Designed especially for neurobiologists, FluoRender is an interactive tool for multi-channel fluorescence microscopy data visualization and analysis.
Deep brain stimulation
BrainStimulator is a set of networks that are used in SCIRun to perform simulations of brain stimulation such as transcranial direct current stimulation (tDCS) and magnetic transcranial stimulation (TMS).
Developing software tools for science has always been a central vision of the SCI Institute.

Scientific Computing

Numerical simulation of real-world phenomena provides fertile ground for building interdisciplinary relationships. The SCI Institute has a long tradition of building these relationships in a win-win fashion – a win for the theoretical and algorithmic development of numerical modeling and simulation techniques and a win for the discipline-specific science of interest. High-order and adaptive methods, uncertainty quantification, complexity analysis, and parallelization are just some of the topics being investigated by SCI faculty. These areas of computing are being applied to a wide variety of engineering applications ranging from fluid mechanics and solid mechanics to bioelectricity.


Martin Berzins

Parallel Computing

Mike Kirby

Finite Element Methods
Uncertainty Quantification

Valerio Pascucci

Scientific Data Management

Chris Johnson

Problem Solving Environments

Amir Arzani

Scientific machine learning
Data-driven fluid flow modeling

Funded Research Projects:

Publications in Scientific Computing:

Accelerating Data-Intensive Seismic Research Through Parallel Workflow Optimization and Federated Cyberinfrastructure
M. Adair, I. Rodero, M. Parashar, D. Melgar. In Proceedings of the SC '23 Workshops of The International Conference on High Performance Computing, Network, Storage, and Analysis, ACM, pp. 1970--1977. 2023.
DOI: 10.1145/3624062.3624276

Earthquake early warning systems use synthetic data from simulation frameworks like MudPy to train models for predicting the magnitudes of large earthquakes. MudPy, although powerful, has limitations: a lengthy simulation time to generate the required data, lack of user-friendliness, and no platform for discovering and sharing its data. We introduce FakeQuakes DAGMan Workflow (FDW), which utilizes Open Science Grid (OSG) for parallel computations to accelerate and streamline MudPy simulations. FDW significantly reduces runtime and increases throughput compared to a single-machine setup. Using FDW, we also explore partitioned parallel HTCondor DAGMan workflows to enhance OSG efficiency. Additionally, we investigate leveraging cyberinfrastructure, such as Virtual Data Collaboratory (VDC), for enhancing MudPy and OSG. Specifically, we simulate using Cloud bursting policies to enforce FDW job-offloading to VDC during OSG peak demand, addressing shared resource issues and user goals; we also discuss VDC’s value in facilitating a platform for broad access to MudPy products.

Deep neural operators as accurate surrogates for shape optimization
K. Shukla, V. Oommen, A. Peyvan, M. Penwarden, N. Plewacki, L. Bravo, A. Ghoshal, R.M. Kirby, G. Karniadakis. In Engineering Applications of Artificial Intelligence, Vol. 129, pp. 107615. 2023.
ISSN: 0952-1976

Deep neural operators, such as DeepONet, have changed the paradigm in high-dimensional nonlinear regression, paving the way for significant generalization and speed-up in computational engineering applications. Here, we investigate the use of DeepONet to infer flow fields around unseen airfoils with the aim of shape constrained optimization, an important design problem in aerodynamics that typically taxes computational resources heavily. We present results that display little to no degradation in prediction accuracy while reducing the online optimization cost by orders of magnitude. We consider NACA airfoils as a test case for our proposed approach, as the four-digit parameterization can easily define their shape. We successfully optimize the constrained NACA four-digit problem with respect to maximizing the lift-to-drag ratio and validate all results by comparing them to a high-order CFD solver. We find that DeepONets have a low generalization error, making them ideal for generating solutions of unseen shapes. Specifically, pressure, density, and velocity fields are accurately inferred at a fraction of a second, hence enabling the use of general objective functions beyond the maximization of the lift-to-drag ratio considered in the current work. Finally, we validate the ability of DeepONet to handle a complex 3D waverider geometry at hypersonic flight by inferring shear stress and heat flux distributions on its surface at unseen angles of attack. The main contribution of this paper is a modular integrated design framework that uses an over-parametrized neural operator as a surrogate model with good generalizability coupled seamlessly with multiple optimization solvers in a plug-and-play mode.

Streaming Factor Trajectory Learning for Temporal Tensor Decomposition
Subtitled “,” S. Fang, X. Yu, S. Li, Z. Wang, R. Kirby, S. Zhe. 2023.

Practical tensor data is often along with time information. Most existing temporal decomposition approaches estimate a set of fixed factors for the objects in each tensor mode, and hence cannot capture the temporal evolution of the objects' representation. More important, we lack an effective approach to capture such evolution from streaming data, which is common in real-world applications. To address these issues, we propose Streaming Factor Trajectory Learning for temporal tensor decomposition. We use Gaussian processes (GPs) to model the trajectory of factors so as to flexibly estimate their temporal evolution. To address the computational challenges in handling streaming data, we convert the GPs into a state-space prior by constructing an equivalent stochastic differential equation (SDE). We develop an efficient online filtering algorithm to estimate a decoupled running posterior of the involved factor states upon receiving new data. The decoupled estimation enables us to conduct standard Rauch-Tung-Striebel smoothing to compute the full posterior of all the trajectories in parallel, without the need for revisiting any previous data. We have shown the advantage of SFTL in both synthetic tasks and real-world applications.

Instance-wise Linearization of Neural Network for Model Interpretation
Subtitled “arXiv:2310.16295v1,” Z. Li, S. Liu, K. Bhavya, T. Bremer, V. Pascucci. 2023.

Neural network have achieved remarkable successes in many scientific fields. However, the interpretability of the neural network model is still a major bottlenecks to deploy such technique into our daily life. The challenge can dive into the non-linear behavior of the neural network, which rises a critical question that how a model use input feature to make a decision. The classical approach to address this challenge is feature attribution, which assigns an important score to each input feature and reveal its importance of current prediction. However, current feature attribution approaches often indicate the importance of each input feature without detail of how they are actually processed by a model internally. These attribution approaches often raise a concern that whether they highlight correct features for a model prediction.

For a neural network model, the non-linear behavior is often caused by non-linear activation units of a model. However, the computation behavior of a prediction from a neural network model is locally linear, because one prediction has only one activation pattern. Base on the observation, we propose an instance-wise linearization approach to reformulates the forward computation process of a neural network prediction. This approach reformulates different layers of convolution neural networks into linear matrix multiplication. Aggregating all layers' computation, a prediction complex convolution neural network operations can be described as a linear matrix multiplication F(x)=Wx+b. This equation can not only provides a feature attribution map that highlights the important of the input features but also tells how each input feature contributes to a prediction exactly. Furthermore, we discuss the application of this technique in both supervise classification and unsupervised neural network learning parametric t-SNE dimension reduction.

Strengthening and Democratizing Artificial Intelligence Research and Development
M. Parashar, T. deBlanc-Knowles, E. Gianchandani, L.E. Parker. In Computer, Vol. 56, No. 11, IEEE, pp. 85-90. 2023.
DOI: 10.1109/MC.2023.3284568

This article summarizes the vision, roadmap, and implementation plan for a National Artificial Intelligence Research Resource that aims to provide a widely accessible cyberinfrastructure for artificial intelligence R&D, with the overarching goal of bridging the resource–access divide.

Energy Stable and Structure-Preserving Schemes for the Stochastic Galerkin Shallow Water Equations
Subtitled “arXiv:2310.06229,” D. Dai, Y. Epshteyn, A. Narayan. 2023.

The shallow water flow model is widely used to describe water flows in rivers, lakes, and coastal areas. Accounting for uncertainty in the corresponding transport-dominated non-linear PDE models presents theoretical and numerical challenges that motivate the central advances of this paper. Starting with a spatially one-dimensional hyperbolicity-preserving, positivity-preserving stochastic Galerkin formulation of the parametric/uncertain shallow water equations, we derive an entropy-entropy flux pair for the system. We exploit this entropy-entropy flux pair to construct structure-preserving second-order energy conservative, and first- and second-order energy stable finite volume schemes for the stochastic Galerkin shallow water system. The performance of the methods is illustrated on several numerical experiments.

Equation Discovery with Bayesian Spike-and-Slab Priors and Efficient Kernels
Subtitled “arXiv:2310.05387v1,” D. Long, W.W. Xing, A.S. Krishnapriyan, R.M. Kirby, S. Zhe, M.W. Mahoney. 2023.

Discovering governing equations from data is important to many scientific and engineering applications. Despite promising successes, existing methods are still challenged by data sparsity as well as noise issues, both of which are ubiquitous in practice. Moreover, state-of-the-art methods lack uncertainty quantification and/or are costly in training. To overcome these limitations, we propose a novel equation discovery method based on Kernel learning and BAyesian Spike-and-Slab priors (KBASS). We use kernel regression to estimate the target function, which is flexible, expressive, and more robust to data sparsity and noises. We combine it with a Bayesian spike-and-slab prior — an ideal Bayesian sparse distribution — for effective operator selection and uncertainty quantification. We develop an expectation propagation expectation-maximization (EP-EM) algorithm for efficient posterior inference and function estimation. To overcome the computational challenge of kernel regression, we place the function values on a mesh and induce a Kronecker product construction, and we use tensor algebra methods to enable efficient computation and optimization. We show the significant advantages of KBASS on a list of benchmark ODE and PDE discovery tasks.

HiPPIS A High-Order Positivity-Preserving Mapping Software for Structured Meshes
Subtitled “arXiv:2310.08818,” T. A. J. Ouermi, R. M Kirby, M. Berzins. 2023.

Polynomial interpolation is an important component of many computational problems. In several of these computational problems, failure to preserve positivity when using polynomials to approximate or map data values between meshes can lead to negative unphysical quantities. Currently, most polynomial-based methods for enforcing positivity are based on splines and polynomial rescaling. The spline-based approaches build interpolants that are positive over the intervals in which they are defined and may require solving a minimization problem and/or system of equations. The linear polynomial rescaling methods allow for high-degree polynomials but enforce positivity only at limited locations (e.g., quadrature nodes). This work introduces open-source software (HiPPIS) for high-order data-bounded interpolation (DBI) and positivity-preserving interpolation (PPI) that addresses the limitations of both the spline and polynomial rescaling methods. HiPPIS is suitable for approximating and mapping physical quantities such as mass, density, and concentration between meshes while preserving positivity. This work provides Fortran and Matlab implementations of the DBI and PPI methods, presents an analysis of the mapping error in the context of PDEs, and uses several 1D and 2D numerical examples to demonstrate the benefits and limitations of HiPPIS.

Multi-Resolution Active Learning of Fourier Neural Operators
Subtitled “arXiv:2309.16971,” S. Li, X. Yu, W. Xing, R.M. Kirby, A. Narayan, S. Zhe. 2023.

Fourier Neural Operator (FNO) is a popular operator learning framework. It not only achieves the state-of-the-art performance in many tasks, but also is highly efficient in training and prediction. However, collecting training data for the FNO can be a costly bottleneck in practice, because it often demands expensive physical simulations. To overcome this problem, we propose Multi-Resolution Active learning of FNO (MRA-FNO), which can dynamically select the input functions and resolutions to lower the data cost as much as possible while optimizing the learning efficiency. Specifically, we propose a probabilistic multi-resolution FNO and use ensemble Monte-Carlo to develop an effective posterior inference algorithm. To conduct active learning, we maximize a utility-cost ratio as the acquisition function to acquire new examples and resolutions at each step. We use moment matching and the matrix determinant lemma to enable tractable, efficient utility computation. Furthermore, we develop a cost annealing framework to avoid over-penalizing high-resolution queries at the early stage. The over-penalization is severe when the cost difference is significant between the resolutions, which renders active learning often stuck at low-resolution queries and inferior performance. Our method overcomes this problem and applies to general multi-fidelity active learning and optimization problems. We have shown the advantage of our method in several benchmark operator learning tasks.

Toward Democratizing Access to Science Data: Introducing the National Data Platform,
M. Parashar, I. Altintas. In IEEE 19th International Conference on e-Science, IEEE, 2023.
DOI: 10.1109/e-Science58273.2023.10254930

Open and equitable access to scientific data is essential to addressing important scientific and societal grand challenges, and to research enterprise more broadly. This paper discusses the importance and urgency of open and equitable data access, and explores the barriers and challenges to such access. It then introduces the vision and architecture of the National Data Platform, a recently launched project aimed at catalyzing an open, equitable and extensible data ecosystem.

Computer Science Abstractions To Help Reason About Decentralized Stablecoin Design
B. Charoenwong, R.M. Kirby, J. Reiter. In IEEE Access, IEEE, 2023.

Computer science as a discipline is known for its penchant for using abstractions as a tool for reasoning. It is no surprise that computer science might have something valuable to lend to the world of decentralized stablecoin design, as it is in fact a “computing" problem. In this paper, we examine the possibility of a decentralized and capital-efficient stablecoin using smart contracts that algorithmically trade to maintain stability and study the potential new functionality that smart contracts enable. By exploiting traditional abstractions from computer science, we show that a capital-efficient algorithmic stablecoin cannot be provably stable. Additionally, we provide a formal exposition of the workings of Central Bank Digital Currencies, connecting this to the space of possible stablecoin designs. We then discuss several outstanding conjectures from both academics and practitioners and finally highlight the regulatory similarities between money-market funds and working stablecoins. Our work builds upon the current and growing interplay between the realms of engineering and financial services, and it also demonstrates how ways of thinking as a computer scientist can aid practitioners. We believe this research is vital for understanding and developing the future of financial technology.

Dynamic Data-Driven Application Systems for Reservoir Simulation-Based Optimization: Lessons Learned and Future Trends,
M. Parashar, T. Kurc, H. Klie, M.F. Wheeler, J.H. Saltz, M. Jammoul, R. Dong. In Handbook of Dynamic Data Driven Applications Systems: Volume 2, Springer International Publishing, pp. 287--330. 2023.
DOI: 10.1007/978-3-031-27986-7_11

Since its introduction in the early 2000s, the Dynamic Data-Driven Applications Systems (DDDAS) paradigm has served as a powerful concept for continuously improving the quality of both models and data embedded in complex dynamical systems. The DDDAS unifying concept enables capabilities to integrate multiple sources and scales of data, mathematical and statistical algorithms, advanced software infrastructures, and diverse applications into a dynamic feedback loop. DDDAS has not only motivated notable scientific and engineering advances on multiple fronts, but it has been also invigorated by the latest technological achievements in artificial intelligence, cloud computing, augmented reality, robotics, edge computing, Internet of Things (IoT), and Big Data. Capabilities to handle more data in a much faster and smarter fashion is paving the road for expanding automation capabilities. The purpose of this chapter is to review the fundamental components that have shaped reservoir-simulation-based optimization in the context of DDDAS. The foundations of each component will be systematically reviewed, followed by a discussion on current and future trends oriented to highlight the outstanding challenges and opportunities of reservoir management problems under the DDDAS paradigm. Moreover, this chapter should be viewed as providing pathways for establishing a synergy between renewable energy and oil and gas industry with the advent of the DDDAS method.

Strengthening the US Department of Energy's Recruitment Pipeline: The DOE/NNSA Predictive Science Academic Alliance Program (PSAAP) Experience
J. K. Holmen, V. G. Vergara Larrea, E. W. Draeger, E. T. Phipps, P. J. Smith, M. Berzins, S. T. Smith, J. N. Thornock, S. Parete-Koon. In Practice and Experience in Advanced Research Computing, ACM, pp. 137--144. 2023.

The US Department of Energy (DOE) oversees a system of 17 national laboratories responsible for developing unique scientific capabilities beyond the scope of academic and industrial institutions. These labs strive to keep America at the forefront of discovery and are home to some of the Nation’s best minds and the world’s best scientific and research facilities. Collaborations between national laboratories and academic institutions are critical to develop and recruit talent for the DOE workforce. Academia’s cooperative education model poses challenges for DOE recruitment pipelines centered around traditional internships. This paper discusses a promising DOE recruitment pipeline, the National Nuclear Security Administration’s (NNSA) Predictive Science Academic Alliance Program (PSAAP) initiative. As a part of this, experiences capturing the successes and challenges faced by the University of Utah’s Carbon Capture Multidisciplinary Simulation Center (CCMSC) through their participation in the PSAAP-II initiative are shared. These experiences demonstrate the success of Utah’s PSAAP center as a recruitment pipeline with approximately 43% of CCMSC students going to a national laboratory after graduation. Potential opportunities to strengthen the DOE’s recruitment pipeline are also discussed.

TEMA: Event Driven Serverless Workflows Platform for Natural Disaster Management
C. Sicari, A. Catalfamo, L. Carnevale, A. Galletta, D. Balouek-Thomert, M. Parashar, M. Villari. In 2023 IEEE Symposium on Computers and Communications (ISCC), pp. 1-6. 2023.
DOI: 10.1109/ISCC58397.2023.10217920

TEMA project is a Horizon Europe funded project that aims at addressing Natural Disaster Management by the use of sophisticated Cloud-Edge Continuum infrastructures by means of data analysis algorithms wrapped in Serverless functions deployed on a distributed infrastructure according to a Federated Learning scheduler that constantly monitors the infrastructure in search of the best way to satisfy required QoS constraints. In this paper, we discuss the advantages of Serverless workflow and how they can be used and monitored to natively trigger complex algorithm pipelines in the continuum, dynamically placing and relocating them taking into account incoming IoT data, QoS constraints, and the current status of the continuum infrastructure. Therefore we presented the Urgent Function Enabler (UFE) platform, a fully distributed architecture able to define, spread, and manage FaaS functions, using local IOT data managed using the Fiware ecosystem and a computing infrastructure composed of mobile and stable nodes.

Optimizing Data Movement for GPU-Based In-Situ Workflow Using GPUDirect RDMA,
B. Zhang, P.E. Davis, N. Morales, Z. Zhang, K. Teranishi, M. Parashar. In Euro-Par 2023: Parallel Processing, Springer Nature Switzerland, pp. 323--338. 2023.
ISBN: 978-3-031-39698-4
DOI: 10.1007/978-3-031-39698-4_22

The extreme-scale computing landscape is increasingly dominated by GPU-accelerated systems. At the same time, in-situ workflows that employ memory-to-memory inter-application data exchanges have emerged as an effective approach for leveraging these extreme-scale systems. In the case of GPUs, GPUDirect RDMA enables third-party devices, such as network interface cards, to access GPU memory directly and has been adopted for intra-application communications across GPUs. In this paper, we present an interoperable framework for GPU-based in-situ workflows that optimizes data movement using GPUDirect RDMA. Specifically, we analyze the characteristics of the possible data movement pathways between GPUs from an in-situ workflow perspective, and design a strategy that maximizes throughput. Furthermore, we implement this approach as an extension of the DataSpaces data staging service, and experimentally evaluate its performance and scalability on a current leadership GPU cluster. The performance results show that the proposed design reduces data-movement time by up to 53% and 40% for the sender and receiver, respectively, and maintains excellent scalability for up to 256 GPUs.

Studying Latency and Throughput Constraints for Geo-Distributed Data in the National Science Data Fabric
J. Luettgau, H. Martinez, G. Tarcea, G. Scorzelli, V. Pascucci, M. Taufer. In Proceedings of the 32nd International Symposium on High-Performance Parallel and Distributed Computing, ACM, pp. 325–326. 2023.
DOI: 10.1145/3588195.3595948

The National Science Data Fabric (NSDF) is our solution to the problem of addressing the data-sharing needs of the growing data science community. NSDF is designed to make sharing data across geographically distributed sites easier for users who lack technical expertise and infrastructure. By developing an easy-to-install software stack, we promote the FAIR data-sharing principles in NSDF while leveraging existing high-speed data transfer infrastructures such as Globus and XRootD. This work shows how we leverage latency and throughput information between geo-distributed NSDF sites with NSDF entry points to optimize the automatic coordination of data placement and transfer across the data fabric, which can further improve the efficiency of data sharing.

Multi-Omic Integration of Blood-Based Tumor-Associated Genomic and Lipidomic Profiles Using Machine Learning Models in Metastatic Prostate Cancer
S. Fang, S. Zhe, H.M. Lin, A.A. Azad, H. Fettke, E.M. Kwan, L. Horvath, B. Mak, T. Zheng, P. Du, S. Jia, R.M. Kirby, M. Kohli. In Clinical Cancer Informatics, 2023.

To determine prognostic and predictive clinical outcomes in metastatic hormone-sensitive prostate cancer (mHSPC) and metastatic castrate-resistant prostate cancer (mCRPC) on the basis of a combination of plasma-derived genomic alterations and lipid features in a longitudinal cohort of patients with advanced prostate cancer.

A multifeature classifier was constructed to predict clinical outcomes using plasma-based genomic alterations detected in 120 genes and 772 lipidomic species as informative features in a cohort of 71 patients with mHSPC and 144 patients with mCRPC. Outcomes of interest were collected over 11 years of follow-up. These included in mHSPC state early failure of androgen-deprivation therapy (ADT) and exceptional responders to ADT; early death (poor prognosis) and long-term survivors in mCRPC state. The approach was to build binary classification models that identified discriminative candidates with optimal weights to predict outcomes. To achieve this, we built multi-omic feature-based classifiers using traditional machine learning (ML) methods, including logistic regression with sparse regularization, multi-kernel Gaussian process regression, and support vector machines.

The levels of specific ceramides (d18:1/14:0 and d18:1/17:0), and the presence of CHEK2 mutations, AR amplification, and RB1 deletion were identified as the most crucial factors associated with clinical outcomes. Using ML models, the optimal multi-omics feature combination determined resulted in AUC scores of 0.751 for predicting mHSPC survival and 0.638 for predicting ADT failure; and in mCRPC state, 0.687 for prognostication and 0.727 for exceptional survival. The models were observed to be superior than using a limited candidate number of features for developing multi-omic prognostic and predictive signatures.

Using a ML approach that incorporates multiple omic features improves the prediction accuracy for metastatic prostate cancer outcomes significantly. Validation of these models will be needed in independent data sets in future.

On the Decentralized Stochastic Gradient Descent with Markov Chain Sampling
T. Sun, D. Li, B. Wang. In IEEE Transactions on Signal Processing, IEEE, July, 2023.

The decentralized stochastic gradient method emerges as a promising solution for solving large-scale machine learning problems. This paper studies the decentralized Markov chain gradient descent (DMGD), a variant of the decentralized stochastic gradient method, which draws random samples along the trajectory of a Markov chain. DMGD arises when obtaining independent samples is costly or impossible, excluding the use of the traditional stochastic gradient algorithms. Specifically, we consider the DMGD over a connected graph, where each node only communicates with its neighbors by sending and receiving the intermediate results. We establish both ergodic and nonergodic convergence rates of DMGD, which elucidate the critical dependencies on the topology of the graph that connects all nodes and the mixing time of the Markov chain. We further numerically verify the sample efficiency of DMGD.

Development of Large-Scale Scientific Cyberinfrastructure and the Growing Opportunity to Democratize Access to Platforms and Data,
J. Luettgau, G. Scorzelli, V. Pascucci, M. Taufer. In Distributed, Ambient and Pervasive Interactions, Springer Nature Switzerland, pp. 378--389. 2023.
ISBN: 978-3-031-34668-2
DOI: 10.1007/978-3-031-34668-2_25

As researchers across scientific domains rapidly adopt advanced scientific computing methodologies, access to advanced cyberinfrastructure (CI) becomes a critical requirement in scientific discovery. Lowering the entry barriers to CI is a crucial challenge in interdisciplinary sciences requiring frictionless software integration, data sharing from many distributed sites, and access to heterogeneous computing platforms. In this paper, we explore how the challenge is not merely a factor of availability and affordability of computing, network, and storage technologies but rather the result of insufficient interfaces with an increasingly heterogeneous mix of computing technologies and data sources. With more distributed computation and data, scientists, educators, and students must invest their time and effort in coordinating data access and movements, often penalizing their scientific research. Investments in the interfaces’ software stack are necessary to help scientists, educators, and students across domains take advantage of advanced computational methods. To this end, we propose developing a science data fabric as the standard scientific discovery interface that seamlessly manages data dependencies within scientific workflows and CI.

Interpreting and generalizing deep learning in physics-based problems with functional linear models
Subtitled “arXiv:2307.04569,” A. Arzani, L. Yuan, P. Newell, B. Wang. 2023.

Although deep learning has achieved remarkable success in various scientific machine learning applications, its black-box nature poses concerns regarding interpretability and generalization capabilities beyond the training data. Interpretability is crucial and often desired in modeling physical systems. Moreover, acquiring extensive datasets that encompass the entire range of input features is challenging in many physics-based learning tasks, leading to increased errors when encountering out-of-distribution (OOD) data. In this work, motivated by the field of functional data analysis (FDA), we propose generalized functional linear models as an interpretable surrogate for a trained deep learning model. We demonstrate that our model could be trained either based on a trained neural network (post-hoc interpretation) or directly from training data (interpretable operator learning). A library of generalized functional linear models with different kernel functions is considered and sparse regression is used to discover an interpretable surrogate model that could be analytically presented. We present test cases in solid mechanics, fluid mechanics, and transport. Our results demonstrate that our model can achieve comparable accuracy to deep learning and can improve OOD generalization while providing more transparency and interpretability. Our study underscores the significance of interpretability in scientific machine learning and showcases the potential of functional linear models as a tool for interpreting and generalizing deep learning.