Maruti Kumar Mudunuru

Maruti K. Mudunuru, Velimir V. Vesselinov, Bulbul Ahmmed

This paper presents a novel ML-based methodology for geothermal exploration towards PFA applications. Our methodology is provided through our open-source ML framework, GeoThermalCloud \url{https://github.com/SmartTensors/GeoThermalCloud.jl}. The GeoThermalCloud uses a series of unsupervised, supervised, and physics-informed ML methods available in SmartTensors AI platform \url{https://github.com/SmartTensors}. Here, the presented analyses are performed using our unsupervised ML algorithm called NMF$k$, which is available in the SmartTensors AI platform. Our ML algorithm facilitates the discovery of new phenomena, hidden patterns, and mechanisms that helps us to make informed decisions. Moreover, the GeoThermalCloud enhances the collected PFA data and discovers signatures representative of geothermal resources. Through GeoThermalCloud, we could identify hidden patterns in the geothermal field data needed to discover blind systems efficiently. Crucial geothermal signatures often overlooked in traditional PFA are extracted using the GeoThermalCloud and analyzed by the subject matter experts to provide ML-enhanced PFA, which is informative for efficient exploration. We applied our ML methodology to various open-source geothermal datasets within the U.S. (some of these are collected by past PFA work). The results provide valuable insights into resource types within those regions. This ML-enhanced workflow makes the GeoThermalCloud attractive for the geothermal community to improve existing datasets and extract valuable information often unnoticed during geothermal exploration.

M. K. Mudunuru, K. B. Nakshatrala

This paper concerns with mesh restrictions that are needed to satisfy several important mathematical properties -- maximum principles, comparison principles, and the non-negative constraint -- for a general linear second-order elliptic partial differential equation. We critically review some recent developments in the field of discrete maximum principles, derive new results, and discuss some possible future research directions in this area. In particular, we derive restrictions for a three-node triangular (T3) element and a four-node quadrilateral (Q4) element to satisfy comparison principles, maximum principles, and the non-negative constraint under the standard single-field Galerkin formulation. Analysis is restricted to uniformly elliptic linear differential operators in divergence form with Dirichlet boundary conditions specified on the entire boundary of the domain. Various versions of maximum principles and comparison principles are discussed in both continuous and discrete settings. In the literature, it is well-known that an acute-angled triangle is sufficient to satisfy the discrete weak maximum principle for pure isotropic diffusion. An iterative algorithm is developed to construct simplicial meshes that preserves discrete maximum principles using existing open source mesh generators. Various numerical examples based on different types of triangulations are presented to show the pros and cons of placing restrictions on a computational mesh. We also quantify local and global mass conservation errors using representative numerical examples, and illustrate the performance of metric-based meshes with respect to mass conservation.

M. K. Mudunuru, K. B. Nakshatrala

This paper deals with the formulation and numerical implementation of a fully coupled continuum model for deformation-diffusion in linearized elastic solids. The mathematical model takes into account the effect of the deformation on the diffusion process, and the affect of the transport of an inert chemical species on the deformation of the solid. We then present a robust computational framework for solving the proposed mathematical model, which consists of coupled non-linear partial differential equations. It should be noted that many popular numerical formulations may produce unphysical negative values for the concentration, particularly, when the diffusion process is anisotropic. The violation of the non-negative constraint by these numerical formulations is not mere numerical noise. In the proposed computational framework we employ a novel numerical formulation that will ensure that the concentration of the diffusant be always non-negative, which is one of the main contributions of this paper. Representative numerical examples are presented to show the robustness, convergence, and performance of the proposed computational framework. Another contribution of this paper is to systematically study the affect of transport of the diffusant on the deformation of the solid and vice-versa, and their implication in modeling degradation/healing of materials. We show that the coupled response is both qualitatively and quantitatively different from the uncoupled response.

Maruti K. Mudunuru, James A. Ang, Mahantesh Halappanavar et al.

Recently, the U.S. Department of Energy (DOE), Office of Science, Biological and Environmental Research (BER), and Advanced Scientific Computing Research (ASCR) programs organized and held the Artificial Intelligence for Earth System Predictability (AI4ESP) workshop series. From this workshop, a critical conclusion that the DOE BER and ASCR community came to is the requirement to develop a new paradigm for Earth system predictability focused on enabling artificial intelligence (AI) across the field, lab, modeling, and analysis activities, called ModEx. The BER's `Model-Experimentation', ModEx, is an iterative approach that enables process models to generate hypotheses. The developed hypotheses inform field and laboratory efforts to collect measurement and observation data, which are subsequently used to parameterize, drive, and test model (e.g., process-based) predictions. A total of 17 technical sessions were held in this AI4ESP workshop series. This paper discusses the topic of the `AI Architectures and Co-design' session and associated outcomes. The AI Architectures and Co-design session included two invited talks, two plenary discussion panels, and three breakout rooms that covered specific topics, including: (1) DOE HPC Systems, (2) Cloud HPC Systems, and (3) Edge computing and Internet of Things (IoT). We also provide forward-looking ideas and perspectives on potential research in this co-design area that can be achieved by synergies with the other 16 session topics. These ideas include topics such as: (1) reimagining co-design, (2) data acquisition to distribution, (3) heterogeneous HPC solutions for integration of AI/ML and other data analytics like uncertainty quantification with earth system modeling and simulation, and (4) AI-enabled sensor integration into earth system measurements and observations. Such perspectives are a distinguishing aspect of this paper.

M. K. Mudunuru, K. Son, P. Jiang et al.

Watershed models such as the Soil and Water Assessment Tool (SWAT) consist of high-dimensional physical and empirical parameters. These parameters need to be accurately calibrated for models to produce reliable predictions for streamflow, evapotranspiration, snow water equivalent, and nutrient loading. Existing parameter estimation methods are time-consuming, inefficient, and computationally intensive, with reduced accuracy when estimating high-dimensional parameters. In this paper, we present a fast, accurate, and reliable methodology to calibrate the SWAT model (i.e., 21 parameters) using deep learning (DL). We develop DL-enabled inverse models based on convolutional neural networks to ingest streamflow data and estimate the SWAT model parameters. Hyperparameter tuning is performed to identify the optimal neural network architecture and the nine next best candidates. We use ensemble SWAT simulations to train, validate, and test the above DL models. We estimated the actual parameters of the SWAT model using observational data. We test and validate the proposed DL methodology on the American River Watershed, located in the Pacific Northwest-based Yakima River basin. Our results show that the DL models-based calibration is better than traditional parameter estimation methods, such as generalized likelihood uncertainty estimation (GLUE). The behavioral parameter sets estimated by DL have narrower ranges than GLUE and produce values within the sampling range even under high relative observational errors. This narrow range of parameters shows the reliability of the proposed workflow to estimate sensitive parameters accurately even under noise. Due to its fast and reasonably accurate estimations of process parameters, the proposed DL workflow is attractive for calibrating integrated hydrologic models for large spatial-scale applications.

Satish Karra, Bulbul Ahmmed, Maruti K. Mudunuru

Physics-informed Machine Learning has recently become attractive for learning physical parameters and features from simulation and observation data. However, most existing methods do not ensure that the physics, such as balance laws (e.g., mass, momentum, energy conservation), are constrained. Some recent works (e.g., physics-informed neural networks) softly enforce physics constraints by including partial differential equation (PDE)-based loss functions but need re-discretization of the PDEs using auto-differentiation. Training these neural nets on observational data showed that one could solve forward and inverse problems in one shot. They evaluate the state variables and the parameters in a PDE. This re-discretization of PDEs is not necessarily an attractive option for domain scientists that work with physics-based codes that have been developed for decades with sophisticated discretization techniques to solve complex process models and advanced equations of state. This paper proposes a physics constrained machine learning framework, AdjointNet, allowing domain scientists to embed their physics code in neural network training workflows. This embedding ensures that physics is constrained everywhere in the domain. Additionally, the mathematical properties such as consistency, stability, and convergence vital to the numerical solution of a PDE are still satisfied. We show that the proposed AdjointNet framework can be used for parameter estimation (and uncertainty quantification by extension) and experimental design using active learning. The applicability of our framework is demonstrated for four flow cases. Results show that AdjointNet-based inversion can estimate process model parameters with reasonable accuracy. These examples demonstrate the applicability of using existing software with no changes in source code to perform accurate and reliable inversion of model parameters.

B. Ahmmed, M. K. Mudunuru, S. Karra et al.

Accurate predictions of reactive mixing are critical for many Earth and environmental science problems. To investigate mixing dynamics over time under different scenarios, a high-fidelity, finite-element-based numerical model is built to solve the fast, irreversible bimolecular reaction-diffusion equations to simulate a range of reactive-mixing scenarios. A total of 2,315 simulations are performed using different sets of model input parameters comprising various spatial scales of vortex structures in the velocity field, time-scales associated with velocity oscillations, the perturbation parameter for the vortex-based velocity, anisotropic dispersion contrast, and molecular diffusion. Outputs comprise concentration profiles of the reactants and products. The inputs and outputs of these simulations are concatenated into feature and label matrices, respectively, to train 20 different machine learning (ML) emulators to approximate system behavior. The 20 ML emulators based on linear methods, Bayesian methods, ensemble learning methods, and multilayer perceptron (MLP), are compared to assess these models. The ML emulators are specifically trained to classify the state of mixing and predict three quantities of interest (QoIs) characterizing species production, decay, and degree of mixing. Linear classifiers and regressors fail to reproduce the QoIs; however, ensemble methods (classifiers and regressors) and the MLP accurately classify the state of reactive mixing and the QoIs. Among ensemble methods, random forest and decision-tree-based AdaBoost faithfully predict the QoIs. At run time, trained ML emulators are $\approx10^5$ times faster than the high-fidelity numerical simulations. Speed and accuracy of the ensemble and MLP models facilitate uncertainty quantification, which usually requires 1,000s of model run, to estimate the uncertainty bounds on the QoIs.

M. K. Mudunuru, S. Karra

This paper presents a physics-informed machine learning (ML) framework to construct reduced-order models (ROMs) for reactive-transport quantities of interest (QoIs) based on high-fidelity numerical simulations. QoIs include species decay, product yield, and degree of mixing. The ROMs for QoIs are applied to quantify and understand how the chemical species evolve over time. First, high-resolution datasets for constructing ROMs are generated by solving anisotropic reaction-diffusion equations using a non-negative finite element formulation for different input parameters. Non-negative finite element formulation ensures that the species concentration is non-negative (which is needed for computing QoIs) on coarse computational grids even under high anisotropy. The reactive-mixing model input parameters are a time-scale associated with flipping of velocity, a spatial-scale controlling small/large vortex structures of velocity, a perturbation parameter of the vortex-based velocity, anisotropic dispersion strength/contrast, and molecular diffusion. Second, random forests, F-test, and mutual information criterion are used to evaluate the importance of model inputs/features with respect to QoIs. Third, Support Vector Machines (SVM) and Support Vector Regression (SVR) are used to construct ROMs based on the model inputs. Then, SVR-ROMs are used to predict scaling of QoIs. Qualitatively, SVR-ROMs are able to describe the trends observed in the scaling law associated with QoIs. Fourth, the scaling law's exponent dependence on model inputs/features are evaluated using $k$-means clustering. Finally, in terms of the computational cost, the proposed SVM-ROMs and SVR-ROMs are $\mathcal{O}(10^7)$ times faster than running a high-fidelity numerical simulation for evaluating QoIs.

B. Yuan, Y. J. Tan, M. K. Mudunuru et al.

We present an approach based on machine learning (ML) to distinguish eruption and precursory signals of Chimayó geyser (New Mexico, USA) under noisy environments. This geyser can be considered as a natural analog of $\mathrm{CO}_2$ intrusion into shallow water aquifers. By studying this geyser, we can understand upwelling of $\mathrm{CO}_2$-rich fluids from depth, which has relevance to leak monitoring in a $\mathrm{CO}_2$ sequestration project. ML methods such as Random Forests (RF) are known to be robust multi-class classifiers and perform well under unfavorable noisy conditions. However, the extent of the RF method's accuracy is poorly understood for this $\mathrm{CO}_2$-driven geysering application. The current study aims to quantify the performance of RF-classifiers to discern the geyser state. Towards this goal, we first present the data collected from the seismometer that is installed near the Chimayó geyser. The seismic signals collected at this site contain different types of noises such as daily temperature variations, seasonal trends, animal movement near the geyser, and human activity. First, we filter the signals from these noises by combining the Butterworth-Highpass filter and an Autoregressive method in a multi-level fashion. We show that by combining these filtering techniques, in a hierarchical fashion, leads to reduction in the noise in the seismic data without removing the precursors and eruption event signals. We then use RF on the filtered data to classify the state of geyser into three classes -- remnant noise, precursor, and eruption states. We show that the classification accuracy using RF on the filtered data is greater than 90\%.These aspects make the proposed ML framework attractive for event discrimination and signal enhancement under noisy conditions, with strong potential for application to monitoring leaks in $\mathrm{CO}_2$ sequestration.

M. K. Mudunuru, N. Panda, S. Karra et al.

In brittle fracture applications, failure paths, regions where the failure occurs and damage statistics, are some of the key quantities of interest (QoI). High-fidelity models for brittle failure that accurately predict these QoI exist but are highly computationally intensive, making them infeasible to incorporate in upscaling and uncertainty quantification frameworks. The goal of this paper is to provide a fast heuristic to reasonably estimate quantities such as failure path and damage in the process of brittle failure. Towards this goal, we first present a method to predict failure paths under tensile loading conditions and low-strain rates. The method uses a $k$-nearest neighbors algorithm built on fracture process zone theory, and identifies the set of all possible pre-existing cracks that are likely to join early to form a large crack. The method then identifies zone of failure and failure paths using weighted graphs algorithms. We compare these failure paths to those computed with a high-fidelity model called the Hybrid Optimization Software Simulation Suite (HOSS). A probabilistic evolution model for average damage in a system is also developed that is trained using 150 HOSS simulations and tested on 40 simulations. A non-parametric approach based on confidence intervals is used to determine the damage evolution over time along the dominant failure path. For upscaling, damage is the key QoI needed as an input by the continuum models. This needs to be informed accurately by the surrogate models for calculating effective modulii at continuum-scale. We show that for the proposed average damage evolution model, the prediction accuracy on the test data is more than 90\%. In terms of the computational time, the proposed models are $\approx \mathcal{O}(10^6)$ times faster compared to high-fidelity HOSS.

A. Hunter, B. A. Moore, M. K. Mudunuru et al.

In this paper, five different approaches for reduced-order modeling of brittle fracture in geomaterials, specifically concrete, are presented and compared. Four of the five methods rely on machine learning (ML) algorithms to approximate important aspects of the brittle fracture problem. In addition to the ML algorithms, each method incorporates different physics-based assumptions in order to reduce the computational complexity while maintaining the physics as much as possible. This work specifically focuses on using the ML approaches to model a 2D concrete sample under low strain rate pure tensile loading conditions with 20 preexisting cracks present. A high-fidelity finite element-discrete element model is used to both produce a training dataset of 150 simulations and an additional 35 simulations for validation. Results from the ML approaches are directly compared against the results from the high-fidelity model. Strengths and weaknesses of each approach are discussed and the most important conclusion is that a combination of physics-informed and data-driven features are necessary for emulating the physics of crack propagation, interaction and coalescence. All of the models presented here have runtimes that are orders of magnitude faster than the original high-fidelity model and pave the path for developing accurate reduced order models that could be used to inform larger length-scale models with important sub-scale physics that often cannot be accounted for due to computational cost.