Alexandre M. Tartakovsky

Chaopeng Shen, Alison P. Appling, Pierre Gentine et al.

Process-Based Modeling (PBM) and Machine Learning (ML) are often perceived as distinct paradigms in the geosciences. Here we present differentiable geoscientific modeling as a powerful pathway toward dissolving the perceived barrier between them and ushering in a paradigm shift. For decades, PBM offered benefits in interpretability and physical consistency but struggled to efficiently leverage large datasets. ML methods, especially deep networks, presented strong predictive skills yet lacked the ability to answer specific scientific questions. While various methods have been proposed for ML-physics integration, an important underlying theme -- differentiable modeling -- is not sufficiently recognized. Here we outline the concepts, applicability, and significance of differentiable geoscientific modeling (DG). "Differentiable" refers to accurately and efficiently calculating gradients with respect to model variables, critically enabling the learning of high-dimensional unknown relationships. DG refers to a range of methods connecting varying amounts of prior knowledge to neural networks and training them together, capturing a different scope than physics-guided machine learning and emphasizing first principles. Preliminary evidence suggests DG offers better interpretability and causality than ML, improved generalizability and extrapolation capability, and strong potential for knowledge discovery, while approaching the performance of purely data-driven ML. DG models require less training data while scaling favorably in performance and efficiency with increasing amounts of data. With DG, geoscientists may be better able to frame and investigate questions, test hypotheses, and discover unrecognized linkages.

Yifei Zong, QiZhi He, Alexandre M. Tartakovsky

In this paper, we develop a physics-informed neural network (PINN) model for parabolic problems with a sharply perturbed initial condition. As an example of a parabolic problem, we consider the advection-dispersion equation (ADE) with a point (Gaussian) source initial condition. In the $d$-dimensional ADE, perturbations in the initial condition decay with time $t$ as $t^{-d/2}$, which can cause a large approximation error in the PINN solution. Localized large gradients in the ADE solution make the (common in PINN) Latin hypercube sampling of the equation's residual highly inefficient. Finally, the PINN solution of parabolic equations is sensitive to the choice of weights in the loss function. We propose a normalized form of ADE where the initial perturbation of the solution does not decrease in amplitude and demonstrate that this normalization significantly reduces the PINN approximation error. We propose criteria for weights in the loss function that produce a more accurate PINN solution than those obtained with the weights selected via other methods. Finally, we proposed an adaptive sampling scheme that significantly reduces the PINN solution error for the same number of the sampling (residual) points. We demonstrate the accuracy of the proposed PINN model for forward, inverse, and backward ADEs.

QiZhi He, Yucheng Fu, Panos Stinis et al.

Numerical modeling and simulation have become indispensable tools for advancing a comprehensive understanding of the underlying mechanisms and cost-effective process optimization and control of flow batteries. In this study, we propose an enhanced version of the physics-constrained deep neural network (PCDNN) approach [1] to provide high-accuracy voltage predictions in the vanadium redox flow batteries (VRFBs). The purpose of the PCDNN approach is to enforce the physics-based zero-dimensional (0D) VRFB model in a neural network to assure model generalization for various battery operation conditions. Limited by the simplifications of the 0D model, the PCDNN cannot capture sharp voltage changes in the extreme SOC regions. To improve the accuracy of voltage prediction at extreme ranges, we introduce a second (enhanced) DNN to mitigate the prediction errors carried from the 0D model itself and call the resulting approach enhanced PCDNN (ePCDNN). By comparing the model prediction with experimental data, we demonstrate that the ePCDNN approach can accurately capture the voltage response throughout the charge--discharge cycle, including the tail region of the voltage discharge curve. Compared to the standard PCDNN, the prediction accuracy of the ePCDNN is significantly improved. The loss function for training the ePCDNN is designed to be flexible by adjusting the weights of the physics-constrained DNN and the enhanced DNN. This allows the ePCDNN framework to be transferable to battery systems with variable physical model fidelity.

Yifei Zong, David Barajas-Solano, Alexandre M. Tartakovsky

We propose a randomized physics-informed neural network (PINN) or rPINN method for uncertainty quantification in inverse partial differential equation (PDE) problems with noisy data. This method is used to quantify uncertainty in the inverse PDE PINN solutions. Recently, the Bayesian PINN (BPINN) method was proposed, where the posterior distribution of the PINN parameters was formulated using the Bayes' theorem and sampled using approximate inference methods such as the Hamiltonian Monte Carlo (HMC) and variational inference (VI) methods. In this work, we demonstrate that HMC fails to converge for non-linear inverse PDE problems. As an alternative to HMC, we sample the distribution by solving the stochastic optimization problem obtained by randomizing the PINN loss function. The effectiveness of the rPINN method is tested for linear and non-linear Poisson equations, and the diffusion equation with a high-dimensional space-dependent diffusion coefficient. The rPINN method provides informative distributions for all considered problems. For the linear Poisson equation, HMC and rPINN produce similar distributions, but rPINN is on average 27 times faster than HMC. For the non-linear Poison and diffusion equations, the HMC method fails to converge because a single HMC chain cannot sample multiple modes of the posterior distribution of the PINN parameters in a reasonable amount of time.

Yuanzhe Wang, Alexandre M. Tartakovsky

We propose an approximate Bayesian method for quantifying the total uncertainty in inverse PDE solutions obtained with machine learning surrogate models, including operator learning models. The proposed method accounts for uncertainty in the observations and PDE and surrogate models. First, we use the surrogate model to formulate a minimization problem in the reduced space for the maximum a posteriori (MAP) inverse solution. Then, we randomize the MAP objective function and obtain samples of the posterior distribution by minimizing different realizations of the objective function. We test the proposed framework by comparing it with the iterative ensemble smoother and deep ensembling methods for a non-linear diffusion equation with an unknown space-dependent diffusion coefficient. Among other problems, this equation describes groundwater flow in an unconfined aquifer. Depending on the training dataset and ensemble sizes, the proposed method provides similar or more descriptive posteriors of the parameters and states than the iterative ensemble smoother method. Deep ensembling underestimates uncertainty and provides less informative posteriors than the other two methods.

Yu-Hong Yeung, David A. Barajas-Solano, Alexandre M. Tartakovsky

We present a model inversion algorithm, CKLEMAP, for data assimilation and parameter estimation in partial differential equation models of physical systems with spatially heterogeneous parameter fields. These fields are approximated using low-dimensional conditional Karhunen-Loéve expansions, which are constructed using Gaussian process regression models of these fields trained on the parameters' measurements. We then assimilate measurements of the state of the system and compute the maximum a posteriori estimate of the CKLE coefficients by solving a nonlinear least-squares problem. When solving this optimization problem, we efficiently compute the Jacobian of the vector objective by exploiting the sparsity structure of the linear system of equations associated with the forward solution of the physics problem. The CKLEMAP method provides better scalability compared to the standard MAP method. In the MAP method, the number of unknowns to be estimated is equal to the number of elements in the numerical forward model. On the other hand, in CKLEMAP, the number of unknowns (CKLE coefficients) is controlled by the smoothness of the parameter field and the number of measurements, and is in general much smaller than the number of discretization nodes, which leads to a significant reduction of computational cost with respect to the standard MAP method. To show its advantage in scalability, we apply CKLEMAP to estimate the transmissivity field in a two-dimensional steady-state subsurface flow model of the Hanford Site by assimilating synthetic measurements of transmissivity and hydraulic head. We find that the execution time of CKLEMAP scales nearly linearly as $N^{1.33}$, where $N$ is the number of discretization nodes, while the execution time of standard MAP scales as $N^{2.91}$. The CKLEMAP method improved execution time without sacrificing accuracy when compared to the standard MAP.

Yu-Hong Yeung, Ramakrishna Tipireddy, David A. Barajas-Solano et al.

We propose a methodology for improving the accuracy of surrogate models of the observable response of physical systems as a function of the systems' spatially heterogeneous parameter fields with applications to uncertainty quantification and parameter estimation in high-dimensional problems. Practitioners often formulate finite-dimensional representations of spatially heterogeneous parameter fields using truncated unconditional Karhunen-Loéve expansions (KLEs) for a certain choice of unconditional covariance kernel and construct surrogate models of the observable response with respect to the random variables in the KLE. When direct measurements of the parameter fields are available, we propose improving the accuracy of these surrogate models by representing the parameter fields via conditional Karhunen-Loéve expansions (CKLEs). CKLEs are constructed by conditioning the covariance kernel of the unconditional expansion on the direct measurements via Gaussian process regression and then truncating the corresponding KLE. We apply the proposed methodology to constructing surrogate models via the Basis Adaptation (BA) method of the stationary hydraulic head response, measured at spatially discrete observation locations, of a groundwater flow model of the Hanford Site, as a function of the 1,000-dimensional representation of the model's log-transmissivity field. We find that BA surrogate models of the hydraulic head based on CKLEs are more accurate than BA surrogate models based on unconditional expansions for forward uncertainty quantification tasks. Furthermore, we find that inverse estimates of the hydraulic transmissivity field computed using CKLE-based BA surrogate models are more accurate than those computed using unconditional BA surrogate models.

Yifei Zong, David Barajas-Solano, Alexandre M. Tartakovsky

We propose a physics-informed machine learning method for uncertainty quantification in high-dimensional inverse problems. In this method, the states and parameters of partial differential equations (PDEs) are approximated with truncated conditional Karhunen-Loève expansions (CKLEs), which, by construction, match the measurements of the respective variables. The maximum a posteriori (MAP) solution of the inverse problem is formulated as a minimization problem over CKLE coefficients where the loss function is the sum of the norm of PDE residuals and the $\ell_2$ regularization term. This MAP formulation is known as the physics-informed CKLE (PICKLE) method. Uncertainty in the inverse solution is quantified in terms of the posterior distribution of CKLE coefficients, and we sample the posterior by solving a randomized PICKLE minimization problem, formulated by adding zero-mean Gaussian perturbations in the PICKLE loss function. We call the proposed approach the randomized PICKLE (rPICKLE) method. For linear and low-dimensional nonlinear problems (15 CKLE parameters), we show analytically and through comparison with Hamiltonian Monte Carlo (HMC) that the rPICKLE posterior converges to the true posterior given by the Bayes rule. For high-dimensional non-linear problems with 2000 CKLE parameters, we numerically demonstrate that rPICKLE posteriors are highly informative--they provide mean estimates with an accuracy comparable to the estimates given by the MAP solution and the confidence interval that mostly covers the reference solution. We are not able to obtain the HMC posterior to validate rPICKLE's convergence to the true posterior due to the HMC's prohibitive computational cost for the considered high-dimensional problems. Our results demonstrate the advantages of rPICKLE over HMC for approximately sampling high-dimensional posterior distributions subject to physics constraints.

Jice Zeng, Yuanzhe Wang, Alexandre M. Tartakovsky et al.

We present a likelihood-free probabilistic inversion method based on normalizing flows for high-dimensional inverse problems. The proposed method is composed of two complementary networks: a summary network for data compression and an inference network for parameter estimation. The summary network encodes raw observations into a fixed-size vector of summary features, while the inference network generates samples of the approximate posterior distribution of the model parameters based on these summary features. The posterior samples are produced in a deep generative fashion by sampling from a latent Gaussian distribution and passing these samples through an invertible transformation. We construct this invertible transformation by sequentially alternating conditional invertible neural network and conditional neural spline flow layers. The summary and inference networks are trained simultaneously. We apply the proposed method to an inversion problem in groundwater hydrology to estimate the posterior distribution of the log-conductivity field conditioned on spatially sparse time-series observations of the system's hydraulic head responses.The conductivity field is represented with 706 degrees of freedom in the considered problem.The comparison with the likelihood-based iterative ensemble smoother PEST-IES method demonstrates that the proposed method accurately estimates the parameter posterior distribution and the observations' predictive posterior distribution at a fraction of the inference time of PEST-IES.

Marta D'Elia, Hang Deng, Cedric Fraces et al.

The "Workshop on Machine learning in heterogeneous porous materials" brought together international scientific communities of applied mathematics, porous media, and material sciences with experts in the areas of heterogeneous materials, machine learning (ML) and applied mathematics to identify how ML can advance materials research. Within the scope of ML and materials research, the goal of the workshop was to discuss the state-of-the-art in each community, promote crosstalk and accelerate multi-disciplinary collaborative research, and identify challenges and opportunities. As the end result, four topic areas were identified: ML in predicting materials properties, and discovery and design of novel materials, ML in porous and fractured media and time-dependent phenomena, Multi-scale modeling in heterogeneous porous materials via ML, and Discovery of materials constitutive laws and new governing equations. This workshop was part of the AmeriMech Symposium series sponsored by the National Academies of Sciences, Engineering and Medicine and the U.S. National Committee on Theoretical and Applied Mechanics.

Yu-Hong Yeung, David A. Barajas-Solano, Alexandre M. Tartakovsky

We develop a physics-informed machine learning approach for large-scale data assimilation and parameter estimation and apply it for estimating transmissivity and hydraulic head in the two-dimensional steady-state subsurface flow model of the Hanford Site given synthetic measurements of said variables. In our approach, we extend the physics-informed conditional Karhunen-Loéve expansion (PICKLE) method for modeling subsurface flow with unknown flux (Neumann) and varying head (Dirichlet) boundary conditions. We demonstrate that the PICKLE method is comparable in accuracy with the standard maximum a posteriori (MAP) method, but is significantly faster than MAP for large-scale problems. Both methods use a mesh to discretize the computational domain. In MAP, the parameters and states are discretized on the mesh; therefore, the size of the MAP parameter estimation problem directly depends on the mesh size. In PICKLE, the mesh is used to evaluate the residuals of the governing equation, while the parameters and states are approximated by the truncated conditional Karhunen-Loéve expansions with the number of parameters controlled by the smoothness of the parameter and state fields, and not by the mesh size. For a considered example, we demonstrate that the computational cost of PICKLE increases near linearly (as $N_{FV}^{1.15}$) with the number of grid points $N_{FV}$, while that of MAP increases much faster as $N_{FV}^{3.28}$. We demonstrated that once trained for one set of Dirichlet boundary conditions (i.e., one river stage), the PICKLE method provides accurate estimates of the hydraulic head for any value of the Dirichlet boundary conditions (i.e., for any river stage).

QiZhi He, Panos Stinis, Alexandre Tartakovsky

In this paper, we present a physics-constrained deep neural network (PCDNN) method for parameter estimation in the zero-dimensional (0D) model of the vanadium redox flow battery (VRFB). In this approach, we use deep neural networks (DNNs) to approximate the model parameters as functions of the operating conditions. This method allows the integration of the VRFB computational models as the physical constraints in the parameter learning process, leading to enhanced accuracy of parameter estimation and cell voltage prediction. Using an experimental dataset, we demonstrate that the PCDNN method can estimate model parameters for a range of operating conditions and improve the 0D model prediction of voltage compared to the 0D model prediction with constant operation-condition-independent parameters estimated with traditional inverse methods. We also demonstrate that the PCDNN approach has an improved generalization ability for estimating parameter values for operating conditions not used in the DNN training.

Amanda A. Howard, Alexandre M. Tartakovsky

Redox flow batteries (RFBs) offer the capability to store large amounts of energy cheaply and efficiently, however, there is a need for fast and accurate models of the charge-discharge curve of a RFB to potentially improve the battery capacity and performance. We develop a multifidelity model for predicting the charge-discharge curve of a RFB. In the multifidelity model, we use the Physics-informed CoKriging (CoPhIK) machine learning method that is trained on experimental data and constrained by the so-called "zero-dimensional" physics-based model. Here we demonstrate that the model shows good agreement with experimental results and significant improvements over existing zero-dimensional models. We show that the proposed model is robust as it is not sensitive to the input parameters in the zero-dimensional model. We also show that only a small amount of high-fidelity experimental datasets are needed for accurate predictions for the range of considered input parameters, which include current density, flow rate, and initial concentrations.

Tong Ma, David Alonso Barajas-Solano, Ramakrishna Tipireddy et al.

Real-time state estimation and forecasting is critical for efficient operation of power grids. In this paper, a physics-informed Gaussian process regression (PhI-GPR) method is presented and used for probabilistic forecasting and estimating the phase angle, angular speed, and wind mechanical power of a three-generator power grid system using sparse measurements. In standard data-driven Gaussian process regression (GPR), parameterized models for the prior statistics are fit by maximizing the marginal likelihood of observed data, whereas in PhI-GPR, we compute the prior statistics by solving stochastic equations governing power grid dynamics. The short-term forecast of a power grid system dominated by wind generation is complicated by the stochastic nature of the wind and the resulting uncertain mechanical wind power. Here, we assume that the power-grid dynamic is governed by the swing equations, and we treat the unknown terms in the swing equations (specifically, the mechanical wind power) as random processes, which turns these equations into stochastic differential equations. We solve these equations for the mean and variance of the power grid system using the Monte Carlo simulations method. We demonstrate that the proposed PhI-GPR method can accurately forecast and estimate both observed and unobserved states, including the mean behavior and associated uncertainty. For observed states, we show that PhI-GPR provides a forecast comparable to the standard data-driven GPR, with both forecasts being significantly more accurate than the autoregressive integrated moving average (ARIMA) forecast. We also show that the ARIMA forecast is much more sensitive to observation frequency and measurement errors than the PhI-GPR forecast.

Daniel Dylewsky, David Barajas-Solano, Tong Ma et al.

Time series forecasting remains a central challenge problem in almost all scientific disciplines. We introduce a novel load forecasting method in which observed dynamics are modeled as a forced linear system using Dynamic Mode Decomposition (DMD) in time delay coordinates. Central to this approach is the insight that grid load, like many observables on complex real-world systems, has an "almost-periodic" character, i.e., a continuous Fourier spectrum punctuated by dominant peaks, which capture regular (e.g., daily or weekly) recurrences in the dynamics. The forecasting method presented takes advantage of this property by (i) regressing to a deterministic linear model whose eigenspectrum maps onto those peaks, and (ii) simultaneously learning a stochastic Gaussian process regression (GPR) process to actuate this system. Our forecasting algorithm is compared against state-of-the-art forecasting techniques not using additional explanatory variables and is shown to produce superior performance. Moreover, its use of linear intrinsic dynamics offers a number of desirable properties in terms of interpretability and parsimony. Results are presented for a test case using load data from an electrical grid. Load forecasting is an essential challenge in power systems engineering, with major implications for real-time control, pricing, maintenance, and security decisions.

Brandon Reyes, Amanda A. Howard, Paris Perdikaris et al.

We extend the physics-informed neural network (PINN) method to learn viscosity models of two non-Newtonian systems (polymer melts and suspensions of particles) using only velocity measurements. The PINN-inferred viscosity models agree with the empirical models for shear rates with large absolute values but deviate for shear rates near zero where the analytical models have an unphysical singularity. Once a viscosity model is learned, we use the PINN method to solve the momentum conservation equation for non-Newtonian fluid flow using only the boundary conditions.

QiZhi He, David Brajas-Solano, Guzel Tartakovsky et al.

Data assimilation for parameter and state estimation in subsurface transport problems remains a significant challenge due to the sparsity of measurements, the heterogeneity of porous media, and the high computational cost of forward numerical models. We present a physics-informed deep neural networks (DNNs) machine learning method for estimating space-dependent hydraulic conductivity, hydraulic head, and concentration fields from sparse measurements. In this approach, we employ individual DNNs to approximate the unknown parameters (e.g., hydraulic conductivity) and states (e.g., hydraulic head and concentration) of a physical system, and jointly train these DNNs by minimizing the loss function that consists of the governing equations residuals in addition to the error with respect to measurement data. We apply this approach to assimilate conductivity, hydraulic head, and concentration measurements for joint inversion of the conductivity, hydraulic head, and concentration fields in a steady-state advection--dispersion problem. We study the accuracy of the physics-informed DNN approach with respect to data size, number of variables (conductivity and head versus conductivity, head, and concentration), DNNs size, and DNN initialization during training. We demonstrate that the physics-informed DNNs are significantly more accurate than standard data-driven DNNs when the training set consists of sparse data. We also show that the accuracy of parameter estimation increases as additional variables are inverted jointly.

Liu Yang, Sean Treichler, Thorsten Kurth et al.

Uncertainty quantification for forward and inverse problems is a central challenge across physical and biomedical disciplines. We address this challenge for the problem of modeling subsurface flow at the Hanford Site by combining stochastic computational models with observational data using physics-informed GAN models. The geographic extent, spatial heterogeneity, and multiple correlation length scales of the Hanford Site require training a computationally intensive GAN model to thousands of dimensions. We develop a hierarchical scheme for exploiting domain parallelism, map discriminators and generators to multiple GPUs, and employ efficient communication schemes to ensure training stability and convergence. We developed a highly optimized implementation of this scheme that scales to 27,500 NVIDIA Volta GPUs and 4584 nodes on the Summit supercomputer with a 93.1% scaling efficiency, achieving peak and sustained half-precision rates of 1228 PF/s and 1207 PF/s.

Tong Ma, Renke Huang, David Barajas-Solano et al.

We propose a new forecasting method for predicting load demand and generation scheduling. Accurate week-long forecasting of load demand and optimal power generation is critical for efficient operation of power grid systems. In this work, we use a synthetic data set describing a power grid with 700 buses and 134 generators over a 365-days period with data synthetically generated at an hourly rate. The proposed approach for week-long forecasting is based on the Gaussian process regression (GPR) method, with prior covariance matrices of the quantities of interest (QoI) computed from ensembles formed by up to twenty preceding weeks of QoI observations. Then, we use these covariances within the GPR framework to forecast the QoIs for the following week. We demonstrate that the the proposed ensemble GPR (EGPR) method is capable of accurately forecasting weekly total load demand and power generation profiles. The EGPR method is shown to outperform traditional forecasting methods including the standard GPR and autoregressive integrated moving average (ARIMA) methods.

Ramakrishna Tipireddy, Paris Perdikaris, Panos Stinis et al.

We investigate the use of discrete and continuous versions of physics-informed neural network methods for learning unknown dynamics or constitutive relations of a dynamical system. For the case of unknown dynamics, we represent all the dynamics with a deep neural network (DNN). When the dynamics of the system are known up to the specification of constitutive relations (that can depend on the state of the system), we represent these constitutive relations with a DNN. The discrete versions combine classical multistep discretization methods for dynamical systems with neural network based machine learning methods. On the other hand, the continuous versions utilize deep neural networks to minimize the residual function for the continuous governing equations. We use the case of a fedbatch bioreactor system to study the effectiveness of these approaches and discuss conditions for their applicability. Our results indicate that the accuracy of the trained neural network models is much higher for the cases where we only have to learn a constitutive relation instead of the whole dynamics. This finding corroborates the well-known fact from scientific computing that building as much structural information is available into an algorithm can enhance its efficiency and/or accuracy.

Xiu Yang, David Barajas-Solano, Guzel Tartakovsky et al.

In this work, we propose a new Gaussian process regression (GPR)-based multifidelity method: physics-informed CoKriging (CoPhIK). In CoKriging-based multifidelity methods, the quantities of interest are modeled as linear combinations of multiple parameterized stationary Gaussian processes (GPs), and the hyperparameters of these GPs are estimated from data via optimization. In CoPhIK, we construct a GP representing low-fidelity data using physics-informed Kriging (PhIK), and model the discrepancy between low- and high-fidelity data using a parameterized GP with hyperparameters identified via optimization. Our approach reduces the cost of optimization for inferring hyperparameters by incorporating partial physical knowledge. We prove that the physical constraints in the form of deterministic linear operators are satisfied up to an error bound. Furthermore, we combine CoPhIK with a greedy active learning algorithm for guiding the selection of additional observation locations. The efficiency and accuracy of CoPhIK are demonstrated for reconstructing the partially observed modified Branin function, reconstructing the sparsely observed state of a steady state heat transport problem, and learning a conservative tracer distribution from sparse tracer concentration measurements.

Xiu Yang, Guzel Tartakovsky, Alexandre Tartakovsky

In this work, we propose a new Gaussian process regression (GPR) method: physics information aided Kriging (PhIK). In the standard data-driven Kriging, the unknown function of interest is usually treated as a Gaussian process with assumed stationary covariance with hyperparameters estimated from data. In PhIK, we compute the mean and covariance function from realizations of available stochastic models, e.g., from realizations of governing stochastic partial differential equations solutions. Such constructed Gaussian process generally is non-stationary, and does not assume a specific form of the covariance function. Our approach avoids the optimization step in data-driven GPR methods to identify the hyperparameters. More importantly, we prove that the physical constraints in the form of a deterministic linear operator are guaranteed in the resulting prediction. We also provide an error estimate in preserving the physical constraints when errors are included in the stochastic model realizations. To reduce the computational cost of obtaining stochastic model realizations, we propose a multilevel Monte Carlo estimate of the mean and covariance functions. Further, we present an active learning algorithm that guides the selection of additional observation locations. The efficiency and accuracy of PhIK are demonstrated for reconstructing a partially known modified Branin function, studying a three-dimensional heat transfer problem and learning a conservative tracer distribution from sparse concentration measurements.

Xiu Yang, Weixuan Li, Alexandre Tartakovsky

Compressive-sensing-based uncertainty quantification methods have become a pow- erful tool for problems with limited data. In this work, we use the sliced inverse regression (SIR) method to provide an initial guess for the alternating direction method, which is used to en- hance sparsity of the Hermite polynomial expansion of stochastic quantity of interest. The sparsity improvement increases both the efficiency and accuracy of the compressive-sensing- based uncertainty quantification method. We demonstrate that the initial guess from SIR is more suitable for cases when the available data are limited (Algorithm 4). We also propose another algorithm (Algorithm 5) that performs dimension reduction first with SIR. Then it constructs a Hermite polynomial expansion of the reduced model. This method affords the ability to approximate the statistics accurately with even less available data. Both methods are non-intrusive and require no a priori information of the sparsity of the system. The effec- tiveness of these two methods (Algorithms 4 and 5) are demonstrated using problems with up to 500 random dimensions.

Panos Stinis, Tobias Hagge, Alexandre M. Tartakovsky et al.

We suggest ways to enforce given constraints in the output of a Generative Adversarial Network (GAN) generator both for interpolation and extrapolation (prediction). For the case of dynamical systems, given a time series, we wish to train GAN generators that can be used to predict trajectories starting from a given initial condition. In this setting, the constraints can be in algebraic and/or differential form. Even though we are predominantly interested in the case of extrapolation, we will see that the tasks of interpolation and extrapolation are related. However, they need to be treated differently. For the case of interpolation, the incorporation of constraints is built into the training of the GAN. The incorporation of the constraints respects the primary game-theoretic setup of a GAN so it can be combined with existing algorithms. However, it can exacerbate the problem of instability during training that is well-known for GANs. We suggest adding small noise to the constraints as a simple remedy that has performed well in our numerical experiments. The case of extrapolation (prediction) is more involved. During training, the GAN generator learns to interpolate a noisy version of the data and we enforce the constraints. This approach has connections with model reduction that we can utilize to improve the efficiency and accuracy of the training. Depending on the form of the constraints, we may enforce them also during prediction through a projection step. We provide examples of linear and nonlinear systems of differential equations to illustrate the various constructions.

Tobias Hagge, Panos Stinis, Enoch Yeung et al.

We solve a system of ordinary differential equations with an unknown functional form of a sink (reaction rate) term. We assume that the measurements (time series) of state variables are partially available, and we use recurrent neural network to "learn" the reaction rate from this data. This is achieved by including a discretized ordinary differential equations as part of a recurrent neural network training problem. We extend TensorFlow's recurrent neural network architecture to create a simple but scalable and effective solver for the unknown functions, and apply it to a fedbatch bioreactor simulation problem. Use of techniques from recent deep learning literature enables training of functions with behavior manifesting over thousands of time steps. Our networks are structurally similar to recurrent neural networks, but differences in design and function require modifications to the conventional wisdom about training such networks.

Ramakrishna Tipireddy, Panos Stinis, Alexandre Tartakovsky

We present a novel uncertainty quantification approach for high-dimensional stochastic partial differential equations that reduces the computational cost of polynomial chaos methods by decomposing the computational domain into non-overlapping subdomains and adapting the stochastic basis in each subdomain so the local solution has a lower dimensional random space representation. The local solutions are coupled using the Neumann-Neumann algorithm, where we first estimate the interface solution then evaluate the interior solution in each subdomain using the interface solution as a boundary condition. The interior solutions in each subdomain are computed independently of each other, which reduces the operation count from $O(N^α)$ to $O(M^α),$ where $N$ is the total number of degrees of freedom, $M$ is the number of degrees of freedom in each subdomain, and the exponent $α>1$ depends on the uncertainty quantification method used. In addition, the localized nature of solutions makes the proposed approach highly parallelizable. We illustrate the accuracy and efficiency of the approach for linear and nonlinear differential equations with random coefficients.