We present a physics-informed deep neural network (DNN) method for estimating hydraulic conductivity in saturated and unsaturated flows governed by Darcy's law. For saturated flow, we approximate hydraulic conductivity and head with two DNNs and use Darcy's law in addition to measurements of hydraulic conductivity and head to train these DNNs. For unsaturated flow, we approximate unsaturated conductivity function and capillary pressure with DNNs and train these DNNs using measurements of capillary pressure and the Richards equation. Because it is difficult to measure unsaturated conductivity in the field, we assume that no measurements of unsaturated conductivity are available. The proposed approach enforces the partial differential equation (PDE) (Darcy or Richards equation) constraints by minimizing the PDE residual at select points in the simulation domain. We demonstrate that physics constraints increase the accuracy of DNN approximations of sparsely observed functions and allow for training DNNs when no direct measurements of the functions of interest are available. For the saturated conductivity estimation problem, we show that the physics-informed DNN method is more accurate than the state-of-the-art maximum a posteriori probability method. For the unsaturated flow in homogeneous porous media, we find that the proposed method can accurately estimate the pressure-conductivity relationship based on the capillary pressure measurements only, even in the presence of measurement noise.

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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.

We present a physics informed deep neural network (DNN) method for estimating parameters and unknown physics (constitutive relationships) in partial differential equation (PDE) models. We use PDEs in addition to measurements to train DNNs to approximate unknown parameters and constitutive relationships as well as states. The proposed approach increases the accuracy of DNN approximations of partially known functions when a limited number of measurements is available and allows for training DNNs when no direct measurements of the functions of interest are available. We employ physics informed DNNs to estimate the unknown space-dependent diffusion coefficient in a linear diffusion equation and an unknown constitutive relationship in a non-linear diffusion equation. For the parameter estimation problem, we assume that partial measurements of the coefficient and states are available and demonstrate that under these conditions, the proposed method is more accurate than state-of-the-art methods. For the non-linear diffusion PDE model with a fully unknown constitutive relationship (i.e., no measurements of constitutive relationship are available), the physics informed DNN method can accurately estimate the non-linear constitutive relationship based on state measurements only. Finally, we demonstrate that the proposed method remains accurate in the presence of measurement noise.

We propose a probabilistic model discovery method for identifying ordinary differential equations governing the dynamics of observed multivariate data. Our method is based on the sparse identification of nonlinear dynamics (SINDy) framework, where models are expressed as sparse linear combinations of pre-specified candidate functions. Promoting parsimony through sparsity leads to interpretable models that generalize to unknown data. Instead of targeting point estimates of the SINDy coefficients, we estimate these coefficients via sparse Bayesian inference. The resulting method, uncertainty quantification SINDy (UQ-SINDy), quantifies not only the uncertainty in the values of the SINDy coefficients due to observation errors and limited data, but also the probability of inclusion of each candidate function in the linear combination. UQ-SINDy promotes robustness against observation noise and limited data, interpretability (in terms of model selection and inclusion probabilities) and generalization capacity for out-of-sample forecast. Sparse inference for UQ-SINDy employs Markov chain Monte Carlo, and we explore two sparsifying priors: the spike and slab prior , and the regularized horseshoe prior . UQ-SINDy is shown to discover accurate models in the presence of noise and with orders-of-magnitude less data than current model discovery methods, thus providing a transformative method for real-world applications which have limited data.