Xihaier Luo

Xihaier Luo, Ahsan Kareem, Shinjae Yoo

Deciding how to optimally deploy sensors in a large, complex, and spatially extended structure is critical to ensure that the surface pressure field is accurately captured for subsequent analysis and design. In some cases, reconstruction of missing data is required in downstream tasks such as the development of digital twins. This paper presents a data-driven sparse sensor selection algorithm, aiming to provide the most information contents for reconstructing aerodynamic characteristics of wind pressures over tall building structures parsimoniously. The algorithm first fits a set of basis functions to the training data, then applies a computationally efficient QR algorithm that ranks existing pressure sensors in order of importance based on the state reconstruction to this tailored basis. The findings of this study show that the proposed algorithm successfully reconstructs the aerodynamic characteristics of tall buildings from sparse measurement locations, generating stable and optimal solutions across a range of conditions. As a result, this study serves as a promising first step toward leveraging the success of data-driven and machine learning algorithms to supplement traditional genetic algorithms currently used in wind engineering.

Xihaier Luo, Sean McCorkle, Gilchan Park et al.

There are various sources of ionizing radiation exposure, where medical exposure for radiation therapy or diagnosis is the most common human-made source. Understanding how gene expression is modulated after ionizing radiation exposure and investigating the presence of any dose-dependent gene expression patterns have broad implications for health risks from radiotherapy, medical radiation diagnostic procedures, as well as other environmental exposure. In this paper, we perform a comprehensive pathway-based analysis of gene expression profiles in response to low-dose radiation exposure, in order to examine the potential mechanism of gene regulation underlying such responses. To accomplish this goal, we employ a statistical framework to determine whether a specific group of genes belonging to a known pathway display coordinated expression patterns that are modulated in a manner consistent with the radiation level. Findings in our study suggest that there exist complex yet consistent signatures that reflect the molecular response to radiation exposure, which differ between low-dose and high-dose radiation.

Gilchan Park, Byung-Jun Yoon, Xihaier Luo et al.

Background: Identification of the interactions and regulatory relations between biomolecules play pivotal roles in understanding complex biological systems and the mechanisms underlying diverse biological functions. However, the collection of such molecular interactions has heavily relied on expert curation in the past, making it labor-intensive and time-consuming. To mitigate these challenges, we propose leveraging the capabilities of large language models (LLMs) to automate genome-scale extraction of this crucial knowledge. Results: In this study, we investigate the efficacy of various LLMs in addressing biological tasks, such as the recognition of protein interactions, identification of genes linked to pathways affected by low-dose radiation, and the delineation of gene regulatory relationships. Overall, the larger models exhibited superior performance, indicating their potential for specific tasks that involve the extraction of complex interactions among genes and proteins. Although these models possessed detailed information for distinct gene and protein groups, they faced challenges in identifying groups with diverse functions and in recognizing highly correlated gene regulatory relationships. Conclusions: By conducting a comprehensive assessment of the state-of-the-art models using well-established molecular interaction and pathway databases, our study reveals that LLMs can identify genes/proteins associated with pathways of interest and predict their interactions to a certain extent. Furthermore, these models can provide important insights, marking a noteworthy stride toward advancing our understanding of biological systems through AI-assisted knowledge discovery.

Alif Bin Abdul Qayyum, Xihaier Luo, Nathan M. Urban et al.

The world is moving towards clean and renewable energy sources, such as wind energy, in an attempt to reduce greenhouse gas emissions that contribute to global warming. To enhance the analysis and storage of wind data, we introduce a deep learning framework designed to simultaneously enable effective dimensionality reduction and continuous representation of multi-altitude wind data from discrete observations. The framework consists of three key components: dimensionality reduction, cross-modal prediction, and super-resolution. We aim to: (1) improve data resolution across diverse climatic conditions to recover high-resolution details; (2) reduce data dimensionality for more efficient storage of large climate datasets; and (3) enable cross-prediction between wind data measured at different heights. Comprehensive testing confirms that our approach surpasses existing methods in both super-resolution quality and compression efficiency.

Vanessa Lopez-Marrero, Patrick R. Johnstone, Gilchan Park et al.

One among several advantages of measure transport methods is that they allow for a unified framework for processing and analysis of data distributed according to a wide class of probability measures. Within this context, we present results from computational studies aimed at assessing the potential of measure transport techniques, specifically, the use of triangular transport maps, as part of a workflow intended to support research in the biological sciences. Scenarios characterized by the availability of limited amount of sample data, which are common in domains such as radiation biology, are of particular interest. We find that when estimating a distribution density function given limited amount of sample data, adaptive transport maps are advantageous. In particular, statistics gathered from computing series of adaptive transport maps, trained on a series of randomly chosen subsets of the set of available data samples, leads to uncovering information hidden in the data. As a result, in the radiation biology application considered here, this approach provides a tool for generating hypotheses about gene relationships and their dynamics under radiation exposure.

Brandon R. Feng, Brian J. Reich, Daniel Beaglehole et al.

The recent developments of complex deep learning models have led to unprecedented ability to accurately predict across multiple data representation types. Conformal prediction for uncertainty quantification of these models has risen in popularity, providing adaptive, statistically-valid prediction sets. For classification tasks, conformal methods have typically focused on utilizing logit scores. For pre-trained models, however, this can result in inefficient, overly conservative set sizes when not calibrated towards the target task. We propose DANCE, a doubly locally adaptive nearest-neighbor based conformal algorithm combining two novel nonconformity scores directly using the data's embedded representation. DANCE first fits a task-adaptive kernel regression model from the embedding layer before using the learned kernel space to produce the final prediction sets for uncertainty quantification. We test against state-of-the-art local, task-adapted and zero-shot conformal baselines, demonstrating DANCE's superior blend of set size efficiency and robustness across various datasets.

Kevin Valencia, Thilina Balasooriya, Xihaier Luo et al.

Multimodal spatiotemporal learning on real-world experimental data is constrained by two challenges: within-modality measurements are sparse, irregular, and noisy (QA/QC artifacts) but cross-modally correlated; the set of available modalities varies across space and time, shrinking the usable record unless models can adapt to arbitrary subsets at train and test time. We propose OmniField, a continuity-aware framework that learns a continuous neural field conditioned on available modalities and iteratively fuses cross-modal context. A multimodal crosstalk block architecture paired with iterative cross-modal refinement aligns signals prior to the decoder, enabling unified reconstruction, interpolation, forecasting, and cross-modal prediction without gridding or surrogate preprocessing. Extensive evaluations show that OmniField consistently outperforms eight strong multimodal spatiotemporal baselines. Under heavy simulated sensor noise, performance remains close to clean-input levels, highlighting robustness to corrupted measurements.

Xihaier Luo, Xiaoning Qian, Byung-Jun Yoon

In this work, we present an arbitrary-scale super-resolution (SR) method to enhance the resolution of scientific data, which often involves complex challenges such as continuity, multi-scale physics, and the intricacies of high-frequency signals. Grounded in operator learning, the proposed method is resolution-invariant. The core of our model is a hierarchical neural operator that leverages a Galerkin-type self-attention mechanism, enabling efficient learning of mappings between function spaces. Sinc filters are used to facilitate the information transfer across different levels in the hierarchy, thereby ensuring representation equivalence in the proposed neural operator. Additionally, we introduce a learnable prior structure that is derived from the spectral resizing of the input data. This loss prior is model-agnostic and is designed to dynamically adjust the weighting of pixel contributions, thereby balancing gradients effectively across the model. We conduct extensive experiments on diverse datasets from different domains and demonstrate consistent improvements compared to strong baselines, which consist of various state-of-the-art SR methods.

Yi-Tang Chen, Haoyu Li, Neng Shi et al.

With the growing computational power available for high-resolution ensemble simulations in scientific fields such as cosmology and oceanology, storage and computational demands present significant challenges. Current surrogate models fall short in the flexibility of point- or region-based predictions as the entire field reconstruction is required for each parameter setting, hence hindering the efficiency of parameter space exploration. Limitations exist in capturing physical attribute distributions and pinpointing optimal parameter configurations. In this work, we propose Explorable INR, a novel implicit neural representation-based surrogate model, designed to facilitate exploration and allow point-based spatial queries without computing full-scale field data. In addition, to further address computational bottlenecks of spatial exploration, we utilize probabilistic affine forms (PAFs) for uncertainty propagation through Explorable INR to obtain statistical summaries, facilitating various ensemble analysis and visualization tasks that are expensive with existing models. Furthermore, we reformulate the parameter exploration problem as optimization tasks using gradient descent and KL divergence minimization that ensures scalability. We demonstrate that the Explorable INR with the proposed approach for spatial and parameter exploration can significantly reduce computation and memory costs while providing effective ensemble analysis.

Ayush Khot, Xihaier Luo, Ai Kagawa et al.

Uncertainty quantification (UQ) methods play an important role in reducing errors in weather forecasting. Conventional approaches in UQ for weather forecasting rely on generating an ensemble of forecasts from physics-based simulations to estimate the uncertainty. However, it is computationally expensive to generate many forecasts to predict real-time extreme weather events. Evidential Deep Learning (EDL) is an uncertainty-aware deep learning approach designed to provide confidence about its predictions using only one forecast. It treats learning as an evidence acquisition process where more evidence is interpreted as increased predictive confidence. We apply EDL to storm forecasting using real-world weather datasets and compare its performance with traditional methods. Our findings indicate that EDL not only reduces computational overhead but also enhances predictive uncertainty. This method opens up novel opportunities in research areas such as climate risk assessment, where quantifying the uncertainty about future climate is crucial.

David Keetae Park, Xihaier Luo, Guang Zhao et al.

Spatiotemporal learning is challenging due to the intricate interplay between spatial and temporal dependencies, the high dimensionality of the data, and scalability constraints. These challenges are further amplified in scientific domains, where data is often irregularly distributed (e.g., missing values from sensor failures) and high-volume (e.g., high-fidelity simulations), posing additional computational and modeling difficulties. In this paper, we present SCENT, a novel framework for scalable and continuity-informed spatiotemporal representation learning. SCENT unifies interpolation, reconstruction, and forecasting within a single architecture. Built on a transformer-based encoder-processor-decoder backbone, SCENT introduces learnable queries to enhance generalization and a query-wise cross-attention mechanism to effectively capture multi-scale dependencies. To ensure scalability in both data size and model complexity, we incorporate a sparse attention mechanism, enabling flexible output representations and efficient evaluation at arbitrary resolutions. We validate SCENT through extensive simulations and real-world experiments, demonstrating state-of-the-art performance across multiple challenging tasks while achieving superior scalability.

Miruna Oprescu, David K. Park, Xihaier Luo et al.

Estimating causal effects from spatiotemporal observational data is essential in public health, environmental science, and policy evaluation, where randomized experiments are often infeasible. Existing approaches, however, either rely on strong structural assumptions or fail to handle key challenges such as interference, spatial confounding, temporal carryover, and time-varying confounding -- where covariates are influenced by past treatments and, in turn, affect future ones. We introduce GST-UNet (G-computation Spatio-Temporal UNet), a theoretically grounded neural framework that combines a U-Net-based spatiotemporal encoder with regression-based iterative G-computation to estimate location-specific potential outcomes under complex intervention sequences. GST-UNet explicitly adjusts for time-varying confounders and captures non-linear spatial and temporal dependencies, enabling valid causal inference from a single observed trajectory in data-scarce settings. We validate its effectiveness in synthetic experiments and in a real-world analysis of wildfire smoke exposure and respiratory hospitalizations during the 2018 California Camp Fire. Together, these results position GST-UNet as a principled and ready-to-use framework for spatiotemporal causal inference, advancing reliable estimation in policy-relevant and scientific domains.

Xihaier Luo, Samuel Lurvey, Yi Huang et al.

High-energy, large-scale particle colliders in nuclear and high-energy physics generate data at extraordinary rates, reaching up to $1$ terabyte and several petabytes per second, respectively. The development of real-time, high-throughput data compression algorithms capable of reducing this data to manageable sizes for permanent storage is of paramount importance. A unique characteristic of the tracking detector data is the extreme sparsity of particle trajectories in space, with an occupancy rate ranging from approximately $10^{-6}$ to $10\%$. Furthermore, for downstream tasks, a continuous representation of this data is often more useful than a voxel-based, discrete representation due to the inherently continuous nature of the signals involved. To address these challenges, we propose a novel approach using implicit neural representations for data learning and compression. We also introduce an importance sampling technique to accelerate the network training process. Our method is competitive with traditional compression algorithms, such as MGARD, SZ, and ZFP, while offering significant speed-ups and maintaining negligible accuracy loss through our importance sampling strategy.

Wei Xu, Derek Freeman DeSantis, Xihaier Luo et al.

Learning a continuous and reliable representation of physical fields from sparse sampling is challenging and it affects diverse scientific disciplines. In a recent work, we present a novel model called MMGN (Multiplicative and Modulated Gabor Network) with implicit neural networks. In this work, we design additional studies leveraging explainability methods to complement the previous experiments and further enhance the understanding of latent representations generated by the model. The adopted methods are general enough to be leveraged for any latent space inspection. Preliminary results demonstrate the contextual information incorporated in the latent representations and their impact on the model performance. As a work in progress, we will continue to verify our findings and develop novel explainability approaches.

Yesom Park, Kelvin Kan, Thomas Flynn et al.

Implicit Neural Representations (INRs) provide a powerful continuous framework for modeling complex visual and geometric signals, but spectral bias remains a fundamental challenge, limiting their ability to capture high-frequency details. Orthogonal to existing remedy strategies, we introduce Dynamical Implicit Neural Representations (DINR), a new INR modeling framework that treats feature evolution as a continuous-time dynamical system rather than a discrete stack of layers. This dynamical formulation mitigates spectral bias by enabling richer, more adaptive frequency representations through continuous feature evolution. Theoretical analysis based on Rademacher complexity and the Neural Tangent Kernel demonstrates that DINR enhances expressivity and improves training dynamics. Moreover, regularizing the complexity of the underlying dynamics provides a principled way to balance expressivity and generalization. Extensive experiments on image representation, field reconstruction, and data compression confirm that DINR delivers more stable convergence, higher signal fidelity, and stronger generalization than conventional static INRs.

Ayush Khot, Miruna Oprescu, Maresa Schröder et al.

Causal inference in spatial domains faces two intertwined challenges: (1) unmeasured spatial factors, such as weather, air pollution, or mobility, that confound treatment and outcome, and (2) interference from nearby treatments that violate standard no-interference assumptions. While existing methods typically address one by assuming away the other, we show they are deeply connected: interference reveals structure in the latent confounder. Leveraging this insight, we propose the Spatial Deconfounder, a two-stage method that reconstructs a substitute confounder from local treatment vectors using a conditional variational autoencoder (CVAE) with a spatial prior, then estimates causal effects via a flexible outcome model. We show that this approach enables nonparametric identification of both direct and spillover effects under weak assumptions--without requiring multiple treatment types or a known model of the latent field. Empirically, we extend SpaCE, a benchmark suite for spatial confounding, to include treatment interference, and show that the Spatial Deconfounder consistently improves effect estimation across real-world datasets in environmental health and social science. By turning interference into a multi-cause signal, our framework bridges spatial and deconfounding literatures to advance robust causal inference in structured data.

David Park, Shuhang Li, Yi Huang et al.

Large language models have revolutionized artificial intelligence by enabling large, generalizable models trained through self-supervision. This paradigm has inspired the development of scientific foundation models (FMs). However, applying this capability to experimental particle physics is challenging due to the sparse, spatially distributed nature of detector data, which differs dramatically from natural language. This work addresses if an FM for particle physics can scale and generalize across diverse tasks. We introduce a new dataset with more than 11 million particle collision events and a suite of downstream tasks and labeled data for evaluation. We propose a novel self-supervised training method for detector data and demonstrate its neural scalability with models that feature up to 188 million parameters. With frozen weights and task-specific adapters, this FM consistently outperforms baseline models across all downstream tasks. The performance also exhibits robust data-efficient adaptation. Further analysis reveals that the representations extracted by the FM are task-agnostic but can be specialized via a single linear mapping for different downstream tasks.

Brandon R. Feng, David Keetae Park, Xihaier Luo et al.

Fitting Gaussian Processes (GPs) provides interpretable aleatoric uncertainty quantification for estimation of spatio-temporal fields. Spatio-temporal deep learning models, while scalable, typically assume a simplistic independent covariance matrix for the response, failing to capture the underlying correlation structure. However, spatio-temporal GPs suffer from issues of scalability and various forms of approximation bias resulting from restrictive assumptions of the covariance kernel function. We propose STACI, a novel framework consisting of a variational Bayesian neural network approximation of non-stationary spatio-temporal GP along with a novel spatio-temporal conformal inference algorithm. STACI is highly scalable, taking advantage of GPU training capabilities for neural network models, and provides statistically valid prediction intervals for uncertainty quantification. STACI outperforms competing GPs and deep methods in accurately approximating spatio-temporal processes and we show it easily scales to datasets with millions of observations.

Guang Zhao, Xihaier Luo, Seungjun Lee et al.

Mesoscale ocean dynamics play a critical role in climate systems, governing heat transport, hurricane genesis, and drought patterns. However, simulating these processes at high resolution remains computationally prohibitive due to their nonlinear, multiscale nature and vast spatiotemporal domains. Implicit neural representations (INRs) reduce the computational costs as resolution-independent surrogates but fail in many-query scenarios (inverse modeling) requiring rapid evaluations across diverse parameters. We present PINROD, a novel framework combining dynamics-aware implicit neural representations with parameterized neural ordinary differential equations to address these limitations. By integrating parametric dependencies into latent dynamics, our method efficiently captures nonlinear oceanic behavior across varying boundary conditions and physical parameters. Experiments on ocean mesoscale activity data show superior accuracy over existing baselines and improved computational efficiency compared to standard numerical simulations.

Yi Huang, Yeonju Go, Jin Huang et al.

High-energy large-scale particle colliders generate data at extraordinary rates. Developing real-time high-throughput data compression algorithms to reduce data volume and meet the bandwidth requirement for storage has become increasingly critical. Deep learning is a promising technology that can address this challenging topic. At the newly constructed sPHENIX experiment at the Relativistic Heavy Ion Collider, a Time Projection Chamber (TPC) serves as the main tracking detector, which records three-dimensional particle trajectories in a volume of a gas-filled cylinder. In terms of occupancy, the resulting data flow can be very sparse reaching $10^{-3}$ for proton-proton collisions. Such sparsity presents a challenge to conventional learning-free lossy compression algorithms, such as SZ, ZFP, and MGARD. In contrast, emerging deep learning-based models, particularly those utilizing convolutional neural networks for compression, have outperformed these conventional methods in terms of compression ratios and reconstruction accuracy. However, research on the efficacy of these deep learning models in handling sparse datasets, like those produced in particle colliders, remains limited. Furthermore, most deep learning models do not adapt their processing speeds to data sparsity, which affects efficiency. To address this issue, we propose a novel approach for TPC data compression via key-point identification facilitated by sparse convolution. Our proposed algorithm, BCAE-VS, achieves a $75\%$ improvement in reconstruction accuracy with a $10\%$ increase in compression ratio over the previous state-of-the-art model. Additionally, BCAE-VS manages to achieve these results with a model size over two orders of magnitude smaller. Lastly, we have experimentally verified that as sparsity increases, so does the model's throughput.

Alif Bin Abdul Qayyum, Xihaier Luo, Nathan M. Urban et al.

World is looking for clean and renewable energy sources that do not pollute the environment, in an attempt to reduce greenhouse gas emissions that contribute to global warming. Wind energy has significant potential to not only reduce greenhouse emission, but also meet the ever increasing demand for energy. To enable the effective utilization of wind energy, addressing the following three challenges in wind data analysis is crucial. Firstly, improving data resolution in various climate conditions to ensure an ample supply of information for assessing potential energy resources. Secondly, implementing dimensionality reduction techniques for data collected from sensors/simulations to efficiently manage and store large datasets. Thirdly, extrapolating wind data from one spatial specification to another, particularly in cases where data acquisition may be impractical or costly. We propose a deep learning based approach to achieve multi-modal continuous resolution wind data prediction from discontinuous wind data, along with data dimensionality reduction.

Xihaier Luo, Wei Xu, Yihui Ren et al.

Reliably reconstructing physical fields from sparse sensor data is a challenge that frequently arises in many scientific domains. In practice, the process generating the data often is not understood to sufficient accuracy. Therefore, there is a growing interest in using the deep neural network route to address the problem. This work presents a novel approach that learns a continuous representation of the physical field using implicit neural representations (INRs). Specifically, after factorizing spatiotemporal variability into spatial and temporal components using the separation of variables technique, the method learns relevant basis functions from sparsely sampled irregular data points to develop a continuous representation of the data. In experimental evaluations, the proposed model outperforms recent INR methods, offering superior reconstruction quality on simulation data from a state-of-the-art climate model and a second dataset that comprises ultra-high resolution satellite-based sea surface temperature fields.

Xihaier Luo, Balasubramanya T. Nadiga, Yihui Ren et al.

Since model bias and associated initialization shock are serious shortcomings that reduce prediction skills in state-of-the-art decadal climate prediction efforts, we pursue a complementary machine-learning-based approach to climate prediction. The example problem setting we consider consists of predicting natural variability of the North Atlantic sea surface temperature on the interannual timescale in the pre-industrial control simulation of the Community Earth System Model (CESM2). While previous works have considered the use of recurrent networks such as convolutional LSTMs and reservoir computing networks in this and other similar problem settings, we currently focus on the use of feedforward convolutional networks. In particular, we find that a feedforward convolutional network with a Densenet architecture is able to outperform a convolutional LSTM in terms of predictive skill. Next, we go on to consider a probabilistic formulation of the same network based on Stein variational gradient descent and find that in addition to providing useful measures of predictive uncertainty, the probabilistic (Bayesian) version improves on its deterministic counterpart in terms of predictive skill. Finally, we characterize the reliability of the ensemble of ML models obtained in the probabilistic setting by using analysis tools developed in the context of ensemble numerical weather prediction.

Xihaier Luo, Ahsan Kareem, Liting Yu et al.

The growing interest in creating a parametric representation of liquid sloshing inside a container stems from its practical applications in modern engineering systems. The resonant excitation, on the other hand, can cause unstable and nonlinear water waves, resulting in chaotic motions and non-Gaussian signals. This paper presents a novel machine learning-based framework for nonlinear liquid sloshing representation learning. The proposed method is a parametric modeling technique that is based on sequential learning and sparse regularization. The dynamics are categorized into two parts: linear evolution and nonlinear forcing. The former advances the dynamical system in time on an embedded manifold, while the latter causes divergent behaviors in temporal evolution, such as bursting and switching. The proposed framework's merit is demonstrated using an experimental dataset of liquid sloshing in a tank under horizontal excitation with a wide frequency range and various vertical slat screen settings.

Wei Xu, Xihaier Luo, Yihui Ren et al.

We present a study using a class of post-hoc local explanation methods i.e., feature importance methods for "understanding" a deep learning (DL) emulator of climate. Specifically, we consider a multiple-input-single-output emulator that uses a DenseNet encoder-decoder architecture and is trained to predict interannual variations of sea surface temperature (SST) at 1, 6, and 9 month lead times using the preceding 36 months of (appropriately filtered) SST data. First, feature importance methods are employed for individual predictions to spatio-temporally identify input features that are important for model prediction at chosen geographical regions and chosen prediction lead times. In a second step, we also examine the behavior of feature importance in a generalized sense by considering an aggregation of the importance heatmaps over training samples. We find that: 1) the climate emulator's prediction at any geographical location depends dominantly on a small neighborhood around it; 2) the longer the prediction lead time, the further back the "importance" extends; and 3) to leading order, the temporal decay of "importance" is independent of geographical location. An ablation experiment is adopted to verify the findings. From the perspective of climate dynamics, these findings suggest a dominant role for local processes and a negligible role for remote teleconnections at the spatial and temporal scales we consider. From the perspective of network architecture, the spatio-temporal relations between the inputs and outputs we find suggest potential model refinements. We discuss further extensions of our methods, some of which we are considering in ongoing work.

Xihaier Luo, Ahsan Kareem

Datasets in engineering applications are often limited and contaminated, mainly due to unavoidable measurement noise and signal distortion. Thus, using conventional data-driven approaches to build a reliable discriminative model, and further applying this identified surrogate to uncertainty analysis remains to be very challenging. A deep learning approach is presented to provide predictions based on limited and noisy data. To address noise perturbation, the Bayesian learning method that naturally facilitates an automatic updating mechanism is considered to quantify and propagate model uncertainties into predictive quantities. Specifically, hierarchical Bayesian modeling (HBM) is first adopted to describe model uncertainties, which allows the prior assumption to be less subjective, while also makes the proposed surrogate more robust. Next, the Bayesian inference is seamlessly integrated into the DL framework, which in turn supports probabilistic programming by yielding a probability distribution of the quantities of interest rather than their point estimates. Variational inference (VI) is implemented for the posterior distribution analysis where the intractable marginalization of the likelihood function over parameter space is framed in an optimization format, and stochastic gradient descent method is applied to solve this optimization problem. Finally, Monte Carlo simulation is used to obtain an unbiased estimator in the predictive phase of Bayesian inference, where the proposed Bayesian deep learning (BDL) scheme is able to offer confidence bounds for the output estimation by analyzing propagated uncertainties. The effectiveness of Bayesian shrinkage is demonstrated in improving predictive performance using contaminated data, and various examples are provided to illustrate concepts, methodologies, and algorithms of this proposed BDL modeling technique.