Cosmin Safta

Reese Jones, Cosmin Safta, Ari Frankel

Many material response functions depend strongly on microstructure, such as inhomogeneities in phase or orientation. Homogenization presents the task of predicting the mean response of a sample of the microstructure to external loading for use in subgrid models and structure-property explorations. Although many microstructural fields have obvious segmentations, learning directly from the graph induced by the segmentation can be difficult because this representation does not encode all the information of the full field. We develop a means of deep learning of hidden features on the reduced graph given the native discretization and a segmentation of the initial input field. The features are associated with regions represented as nodes on the reduced graph. This reduced representation is then the basis for the subsequent multi-level/scale graph convolutional network model. There are a number of advantages of reducing the graph before fully processing with convolutional layers it, such as interpretable features and efficiency on large meshes. We demonstrate the performance of the proposed network relative to convolutional neural networks operating directly on the native discretization of the data using three physical exemplars.

Cosmin Safta, Habib N. Najm

This report examines numerical aspects of constructing Karhunen-Loève expansions (KLEs) for second-order stochastic processes. The KLE relies on the spectral decomposition of the covariance operator via the Fredholm integral equation of the second kind, which is then discretized on a computational grid, leading to an eigendecomposition task. We derive the algebraic equivalence between this Fredholm-based eigensolution and the singular value decomposition of the weight-scaled sample matrix, yielding consistent solutions for both model-based and data-driven KLE construction. Analytical eigensolutions for exponential and squared-exponential covariance kernels serve as reference benchmarks to assess numerical consistency and accuracy in 1D settings. The convergence of SVD-based eigenvalue estimates and of the empirical distributions of the KL coefficients to their theoretical $\mathcal{N}(0,1)$ target are characterized as a function of sample count. Higher-dimensional configurations include a two-dimensional irregular domain discretized by unstructured triangular meshes with two refinement levels, and a three-dimensional toroidal domain whose non-simply-connected topology motivates a comparison between Euclidean and shortest interior path distances between the grid points. The numerical results highlight the interplay between the discretization strategy, quadrature rule, and sample count, and their impact on the KLE results.

Christophe Bonneville, Nathan Bieberdorf, Pieterjan Robbe et al.

Phase-field simulations of liquid metal dealloying (LMD) can capture complex microstructural evolutions but can be prohibitively expensive for large domains and long time horizons. In this paper, we introduce a fully convolutional, conditionally parameterized U-Net surrogate designed to extrapolate far beyond its training data in both space and time. The architecture integrates convolutional self-attention, physically informed padding, and a flood-fill corrector method to maintain accuracy under extreme extrapolation, while conditioning on simulation parameters allows for flexible time-step skipping and adaptation to varying alloy compositions. To remove the need for costly solver-based initialization, we couple the surrogate with a conditional diffusion model that generates synthetic, physically consistent initial conditions. We train our surrogate on simulations generated over small domain sizes and short time spans, but, by taking advantage of the convolutional nature of U-Nets, we are able to run and extrapolate surrogate simulations for longer time horizons than what would be achievable with classic numerical solvers. Across multiple alloy compositions, the framework is able to reproduce the LMD physics accurately. It predicts key quantities of interest and spatial statistics with relative errors typically below 5% in the training regime and under 15% during large-scale, long time-horizon extrapolations. Our framework can also deliver speed-ups of up to 36,000 times, bringing the time to run weeks-long simulations down to a few seconds. This work is a first stepping stone towards high-fidelity extrapolation in both space and time of phase-field simulation for LMD.

Aniket Jivani, Cosmin Safta, Beckett Y. Zhou et al.

We present a bifidelity Karhunen-Loève expansion (KLE) surrogate model for field-valued quantities of interest (QoIs) under uncertain inputs. The approach combines the spectral efficiency of the KLE with polynomial chaos expansions (PCEs) to preserve an explicit mapping between input uncertainties and output fields. By coupling inexpensive low-fidelity (LF) simulations that capture dominant response trends with a limited number of high-fidelity (HF) simulations that correct for systematic bias, the proposed method enables accurate and computationally affordable surrogate construction. To further improve surrogate accuracy, we form an active learning strategy that adaptively selects new HF evaluations based on the surrogate's generalization error, estimated via cross-validation and modeled using Gaussian process regression. New HF samples are then acquired by maximizing an expected improvement criterion, targeting regions of high surrogate error. The resulting BF-KLE-AL framework is demonstrated on three examples of increasing complexity: a one-dimensional analytical benchmark, a two-dimensional convection-diffusion system, and a three-dimensional turbulent round jet simulation based on Reynolds-averaged Navier--Stokes (RANS) and enhanced delayed detached-eddy simulations (EDDES). Across these cases, the method achieves consistent improvements in predictive accuracy and sample efficiency relative to single-fidelity and random-sampling approaches.

Govinda Anantha Padmanabha, Cosmin Safta, Nikolaos Bouklas et al.

We propose a Stein variational gradient descent method to concurrently sparsify, train, and provide uncertainty quantification of a complexly parameterized model such as a neural network. It employs a graph reconciliation and condensation process to reduce complexity and increase similarity in the Stein ensemble of parameterizations. Therefore, the proposed condensed Stein variational gradient (cSVGD) method provides uncertainty quantification on parameters, not just outputs. Furthermore, the parameter reduction speeds up the convergence of the Stein gradient descent as it reduces the combinatorial complexity by aligning and differentiating the sensitivity to parameters. These properties are demonstrated with an illustrative example and an application to a representation problem in solid mechanics.

Jeremiah Hauth, Cosmin Safta, Xun Huan et al.

The application of neural network models to scientific machine learning tasks has proliferated in recent years. In particular, neural network models have proved to be adept at modeling processes with spatial-temporal complexity. Nevertheless, these highly parameterized models have garnered skepticism in their ability to produce outputs with quantified error bounds over the regimes of interest. Hence there is a need to find uncertainty quantification methods that are suitable for neural networks. In this work we present comparisons of the parametric uncertainty quantification of neural networks modeling complex spatial-temporal processes with Hamiltonian Monte Carlo and Stein variational gradient descent and its projected variant. Specifically we apply these methods to graph convolutional neural network models of evolving systems modeled with recurrent neural network and neural ordinary differential equations architectures. We show that Stein variational inference is a viable alternative to Monte Carlo methods with some clear advantages for complex neural network models. For our exemplars, Stein variational interference gave similar uncertainty profiles through time compared to Hamiltonian Monte Carlo, albeit with generally more generous variance.Projected Stein variational gradient descent also produced similar uncertainty profiles to the non-projected counterpart, but large reductions in the active weight space were confounded by the stability of the neural network predictions and the convoluted likelihood landscape.

Cosmin Safta, Reese E. Jones, Ravi G. Patel et al.

We propose a scalable, approximate inference hypernetwork framework for a general model of history-dependent processes. The flexible data model is based on a neural ordinary differential equation (NODE) representing the evolution of internal states together with a trainable observation model subcomponent. The posterior distribution corresponding to the data model parameters (weights and biases) follows a stochastic differential equation with a drift term related to the score of the posterior that is learned jointly with the data model parameters. This Langevin sampling approach offers flexibility in balancing the computational budget between the evaluation cost of the data model and the approximation of the posterior density of its parameters. We demonstrate performance of the ensemble sampling hypernetwork on chemical reaction and material physics data and compare it to standard variational inference.

Ravi Patel, Cosmin Safta, Reese E. Jones

Composite materials with different microstructural material symmetries are common in engineering applications where grain structure, alloying and particle/fiber packing are optimized via controlled manufacturing. In fact these microstructural tunings can be done throughout a part to achieve functional gradation and optimization at a structural level. To predict the performance of particular microstructural configuration and thereby overall performance, constitutive models of materials with microstructure are needed. In this work we provide neural network architectures that provide effective homogenization models of materials with anisotropic components. These models satisfy equivariance and material symmetry principles inherently through a combination of equivariant and tensor basis operations. We demonstrate them on datasets of stochastic volume elements with different textures and phases where the material undergoes elastic and plastic deformation, and show that the these network architectures provide significant performance improvements.

Wyatt Bridgman, Uma Balakrishnan, Reese Jones et al.

In the field of surrogate modeling, polynomial chaos expansion (PCE) allows practitioners to construct inexpensive yet accurate surrogates to be used in place of the expensive forward model simulations. For black-box simulations, non-intrusive PCE allows the construction of these surrogates using a set of simulation response evaluations. In this context, the PCE coefficients can be obtained using linear regression, which is also known as point collocation or stochastic response surfaces. Regression exhibits better scalability and can handle noisy function evaluations in contrast to other non-intrusive approaches, such as projection. However, since over-sampling is generally advisable for the linear regression approach, the simulation requirements become prohibitive for expensive forward models. We propose to leverage transfer learning whereby knowledge gained through similar PCE surrogate construction tasks (source domains) is transferred to a new surrogate-construction task (target domain) which has a limited number of forward model simulations (training data). The proposed transfer learning strategy determines how much, if any, information to transfer using new techniques inspired by Bayesian modeling and data assimilation. The strategy is scrutinized using numerical investigations and applied to an engineering problem from the oil and gas industry.

Christophe Bonneville, Nathan Bieberdorf, Pieterjan Robbe et al.

Phase-field models of liquid metal dealloying (LMD) can resolve rich microstructural dynamics but become intractable for large domains or long time horizons. We present a conditionally parameterized, fully convolutional U-Net surrogate that generalizes far beyond its training window in both space and time. The design integrates convolutional self-attention and physics-aware padding, while parameter conditioning enables variable time-step skipping and adaptation to diverse alloy systems. Although trained only on short, small-scale simulations, the surrogate exploits the translational invariance of convolutions to extend predictions to much longer horizons than traditional solvers. It accurately reproduces key LMD physics, with relative errors typically under 5% within the training regime and below 10% when extrapolating to larger domains and later times. The method accelerates computations by up to 16,000 times, cutting weeks of simulation down to seconds, and marks an early step toward scalable, high-fidelity extrapolation of LMD phase-field models.

Pieterjan Robbe, Andre Ruybalid, Arun Hegde et al.

Mechanistic microstructure-informed constitutive models for the mechanical response of polycrystals are a cornerstone of computational materials science. However, as these models become increasingly more complex - often involving coupled differential equations describing the effect of specific deformation modes - their associated computational costs can become prohibitive, particularly in optimization or uncertainty quantification tasks that require numerous model evaluations. To address this challenge, surrogate constitutive models that balance accuracy and computational efficiency are highly desirable. Data-driven surrogate models, that learn the constitutive relation directly from data, have emerged as a promising solution. In this work, we develop two local surrogate models for the viscoplastic response of a steel: a piecewise response surface method and a mixture of experts model. These surrogates are designed to adapt to complex material behavior, which may vary with material parameters or operating conditions. The surrogate constitutive models are applied to creep simulations of HT-9 steel, an alloy of considerable interest to the nuclear energy sector due to its high tolerance to radiation damage, using training data generated from viscoplastic self-consistent (VPSC) simulations. We define a set of test metrics to numerically assess the accuracy of our surrogate models for predicting viscoplastic material behavior, and show that the mixture of experts model outperforms the piecewise response surface method in terms of accuracy.

Connor Robertson, Cosmin Safta, Nicholson Collier et al.

Accurate calibration of stochastic agent-based models (ABMs) in epidemiology is crucial to make them useful in public health policy decisions and interventions. Traditional calibration methods, e.g., Markov Chain Monte Carlo (MCMC), that yield a probability density function for the parameters being calibrated, are often computationally expensive. When applied to ABMs which are highly parametrized, the calibration process becomes computationally infeasible. This paper investigates the utility of Stein Variational Inference (SVI) as an alternative calibration technique for stochastic epidemiological ABMs approximated by Gaussian process (GP) surrogates. SVI leverages gradient information to iteratively update a set of particles in the space of parameters being calibrated, offering potential advantages in scalability and efficiency for high-dimensional ABMs. The ensemble of particles yields a joint probability density function for the parameters and serves as the calibration. We compare the performance of SVI and MCMC in calibrating CityCOVID, a stochastic epidemiological ABM, focusing on predictive accuracy and calibration effectiveness. Our results demonstrate that SVI maintains predictive accuracy and calibration effectiveness comparable to MCMC, making it a viable alternative for complex epidemiological models. We also present the practical challenges of using a gradient-based calibration such as SVI which include careful tuning of hyperparameters and monitoring of the particle dynamics.

Govinda Anantha Padmanabha, Jan Niklas Fuhg, Cosmin Safta et al.

Most scientific machine learning (SciML) applications of neural networks involve hundreds to thousands of parameters, and hence, uncertainty quantification for such models is plagued by the curse of dimensionality. Using physical applications, we show that $L_0$ sparsification prior to Stein variational gradient descent ($L_0$+SVGD) is a more robust and efficient means of uncertainty quantification, in terms of computational cost and performance than the direct application of SGVD or projected SGVD methods. Specifically, $L_0$+SVGD demonstrates superior resilience to noise, the ability to perform well in extrapolated regions, and a faster convergence rate to an optimal solution.

Connor Robertson, Cosmin Safta, Nicholson Collier et al.

Agent-based models (ABM) provide an excellent framework for modeling outbreaks and interventions in epidemiology by explicitly accounting for diverse individual interactions and environments. However, these models are usually stochastic and highly parametrized, requiring precise calibration for predictive performance. When considering realistic numbers of agents and properly accounting for stochasticity, this high dimensional calibration can be computationally prohibitive. This paper presents a random forest based surrogate modeling technique to accelerate the evaluation of ABMs and demonstrates its use to calibrate an epidemiological ABM named CityCOVID via Markov chain Monte Carlo (MCMC). The technique is first outlined in the context of CityCOVID's quantities of interest, namely hospitalizations and deaths, by exploring dimensionality reduction via temporal decomposition with principal component analysis (PCA) and via sensitivity analysis. The calibration problem is then presented and samples are generated to best match COVID-19 hospitalization and death numbers in Chicago from March to June in 2020. These results are compared with previous approximate Bayesian calibration (IMABC) results and their predictive performance is analyzed showing improved performance with a reduction in computation.

Christophe Bonneville, Nathan Bieberdorf, Arun Hegde et al.

Prolonged contact between a corrosive liquid and metal alloys can cause progressive dealloying. For such liquid-metal dealloying (LMD) process, phase field models have been developed. However, the governing equations often involve coupled non-linear partial differential equations (PDE), which are challenging to solve numerically. In particular, stiffness in the PDEs requires an extremely small time steps (e.g. $10^{-12}$ or smaller). This computational bottleneck is especially problematic when running LMD simulation until a late time horizon is required. This motivates the development of surrogate models capable of leaping forward in time, by skipping several consecutive time steps at-once. In this paper, we propose U-Shaped Adaptive Fourier Neural Operators (U-AFNO), a machine learning (ML) model inspired by recent advances in neural operator learning. U-AFNO employs U-Nets for extracting and reconstructing local features within the physical fields, and passes the latent space through a vision transformer (ViT) implemented in the Fourier space (AFNO). We use U-AFNOs to learn the dynamics mapping the field at a current time step into a later time step. We also identify global quantities of interest (QoI) describing the corrosion process (e.g. the deformation of the liquid-metal interface) and show that our proposed U-AFNO model is able to accurately predict the field dynamics, in-spite of the chaotic nature of LMD. Our model reproduces the key micro-structure statistics and QoIs with a level of accuracy on-par with the high-fidelity numerical solver. We also investigate the opportunity of using hybrid simulations, in which we alternate forward leap in time using the U-AFNO with high-fidelity time stepping. We demonstrate that while advantageous for some surrogate model design choices, our proposed U-AFNO model in fully auto-regressive settings consistently outperforms hybrid schemes.

Ari Frankel, Cosmin Safta, Coleman Alleman et al.

Predicting the evolution of a representative sample of a material with microstructure is a fundamental problem in homogenization. In this work we propose a graph convolutional neural network that utilizes the discretized representation of the initial microstructure directly, without segmentation or clustering. Compared to feature-based and pixel-based convolutional neural network models, the proposed method has a number of advantages: (a) it is deep in that it does not require featurization but can benefit from it, (b) it has a simple implementation with standard convolutional filters and layers, (c) it works natively on unstructured and structured grid data without interpolation (unlike pixel-based convolutional neural networks), and (d) it preserves rotational invariance like other graph-based convolutional neural networks. We demonstrate the performance of the proposed network and compare it to traditional pixel-based convolution neural network models and feature-based graph convolutional neural networks on multiple large datasets.

Laura Swiler, Mamikon Gulian, Ari Frankel et al.

Gaussian process regression is a popular Bayesian framework for surrogate modeling of expensive data sources. As part of a broader effort in scientific machine learning, many recent works have incorporated physical constraints or other a priori information within Gaussian process regression to supplement limited data and regularize the behavior of the model. We provide an overview and survey of several classes of Gaussian process constraints, including positivity or bound constraints, monotonicity and convexity constraints, differential equation constraints provided by linear PDEs, and boundary condition constraints. We compare the strategies behind each approach as well as the differences in implementation, concluding with a discussion of the computational challenges introduced by constraints.

Panagiotis Tsilifis, Xun Huan, Cosmin Safta et al.

Basis adaptation in Homogeneous Chaos spaces rely on a suitable rotation of the underlying Gaussian germ. Several rotations have been proposed in the literature resulting in adaptations with different convergence properties. In this paper we present a new adaptation mechanism that builds on compressive sensing algorithms, resulting in a reduced polynomial chaos approximation with optimal sparsity. The developed adaptation algorithm consists of a two-step optimization procedure that computes the optimal coefficients and the input projection matrix of a low dimensional chaos expansion with respect to an optimally rotated basis. We demonstrate the attractive features of our algorithm through several numerical examples including the application on Large-Eddy Simulation (LES) calculations of turbulent combustion in a HIFiRE scramjet engine.