's prediction accuracy for new observations improves with the number of observed data used to train the model.

First-principles quantum Monte Carlo study of charge-carrier mobility in organic molecular semiconductors

We present a first-principles numerical study of charge transport in a realistic two-dimensional tight-binding model of organic molecular semiconductors. We use the hybrid Monte Carlo (HMC) algorithm to simulate the full quantum dynamics of phonons and either single or multiple charge carriers without any tunable parameters. We introduce a number of algorithmic improvements, including efficient Metropolis updates for phonon fields based on analytical insights, which lead to negligible autocorrelation times and allow sub-per-mille precisions to be reached at a low computational cost of O(1) CPU hours. Our simulations produce charge-mobility estimates that are in good agreement with experiments and that also justify the phenomenological transient localization approach. Published by the American Physical Society 2024

Delayed rejection Hamiltonian Monte Carlo for sampling multiscale distributions

The efficiency of Hamiltonian Monte Carlo (HMC) can suffer when sampling a distribution with a wide range of length scales, because the small step sizes needed for stability in high-curvature regions are inefficient elsewhere. To address this we present a delayed rejection (DR) variant: if an initial HMC trajectory is rejected, we make one or more subsequent proposals each using a step size geometrically smaller than the last. To reduce the cost of DR approaches, we extend the standard delayed rejection to a probabilistic framework wherein we do not make multiple proposals at every rejection, but allow the probability of a retry to depend on the probability of accepting the previous proposal. We test the scheme in several sampling tasks, including statistical applications and multiscale model distributions such as Neal’s funnel. Delayed rejection enables sampling multiscale distributions for which standard approaches such as HMC fail to explore the tails, and improves performance five-fold over optimally-tuned HMC as measured by effective sample size per gradient evaluation. Even for simpler distributions, delayed rejection provides increased robustness to step size misspecification.

Reliability analysis of a grid-connected hybrid renewable energy system using hybrid Monte-Carlo and Newton Raphson methods

The integration of renewable energy sources into the power grid is essential for sustainable development, yet it presents significant dependability challenges, particularly in terms of reliability, stability, and robustness due to the inherent variability of these sources. This research introduces a novel hybrid methodology that combines Monte Carlo simulation with Newton-Raphson power flow analysis to enhance the reliability assessment of grid-connected hybrid renewable energy systems. This innovative approach uniquely addresses the limitations of existing methodologies by merging the probabilistic handling of uncertainties with precise deterministic power flow analysis. Our hybrid method significantly reduces the Loss of Load Expectation (LOLE) to 5 h per year and the Loss of Load Energy Expectation (LOEE) to 200 MWh per year, outperforming traditional methods which typically report LOLEs of 2020 h/year and LOEEs of 10001000 MWh/year. Additionally, the hybrid method achieves a reduction in power losses to 1.2%, showcasing its superior efficiency compared to the 2.5% losses seen with standalone Monte Carlo methods. Real-time validation using the IEEE-30 bus model further confirms the practical applicability and robustness of our approach, making it a pivotal tool for enhancing grid stability and optimizing renewable energy integration. This research not only advances the methodology for reliability assessment but also sets a new standard for balancing accuracy and computational efficiency in energy system management. The implications of this work are far-reaching, offering significant contributions to both grid reliability and the sustainable management of renewable energy resources.

Bayesian Modeling for Nonstationary Spatial Point Process via Spatial Deformations.

Many techniques have been proposed to model space-varying observation processes with a nonstationary spatial covariance structure and/or anisotropy, usually on a geostatistical framework. Nevertheless, there is an increasing interest in point process applications, and methodologies that take nonstationarity into account are welcomed. In this sense, this work proposes an extension of a class of spatial Cox process using spatial deformation. The proposed method enables the deformation behavior to be data-driven, through a multivariate latent Gaussian process. Inference leads to intractable posterior distributions that are approximated via MCMC. The convergence of algorithms based on the Metropolis-Hastings steps proved to be slow, and the computational efficiency of the Bayesian updating scheme was improved by adopting Hamiltonian Monte Carlo (HMC) methods. Our proposal was also compared against an alternative anisotropic formulation. Studies based on synthetic data provided empirical evidence of the benefit brought by the adoption of nonstationarity through our anisotropic structure. A real data application was conducted on the spatial spread of the Spodoptera frugiperda pest in a corn-producing agricultural area in southern Brazil. Once again, the proposed method demonstrated its benefit over alternatives.

Quantum dynamical Hamiltonian Monte Carlo

One of the open challenges in quantum computing is to find meaningful and practical methods to leverage quantum computation to accelerate classical machine-learning workflows. A ubiquitous problem in machine-learning workflows is sampling from probability distributions that we only have access to via their log probability. To this end, we extend the well-known Hamiltonian Monte Carlo (HMC) method for Markov chain Monte Carlo (MCMC) sampling to leverage quantum computation in a hybrid manner as a proposal function. Our new algorithm, Quantum Dynamical Hamiltonian Monte Carlo (QD-HMC), replaces the classical symplectic integration proposal step with simulations of quantum-coherent continuous-space dynamics on digital or analog quantum computers. We show that QD-HMC maintains key characteristics of HMC, such as maintaining the detailed balanced condition with momentum inversion, while also having the potential for polynomial speedups over its classical counterpart in certain scenarios. As sampling is a core subroutine in many forms of probabilistic inference, and MCMC in continuously parametrized spaces covers a large class of potential applications, this work widens the areas of applicability of quantum devices. Published by the American Physical Society 2024

Learning variational autoencoders via MCMC speed measures

Variational autoencoders (VAEs) are popular likelihood-based generative models which can be efficiently trained by maximising an evidence lower bound. There has been much progress in improving the expressiveness of the variational distribution to obtain tighter variational bounds and increased generative performance. Whilst previous work has leveraged Markov chain Monte Carlo methods for constructing variational densities, gradient-based methods for adapting the proposal distributions for deep latent variable models have received less attention. This work suggests an entropy-based adaptation for a short-run metropolis-adjusted Langevin or Hamiltonian Monte Carlo (HMC) chain while optimising a tighter variational bound to the log-evidence. Experiments show that this approach yields higher held-out log-likelihoods as well as improved generative metrics. Our implicit variational density can adapt to complicated posterior geometries of latent hierarchical representations arising in hierarchical VAEs.

A Method of Rapidly Deriving Late-type Contact Binary Parameters and Its Application in the Catalina Sky Survey

With the continuous development of large optical surveys, a large number of light curves of late-type contact binary systems (CBs) have been released. Deriving parameters for CBs using the the Wilson–Devinney program and the PHOEBE program poses a challenge. Therefore, this study developed a method for rapidly deriving light curves based on the Neural Networks model combined with the Hamiltonian Monte Carlo (HMC) algorithm (NNHMC). The neural network was employed to establish the mapping relationship between the parameters and the pregenerated light curves by the PHOEBE program, and the HMC algorithm was used to obtain the posterior distribution of the parameters. The NNHMC method was applied to a large contact binary sample from the Catalina Sky Survey, and a total of 19,104 late-type contact binary parameters were derived. Among them, 5172 have an inclination greater than 70° and a temperature difference less than 400 K. The obtained results were compared with the previous studies for 30 CBs, and there was an essentially consistent goodness-of-fit (R 2) distribution between them. The NNHMC method possesses the capability to simultaneously derive parameters for a vast number of targets. Furthermore, it can provide an extremely efficient tool for the rapid derivation of parameters in future sky surveys involving large samples of CBs.

Hybrid simulation of a capacitive Ar/SiH4 discharge driven by electrically asymmetric voltage waveforms

Voltage waveforms associated with the electrical asymmetry effect (EAE) have the potential to be used in the deposition of the silicon-based film, since they are expected to decouple ion energy and flux at the wafer surface, and further facilitate control of the process. In this study, a one-dimensional fluid/electron Monte Carlo hybrid model is employed to examine the EAE in a capacitively coupled argon-silane discharge, encompassing both amplitude asymmetry effect (AAE) and slope asymmetry effect (SAE). In the case of AAE, with the increasing pressure, the discharge electronegativity gradually intensifies, in conjunction with a transition of the electron heating mode from α to drift-ambipolar, a reduction of the absolute value of the DC self-bias voltage, and a decrease in Ar+ content, with an increase in SiH3 + content. For SAE, the trend in the discharge characteristics with the increasing pressure is similar to that for AAE, but the details are different. In SAE, the electronegativity and bulk electric field are much enhanced, resulting in higher content of high-energy electrons and Ar+ in the bulk. In addition, the absolute value of the self-bias is lower, but shows a fewer decline with the increasing pressure. The deposition rate is lower in SAE, due to the lower electron heating efficiency. However, larger voltage drop difference between two sheaths leads to a wider range of ion energy modulation at higher pressures. This study systematically investigates and compares Ar/SiH4 discharges driven by two electrically asymmetric voltage waveforms across various parameters including electron dynamics, ion and neutral transport properties, and deposition rates, with the aim of providing valuable insights and a reference for industrial applications.

Resetting the environmental reservoir; evaluating the impact of a new hospital building on Clostridioides difficile infection

Background: Prior research has implicated contaminated surfaces in the transmission of Clostridioides difficile within the hospital. To reduce the risk of transmission, enhanced environmental hygiene is performed in rooms of patients with known C.difficile infection (CDI). We wished to evaluate the residual impact of environmental surfaces on hospital-onset CDI (HO-CDI) by comparing HO-CDI rates before and after the opening of a new 504-bed hospital building, HUP Pavilion (PAV). We hypothesized that we would observe a reduction in HO-CDI after opening of PAV due to a reduced burden of C.difficile spores in the environment. Methods: We included NHSN reported HO-CDI rates for 28 months prior and 24 months after opening of PAV. Upon opening, patients were divided between the old building (HUP) and PAV. We included all patient units before and after opening. We created hierarchical models of HO-CDI rates using Stan Hamiltonian Monte Carlo (HMC) version 2.30.1, via the “cmdstanr” and “brms” packages with a GAM smooth function by month and intervention period with default, weakly-informative priors. Results: At baseline, there was an average of approximately 20,100 patient days per month, subsequently divided between HUP and PAV (mean 10,100 and 12,100 patient days per month). After opening of PAV, we observed a reduced HO-CDI rate (mean 0.21 vs 0.31 per 1000 patient days, P=0.01). When comparing the two specific buildings after opening of PAV, there was a greater reduction noticed in the old building (HUP) as compared to the new building (PAV) (0.12 vs 0.29 per 1000 patient days) (Figure 1). The predicted contrast in HO-CDI rate (Figure 2), shows no immediate change in HO-CDI after opening, however a sustained reduction estimated at 0.1 HO-CDI events per 1000 patient days for the duration of follow-up. Conclusions: We observed a reduction in HO-CDI rates after the opening of a new hospital building. The difference in HO-CDI rates between hospital buildings after the move is likely due to the concentration of high-risk patient cohorts within this building. Our findings suggests that there remains an opportunity to reduce HO-CDI through environmental hygiene. However, it is possible that other factors beyond surface environment contributed to an observed reduction in HO-CDI, including other concurrent infection control interventions that focused on smaller populations within the hospital. In future work we will investigate the durability of this observed effect with additional analyses including patient-level risk for HO-CDI.

Bayesian Parameter Estimation of Vibrating Systems via Physics-Based Biases from Neural Networks

Often, Structural Health Monitoring (SHM) campaigns draw damage-identifiers from vibration-based monitoring data. One of the downstream tasks of vibration-based SHM is that of system identification, i.e., inference of a model which is able to describe the system. For vibration-based monitoring, such models often rely on partial-differential equation formulations, where the identification task may be tied to inference of the parameters of such models. The probabilistic nature of SHM incentivises the extraction of stochastic estimates of these parameters, allowing for propagation of uncertainties associated to measurement noise or model imprecision. Common approaches to generating such stochastic estimates require an assumption on the shape and independence of the posterior distributions. To circumvent these assumptions, the Hamiltonian Monte-Carlo (HMC) sampling approach simulates the posterior distribution, allowing for freedom in the posterior. However, this requires intensive computational effort, especially when the governing equations have increased complexities, such as nonlinearities. Physics-informed neural networks (PINNs) utilise physics-based objective functions in the form of known PDEs, exploiting the automatic differentiation of NN architectures. This implies that additional terms in the PDE, such as nonlinearities, or boundary conditions can be accounted for. When the domain of observations encapsulates the collocation domain of the PINN, they can be exploited for parameter identification, by optimising over these parameters in addition to the network weights. However, classic objective functions in the form of mean-squared-norms underperform on noisy data, as a consequence of their deterministic nature. In this work, PINNs are combined with a HMC sampler to estimate the posterior distribution of the parameters of dynamic systems with noisy measurements. Multiple types of dynamic systems are simulated, varying the system complexity in terms of type and strength, and with different levels of noise. The results are assessed against the known values, and the codependent posterior distributions of each system are discussed.

Adaptive parameters tuning based on energy-preserving splitting integration for Hamiltonian Monte Carlo Method

Splitting schemes, a class of numerical integrators for Hamiltonian problems, offer a favorable alternative to the Störmer–Verlet method in Hamiltonian Monte Carlo (HMC) methodology. However, the performance of HMC is highly sensitive to the adopted step size. In this paper, we propose a novel approach for selecting the step size h for advancing with the method defined by the free parameter b, within the family of one-parameter second order splitting procedures. Our methodology utilizes a designated function hb of the parameter b to determine the step size, i.e. h=hb(b). By appropriately restricting the domain of hb to a suitable interval I, the pairs (b,hb) with b∈I ensure both stability and Hamiltonian preservation when sampling from Gaussian distributions. As a result, our technique never rejects a sample within the HMC process, and this characteristic is the key factor behind its superior performance compared to similar methods recently introduced in other studies. Additionally, we assess the effectiveness of the methods defined by the pairs (b,hb) for general not-Gaussian distribution sampling. In this case we also present a technique based on an adaptive selection of the b parameter for improving the HMC performance. The effectiveness of the proposed approach is evaluated through benchmark examples from literature and experiments involving the Log-Gaussian Cox process and Bayesian Logistic Regression.

A robust multi-risk model and its reliability relevance: A Bayes study with Hamiltonian Monte Carlo methodology

This paper proposes a robust model that describes complex time-to-failure and competing risk datasets found with bathtub curve hazard rate (BCHR). The resulting model, named the flexible Dhillon–Weibull competing risk (FDWCR) model, hybridized the Weibull and flexible Dhillon distributions. It is built with the assumptions of competing risk data characterized by monotone and non-monotone hazard rates. In contrast to most recent additive or competing risk models, the FDWCR model can analyze not only time-to-failure datasets identified with increasing hazard rates and BCHR but also effectively describe time-to-failure data associated with decreasing, upside-down bathtub and modified bathtub hazard rates. Several reliability and structural properties of the model are discussed. We present a complete Bayesian paradigm for FDWCR and suggest employing the Hamiltonian Monte Carlo (HMC) algorithm to enhance precision and expedite inference. The maximum likelihood estimators are also presented. We evaluate the appropriateness of the proposed estimators via simulation experiments, considering both techniques. We offer three case studies to demonstrate the practical utility of the proposed model. The results outperform those produced by other alternative models, indicating a reliable approach for modeling diverse competing risk problems, particularly in the context of engineering reliability studies or other domains of quantitative research.

Bayesian Uncertainty Estimation by Hamiltonian Monte Carlo: Applications to Cardiac MRI Segmentation

Deep learning (DL)-based methods have achieved state-of-the-art performance for many medical image segmentation tasks. Nevertheless, recent studies show that deep neural net- works (DNNs) can be miscalibrated and overconfident, leading to ”silent failures” that are risky for clinical applications. Bayesian DL provides an intuitive approach to DL failure de- tection, based on posterior probability estimation. However, the posterior is intractable for large medical image segmentation DNNs. To tackle this challenge, we propose a Bayesian learning framework using Hamiltonian Monte Carlo (HMC), tempered by cold posterior (CP) to accommodate medical data augmentation, named HMC-CP. For HMC compu- tation, we further propose a cyclical annealing strategy, capturing both local and global geometries of the posterior distribution, enabling highly efficient Bayesian DNN training with the same computational budget as training a single DNN. The resulting Bayesian DNN outputs an ensemble segmentation along with the segmentation uncertainty. We evaluate the proposed HMC-CP extensively on cardiac magnetic resonance image (MRI) segmentation, using in-domain steady-state free precession (SSFP) cine images as well as out-of-domain datasets of quantitative T1 and T2 mapping. Our results show that the proposed method improves both segmentation accuracy and uncertainty estimation for in- and out-of-domain data, compared with well-established baseline methods such as Monte Carlo Dropout and Deep Ensembles. Additionally, we establish a conceptual link between HMC and the commonly known stochastic gradient descent (SGD) and provide general insight into the uncertainty of DL. This uncertainty is implicitly encoded in the training dynamics but often overlooked. With reliable uncertainty estimation, our method provides a promising direction toward trustworthy DL in clinical applications. We release our code in <a href='https://gitlab.tudelft.nl/yidongzhao/hmc_uncertainty'>https://gitlab.tudelft.nl/yidongzhao/hmc_uncertainty</a>

Bayesian Estimation of Simultaneous Regression Quantiles Using Hamiltonian Monte Carlo

The simultaneous estimation of multiple quantiles is a crucial statistical task that enables a thorough understanding of data distribution for robust analysis and decision-making. In this study, we adopt a Bayesian approach to tackle this critical task, employing the asymmetric Laplace distribution (ALD) as a flexible framework for quantile modeling. Our methodology implementation involves the Hamiltonian Monte Carlo (HMC) algorithm, building on the foundation laid in priorwork, where the error term is assumed to follow an ALD. Capitalizing on the interplay between two distinct quantiles of this distribution, we endorse a straightforward and fully Bayesian method that adheres to the non-crossing property of quantiles. Illustrated through simulated scenarios, we showcase the effectiveness of our approach in quantile estimation, enhancing precision via the HMC algorithm. The proposed method proves versatile, finding application in finance, environmental science, healthcare, and manufacturing, and contributing to sustainable development goals by fostering innovation and enhancing decision-making in diverse fields.

Hierarchical modeling of irregularly spaced financial returns

Volatility modeling is crucial in finance, especially when dealing with intraday transaction‐level asset returns. The irregular and high‐frequency nature of the data presents unique challenges. While stochastic volatility (SV) models are widely used for understanding patterns in volatility of daily stock returns which constitute regularly spaced time series, new classes of models must be introduced for analyzing volatility in irregularly spaced intraday data. Specifically these models must accommodate the random gaps between successive transactional events. By modeling the gaps using autoregressive conditional duration (ACD) models, we describe a hierarchical irregular SV autoregressive conditional duration (IR‐SV‐ACD) model for estimating and forecasting intertransaction gaps and the volatility of log‐returns. We carry out the analysis in the Bayesian framework via the Hamiltonian Monte Carlo (HMC) algorithm with No‐U‐turn sampler (NUTS) in R using the cmdstanr package. The fits and forecasts are obtained using Monte Carlo averages based on the posterior samples. We illustrate this approach using simulation studies and real data analysis for intraday prices available at microseconds level of health stocks traded on the New York Stock Exchange (NYSE). The log‐returns and gaps are calculated for the stocks and are used for modeling.

Time-Varying Zero-Adjusted Poisson Distribution for Modeling Count Time Series

Many studies have used extensions of ARMA models for the analysis of non-Gaussian time series. One of them is the Generalized Autoregressive Moving Average, GARMA, enabling the modeling of count time series with distributions such as Poisson. The GARMA class is being expanded to accommodate other distributions, aiming to capture the typical characteristics of count data, including under or overdispersion and excess zeros. This study aims to propose an approach based on the GARMA class in order to analyze count time series with excess zeros, assuming a time-varying zero-adjusted Poisson distribution. This approach allows for capturing serial correlation, forecasting the future values, and estimating the future probability of zeros. For inference, a Bayesian analysis was adopted using the Hamiltonian Monte Carlo (HMC) algorithm for sampling from the joint posterior distribution. We conducted a simulation study and presented an application to influenza mortality reported in Brazil. Our findings demonstrated the usefulness of the model in estimating the probability of non-occurrence and the number of counts in future periods.

Assessing the E-bike trends and impact on sustainable mobility: A national-level study in the Netherlands

Over the past decade, e-bikes have become increasingly popular, sparking interest in their potential replacement for car use and benefit for the environment. However, many studies on e-bike development and their substitution effects exhibit limitations. These include a lack of modeling on e-bike trend development, inadequate assessments of their impact on national-level mobility, a predominant focus on commuting, and a lack of foresight into future e-bike substitution effects. Our research introduces an innovative approach to model e-bike development, employing a multilevel Richards growth curve model fitted within a hierarchical Bayesian framework using the Hamiltonian Monte Carlo (HMC) method. Further, we incorporate an intention-based method to delve into the potential of e-bikes in stimulating sustainable mobility in the Netherlands. Our findings highlight an ongoing increase in e-bike distance share, with marked gender and generational differences in growth patterns. Notably, women have higher e-bike usage than men, and this gap is narrowing for older age groups while widening among younger demographics, suggesting that younger people may adopt e-bike usage differently than older generation. E-bike ownership strongly reduces the conventional bicycle use and, to a lesser extent, car and public transport use, especially for commuting. This study provides insight into whether and to what extent e-bikes substitute for car use and other modes of transportation, and how the expected growth in e-bike use in coming years may impact national mobility in the Netherlands.

Efficient Bayesian Physics Informed Neural Networks for inverse problems via Ensemble Kalman Inversion

Bayesian Physics Informed Neural Networks (B-PINNs) have gained significant attention for inferring physical parameters and learning the forward solutions for problems based on partial differential equations. However, the overparameterized nature of neural networks poses a computational challenge for high-dimensional posterior inference. Existing inference approaches, such as particle-based or variance inference methods, are either computationally expensive for high-dimensional posterior inference or provide unsatisfactory uncertainty estimates. In this paper, we present a new efficient inference algorithm for B-PINNs that uses Ensemble Kalman Inversion (EKI) for high-dimensional inference tasks. By reframing the setup of B-PINNs as a traditional Bayesian inverse problem, we can take advantage of EKI's key features: (1) gradient-free, (2) computational complexity scales linearly with the dimension of the parameter spaces, and (3) rapid convergence with typically O(100) iterations. We demonstrate the applicability and performance of the proposed method through various types of numerical examples. We find that our proposed method can achieve inference results with informative uncertainty estimates comparable to Hamiltonian Monte Carlo (HMC)-based B-PINNs with a much reduced computational cost. These findings suggest that our proposed approach has great potential for uncertainty quantification in physics-informed machine learning for practical applications.