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Michael D. Shields

Publications and source records attributed to Michael D. Shields.

At least 19 recordsLinked to original sources
arXiv2026

SuSIE: Subset Simulation with Intrepid Exploration

This work explores the challenges associated with the subset simulation framework - a well-established algorithm for estimating structural reliability - in settings involving multiple, possibly disconnected regions of failure, or involving discontinuous or sharply changing performance functions. We demonstrate that these drawbacks of subset simulation stem from the limitations of the Markov chain Monte Carlo (MCMC) sampler employed within the method, not from the subset simulation framework itself. Traditional random-walk Metropolis algorithms, which are known to struggle severely with sampling from multimodal distributions, are conventionally applied within subset simulation, which leads to inaccurate failure probability estimates in such cases. In this work, we instead utilize a modified version of the Intrepid MCMC sampler, which has recently been shown to be more effective than vanilla random-walk Metropolis algorithms in sampling from multimodal probability distributions. The subset simulation method with the proposed Intrepid sampler is demonstrated to address these complicating features. Several illustrative examples are considered, ranging from 2 to 1003 dimensions and exhibiting multiple failure regions or highly nonlinear performance functions, including both analytical problems and structural engineering applications.

stat.CO↗
arXiv2026

$\phi-$DeepONet: A Discontinuity Capturing Neural Operator

We present $\phi-$DeepONet, a physics-informed neural operator designed to learn mappings between function spaces that may contain discontinuities or exhibit non-smooth behavior. Classical neural operators are based on the universal approximation theorem which assumes that both the operator and the functions it acts on are continuous. However, many scientific and engineering problems involve naturally discontinuous input fields as well as strong and weak discontinuities in the output fields caused by material interfaces. In $\phi$-DeepONet, discontinuities in the input are handled using multiple branch networks, while discontinuities in the output are learned through a nonlinear latent embedding of the interface. This embedding is constructed from a {\it one-hot} representation of the domain decomposition that is combined with the spatial coordinates in a modified trunk network. The outputs of the branch and trunk networks are then combined through a dot product to produce the final solution, which is trained using a physics- and interface-informed loss function. We evaluate $\phi$-DeepONet on several one- and two-dimensional benchmark problems and demonstrate that it delivers accurate and stable predictions even in the presence of strong interface-driven discontinuities.

cs.CE↗
arXiv2026

Physics-constrained Gaussian Processes for Predicting Shockwave Hugoniot Curves

A physics-constrained Gaussian Process regression framework is developed for predicting shocked material states and their associated uncertainties along the Hugoniot curve using data from a small number of shockwave simulations. The proposed Gaussian process is constrained by the Rankine-Hugoniot jump conditions between the various shocked material states to construct a thermodynamically consistent covariance function. This leads to the formulation of an optimization problem over a small number of interpretable hyperparameters and enables the identification of regime transitions, from a leading elastic wave to trailing plastic and phase transformation waves. Shock Hugoniots are an important measure for understanding material behavior under extreme conditions, including for the development of equations of state and determining material properties such as the Hugoniot Elastic Limit, but they are costly to generate through large-scale molecular dynamics simulations or shock experiments. Under these constraints, the proposed methodology establishes Hugoniot curves from a limited number of molecular dynamics simulations. We consider silicon carbide as a representative material and Molecular Dynamics simulations are performed using a reverse ballistic approach. The framework reproduces the Hugoniot curve with satisfactory accuracy while also quantifying the uncertainty in the predictions using the Gaussian Process posterior. These uncertain Hugoniot predictions can then be used to calibrate equation of state models, estimate material properties, or inform future experimental and/or simulation campaigns.

cs.CE↗
arXiv2026

Physics-Informed Gaussian Process Regression for the Constitutive Modeling of Concrete: A Data-Driven Improvement to Phenomenological Models

Understanding and modeling the constitutive behavior of concrete is crucial for civil and defense applications, yet widely used phenomenological models such as Karagozian \& Case concrete (KCC) model depend on empirically calibrated failure surfaces that lack flexibility in model form and associated uncertainty quantification. This work develops a physics-informed framework that retains the modular elastoplastic structure of KCC model while replacing its empirical failure surface with a constrained Gaussian Process Regression (GPR) surrogate that can be learned directly from experimentally accessible observables. Triaxial compression data under varying confinement levels are used for training, and the surrogate is then evaluated at confinement levels not included in the training set to assess its generalization capability. Results show that an unconstrained GPR interpolates well near training conditions but deteriorates and violates essential physical constraints under extrapolation, even when augmented with simulated data. In contrast, a physics-informed GPR that incorporates derivative-based constraints aligned with known material behavior yields markedly better accuracy and reliability, including at higher confinement levels beyond the training range. Probabilistic enforcement of these constraints also reduces predictive variance, producing tighter confidence intervals in data-scarce regimes. Overall, the proposed approach delivers a robust, uncertainty-aware surrogate that improves generalization and streamlines calibration without sacrificing the interpretability and numerical efficiency of the KCC model, offering a practical path toward an improved constitutive models for concrete.

cs.LG↗
arXiv2025

Physics-informed Polynomial Chaos Expansion with Enhanced Constrained Optimization Solver and D-optimal Sampling

Physics-informed polynomial chaos expansions (PC$^2$) provide an efficient physically constrained surrogate modeling framework by embedding governing equations and other physical constraints into the standard data-driven polynomial chaos expansions (PCE) and solving via the Karush-Kuhn-Tucker (KKT) conditions. This approach improves the physical interpretability of surrogate models while achieving high computational efficiency and accuracy. However, the performance and efficiency of PC$^2$ can still be degraded with high-dimensional parameter spaces, limited data availability, or unrepresentative training data. To address this problem, this study explores two complementary enhancements to the PC$^2$ framework. First, a numerically efficient constrained optimization solver, straightforward updating of Lagrange multipliers (SULM), is adopted as an alternative to the conventional KKT solver. The SULM method significantly reduces computational cost when solving physically constrained problems with high-dimensionality and derivative boundary conditions that require a large number of virtual points. Second, a D-optimal sampling strategy is utilized to select informative virtual points to improve the stability and achieve the balance of accuracy and efficiency of the PC$^2$. The proposed methods are integrated into the PC$^2$ framework and evaluated through numerical examples of representative physical systems governed by ordinary or partial differential equations. The results demonstrate that the enhanced PC$^2$ has better comprehensive capability than standard PC$^2$, and is well-suited for high-dimensional uncertainty quantification tasks.

stat.ML↗
arXiv2025

A Physics-informed Multi-resolution Neural Operator

The predictive accuracy of operator learning frameworks depends on the quality and quantity of available training data (input-output function pairs), often requiring substantial amounts of high-fidelity data, which can be challenging to obtain in some real-world engineering applications. These datasets may be unevenly discretized from one realization to another, with the grid resolution varying across samples. In this study, we introduce a physics-informed operator learning approach by extending the Resolution Independent Neural Operator (RINO) framework to a fully data-free setup, addressing both challenges simultaneously. Here, the arbitrarily (but sufficiently finely) discretized input functions are projected onto a latent embedding space (i.e., a vector space of finite dimensions), using pre-trained basis functions. The operator associated with the underlying partial differential equations (PDEs) is then approximated by a simple multi-layer perceptron (MLP), which takes as input a latent code along with spatiotemporal coordinates to produce the solution in the physical space. The PDEs are enforced via a finite difference solver in the physical space. The validation and performance of the proposed method are benchmarked on several numerical examples with multi-resolution data, where input functions are sampled at varying resolutions, including both coarse and fine discretizations.

cs.LG↗
arXiv2025

Polynomial Chaos Expansion for Operator Learning

Operator learning (OL) has emerged as a powerful tool in scientific machine learning (SciML) for approximating mappings between infinite-dimensional functional spaces. One of its main applications is learning the solution operator of partial differential equations (PDEs). While much of the progress in this area has been driven by deep neural network-based approaches such as Deep Operator Networks (DeepONet) and Fourier Neural Operator (FNO), recent work has begun to explore traditional machine learning methods for OL. In this work, we introduce polynomial chaos expansion (PCE) as an OL method. PCE has been widely used for uncertainty quantification (UQ) and has recently gained attention in the context of SciML. For OL, we establish a mathematical framework that enables PCE to approximate operators in both purely data-driven and physics-informed settings. The proposed framework reduces the task of learning the operator to solving a system of equations for the PCE coefficients. Moreover, the framework provides UQ by simply post-processing the PCE coefficients, without any additional computational cost. We apply the proposed method to a diverse set of PDE problems to demonstrate its capabilities. Numerical results demonstrate the strong performance of the proposed method in both OL and UQ tasks, achieving excellent numerical accuracy and computational efficiency.

stat.ML↗
arXiv2025

Accelerating Hamiltonian Monte Carlo for Bayesian Inference in Neural Networks and Neural Operators

Hamiltonian Monte Carlo (HMC) is a powerful and accurate method to sample from the posterior distribution in Bayesian inference. However, HMC techniques are computationally demanding for Bayesian neural networks due to the high dimensionality of the network's parameter space and the non-convexity of their posterior distributions. Therefore, various approximation techniques, such as variational inference (VI) or stochastic gradient MCMC, are often employed to infer the posterior distribution of the network parameters. Such approximations introduce inaccuracies in the inferred distributions, resulting in unreliable uncertainty estimates. In this work, we propose a hybrid approach that combines inexpensive VI and accurate HMC methods to efficiently and accurately quantify uncertainties in neural networks and neural operators. The proposed approach leverages an initial VI training on the full network. We examine the influence of individual parameters on the prediction uncertainty, which shows that a large proportion of the parameters do not contribute substantially to uncertainty in the network predictions. This information is then used to significantly reduce the dimension of the parameter space, and HMC is performed only for the subset of network parameters that strongly influence prediction uncertainties. This yields a framework for accelerating the full batch HMC for posterior inference in neural networks. We demonstrate the efficiency and accuracy of the proposed framework on deep neural networks and operator networks, showing that inference can be performed for large networks with tens to hundreds of thousands of parameters. We show that this method can effectively learn surrogates for complex physical systems by modeling the operator that maps from upstream conditions to wall-pressure data on a cone in hypersonic flow.

stat.ML↗
arXiv2025

Joint space-time wind field data extrapolation and uncertainty quantification using nonparametric Bayesian dictionary learning

A methodology is developed, based on nonparametric Bayesian dictionary learning, for joint space-time wind field data extrapolation and estimation of related statistics by relying on limited/incomplete measurements. Specifically, utilizing sparse/incomplete measured data, a time-dependent optimization problem is formulated for determining the expansion coefficients of an associated low-dimensional representation of the stochastic wind field. Compared to an alternative, standard, compressive sampling treatment of the problem, the developed methodology exhibits the following advantages. First, the Bayesian formulation enables also the quantification of the uncertainty in the estimates. Second, the requirement in standard CS-based applications for an a priori selection of the expansion basis is circumvented. Instead, this is done herein in an adaptive manner based on the acquired data. Overall, the methodology exhibits enhanced extrapolation accuracy, even in cases of high-dimensional data of arbitrary form, and of relatively large extrapolation distances. Thus, it can be used, potentially, in a wide range of wind engineering applications where various constraints dictate the use of a limited number of sensors. The efficacy of the methodology is demonstrated by considering two case studies. The first relates to the extrapolation of simulated wind velocity records consistent with a prescribed joint wavenumber-frequency power spectral density in a three-dimensional domain (2D and time). The second pertains to the extrapolation of four-dimensional (3D and time) boundary layer wind tunnel experimental data that exhibit significant spatial variability and non-Gaussian characteristics.

stat.ML↗
arXiv2025

The tail wags the distribution: Only sample the tails for efficient reliability analysis

To ensure that real-world infrastructure is safe and durable, systems are designed to not fail for any but the most rarely occurring parameter values. By only happening deep in the tails of the parameter distribution, failure probabilities are kept small. At the same time, it is essential to understand the risk associated with the failure of a system, no matter how unlikely. However, estimating such small failure probabilities is challenging; numerous system performance evaluations are necessary to produce even a single system state corresponding to failure, and each such evaluation is usually significantly computationally expensive. To alleviate this difficulty, we propose the Tail Stratified Sampling (TSS) estimator - an intuitive stratified sampling estimator for the failure probability that successively refines the tails of the system parameter distribution, enabling direct sampling of the tails, where failure is expected to occur. The most general construction of TSS is presented, highlighting its versatility and robustness for a variety of applications. The intuitions behind the formulation are explained, followed by a discussion of the theoretical and practical benefits of the method. Various details of the implementation are presented. The performance of the algorithm is then showcased through a host of analytical examples with varying failure domain geometries and failure probabilities as well as multiple numerical case studies of moderate and high dimensionality. To conclude, a qualitative comparison of TSS against the existing foundational variance-reduction methods for reliability analysis is presented, along with suggestions for future developments.

stat.ME↗
arXiv2025

Neural Operators for Stochastic Modeling of Nonlinear Structural System Response to Natural Hazards

Traditionally, neural networks have been employed to learn the mapping between finite-dimensional Euclidean spaces. However, recent research has opened up new horizons, focusing on the utilization of deep neural networks to learn operators capable of mapping infinite-dimensional function spaces. In this work, we employ two state-of-the-art neural operators, the deep operator network (DeepONet) and the Fourier neural operator (FNO) for the prediction of the nonlinear time history response of structural systems exposed to natural hazards, such as earthquakes and wind. Specifically, we propose two architectures, a self-adaptive FNO and a Fast Fourier Transform-based DeepONet (DeepFNOnet), where we employ a FNO beyond the DeepONet to learn the discrepancy between the ground truth and the solution predicted by the DeepONet. To demonstrate the efficiency and applicability of the architectures, two problems are considered. In the first, we use the proposed model to predict the seismic nonlinear dynamic response of a six-story shear building subject to stochastic ground motions. In the second problem, we employ the operators to predict the wind-induced nonlinear dynamic response of a high-rise building while explicitly accounting for the stochastic nature of the wind excitation. In both cases, the trained metamodels achieve high accuracy while being orders of magnitude faster than their corresponding high-fidelity models.

cs.LG↗
arXiv2024

Intrepid MCMC: Metropolis-Hastings with Exploration

In engineering examples, one often encounters the need to sample from unnormalized distributions with complex shapes that may also be implicitly defined through a physical or numerical simulation model, making it computationally expensive to evaluate the associated density function. For such cases, MCMC has proven to be an invaluable tool. Random-walk Metropolis Methods (also known as Metropolis-Hastings (MH)), in particular, are highly popular for their simplicity, flexibility, and ease of implementation. However, most MH algorithms suffer from significant limitations when attempting to sample from distributions with multiple modes (particularly disconnected ones). In this paper, we present Intrepid MCMC - a novel MH scheme that utilizes a simple coordinate transformation to significantly improve the mode-finding ability and convergence rate to the target distribution of random-walk Markov chains while retaining most of the simplicity of the vanilla MH paradigm. Through multiple examples, we showcase the improvement in the performance of Intrepid MCMC over vanilla MH for a wide variety of target distribution shapes. We also provide an analysis of the mixing behavior of the Intrepid Markov chain, as well as the efficiency of our algorithm for increasing dimensions. A thorough discussion is presented on the practical implementation of the Intrepid MCMC algorithm. Finally, its utility is highlighted through a Bayesian parameter inference problem for a two-degree-of-freedom oscillator under free vibration.

stat.ME↗
arXiv2024

Covariance-free Bi-fidelity Control Variates Importance Sampling for Rare Event Reliability Analysis

Multifidelity modeling has been steadily gaining attention as a tool to address the problem of exorbitant model evaluation costs that makes the estimation of failure probabilities a significant computational challenge for complex real-world problems, particularly when failure is a rare event. To implement multifidelity modeling, estimators that efficiently combine information from multiple models/sources are necessary. In past works, the variance reduction techniques of Control Variates (CV) and Importance Sampling (IS) have been leveraged for this task. In this paper, we present the CVIS framework; a creative take on a coupled CV and IS estimator for bifidelity reliability analysis. The framework addresses some of the practical challenges of the CV method by using an estimator for the control variate mean and side-stepping the need to estimate the covariance between the original estimator and the control variate through a clever choice for the tuning constant. The task of selecting an efficient IS distribution is also considered, with a view towards maximally leveraging the bifidelity structure and maintaining expressivity. Additionally, a diagnostic is provided that indicates both the efficiency of the algorithm as well as the relative predictive quality of the models utilized. Finally, the behavior and performance of the framework is explored through analytical and numerical examples.

stat.ME↗
arXiv2024

Stochastic evolution elasto-plastic modeling of a metallic glass

This paper develops a general data-driven approach to stochastic elastoplastic modelling that leverages atomistic simulation data directly rather than by fitting parameters. The approach is developed in the context of metallic glasses, which present inherent complexities due to their disordered structure. By harvesting statistics from simulated metallic glass shear response histories, the material state is mapped onto a two-dimensional state space consisting of the shear stress and the inelastic contribution to the potential energy. The resulting elastoplastic model is intrinsically stochastic and represented as a non-deterministic dynamical map. The state space statistics provide insights into the deformation physics of metallic glasses, revealing that two state variables are sufficient to describe the main features of the elastoplastic response. In this two-dimensional state space, the gradually quenched metallic glass rejuvenates during the initial quasi-elastic shearing, ultimately reaching a steady state that fluctuates about a fixed point in the state space as rejuvenation and aging balance.

cond-mat.stat-mech↗
arXiv2024

Synergistic Learning with Multi-Task DeepONet for Efficient PDE Problem Solving

Multi-task learning (MTL) is an inductive transfer mechanism designed to leverage useful information from multiple tasks to improve generalization performance compared to single-task learning. It has been extensively explored in traditional machine learning to address issues such as data sparsity and overfitting in neural networks. In this work, we apply MTL to problems in science and engineering governed by partial differential equations (PDEs). However, implementing MTL in this context is complex, as it requires task-specific modifications to accommodate various scenarios representing different physical processes. To this end, we present a multi-task deep operator network (MT-DeepONet) to learn solutions across various functional forms of source terms in a PDE and multiple geometries in a single concurrent training session. We introduce modifications in the branch network of the vanilla DeepONet to account for various functional forms of a parameterized coefficient in a PDE. Additionally, we handle parameterized geometries by introducing a binary mask in the branch network and incorporating it into the loss term to improve convergence and generalization to new geometry tasks. Our approach is demonstrated on three benchmark problems: (1) learning different functional forms of the source term in the Fisher equation; (2) learning multiple geometries in a 2D Darcy Flow problem and showcasing better transfer learning capabilities to new geometries; and (3) learning 3D parameterized geometries for a heat transfer problem and demonstrate the ability to predict on new but similar geometries. Our MT-DeepONet framework offers a novel approach to solving PDE problems in engineering and science under a unified umbrella based on synergistic learning that reduces the overall training cost for neural operators.

cs.LG↗
arXiv2024

A Resolution Independent Neural Operator

The Deep Operator Network (DeepONet) is a powerful neural operator architecture that uses two neural networks to map between infinite-dimensional function spaces. This architecture allows for the evaluation of the solution field at any location within the domain but requires input functions to be discretized at identical locations, limiting practical applications. We introduce a general framework for operator learning from input-output data with arbitrary sensor locations and counts. This begins by introducing a resolution-independent DeepONet (RI-DeepONet), which handles input functions discretized arbitrarily but sufficiently finely. To achieve this, we propose two dictionary learning algorithms that adaptively learn continuous basis functions, parameterized as implicit neural representations (INRs), from correlated signals on arbitrary point clouds. These basis functions project input function data onto a finite-dimensional embedding space, making it compatible with DeepONet without architectural changes. We specifically use sinusoidal representation networks (SIRENs) as trainable INR basis functions. Similarly, the dictionary learning algorithms identify basis functions for output data, defining a new neural operator architecture: the Resolution Independent Neural Operator (RINO). In RINO, the operator learning task reduces to mapping coefficients of input basis functions to output basis functions. We demonstrate RINO's robustness and applicability in handling arbitrarily sampled input and output functions during both training and inference through several numerical examples.

cs.LG↗
arXiv2024

Bayesian neural networks for predicting uncertainty in full-field material response

Stress and material deformation field predictions are among the most important tasks in computational mechanics. These predictions are typically made by solving the governing equations of continuum mechanics using finite element analysis, which can become computationally prohibitive considering complex microstructures and material behaviors. Machine learning (ML) methods offer potentially cost effective surrogates for these applications. However, existing ML surrogates are either limited to low-dimensional problems and/or do not provide uncertainty estimates in the predictions. This work proposes an ML surrogate framework for stress field prediction and uncertainty quantification for diverse materials microstructures. A modified Bayesian U-net architecture is employed to provide a data-driven image-to-image mapping from initial microstructure to stress field with prediction (epistemic) uncertainty estimates. The Bayesian posterior distributions for the U-net parameters are estimated using three state-of-the-art inference algorithms: the posterior sampling-based Hamiltonian Monte Carlo method and two variational approaches, the Monte-Carlo Dropout method and the Bayes by Backprop algorithm. A systematic comparison of the predictive accuracy and uncertainty estimates for these methods is performed for a fiber reinforced composite material and polycrystalline microstructure application. It is shown that the proposed methods yield predictions of high accuracy compared to the FEA solution, while uncertainty estimates depend on the inference approach. Generally, the Hamiltonian Monte Carlo and Bayes by Backprop methods provide consistent uncertainty estimates. Uncertainty estimates from Monte Carlo Dropout, on the other hand, are more difficult to interpret and depend strongly on the method's design.

stat.ML↗
arXiv2024

Physics-constrained polynomial chaos expansion for scientific machine learning and uncertainty quantification

We present a novel physics-constrained polynomial chaos expansion as a surrogate modeling method capable of performing both scientific machine learning (SciML) and uncertainty quantification (UQ) tasks. The proposed method possesses a unique capability: it seamlessly integrates SciML into UQ and vice versa, which allows it to quantify the uncertainties in SciML tasks effectively and leverage SciML for improved uncertainty assessment during UQ-related tasks. The proposed surrogate model can effectively incorporate a variety of physical constraints, such as governing partial differential equations (PDEs) with associated initial and boundary conditions constraints, inequality-type constraints (e.g., monotonicity, convexity, non-negativity, among others), and additional a priori information in the training process to supplement limited data. This ensures physically realistic predictions and significantly reduces the need for expensive computational model evaluations to train the surrogate model. Furthermore, the proposed method has a built-in uncertainty quantification (UQ) feature to efficiently estimate output uncertainties. To demonstrate the effectiveness of the proposed method, we apply it to a diverse set of problems, including linear/non-linear PDEs with deterministic and stochastic parameters, data-driven surrogate modeling of a complex physical system, and UQ of a stochastic system with parameters modeled as random fields.

stat.ML↗