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Register a new userPhysics-Informed Neural Networks for Multiphysics Data Assimilation with Application to Subsurface Transport
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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.
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Physics-Informed Neural Networks (PINNs) have recently shown great promise as a way of incorporating physics-based domain knowledge, including fundamental governing equations, into neural network models for many complex engineering systems. They have been particularly effective in the area of inverse problems, where boundary conditions may be ill-defined, and data-absent scenarios, where typical supervised learning approaches will fail. Here, we further explore the use of this modeling methodology to surrogate modeling of a fluid dynamical system, and demonstrate additional undiscussed and interesting advantages of such a modeling methodology over conventional data-driven approaches: 1) improving the models predictive performance even with incomplete description of the underlying physics; 2) improving the robustness of the model to noise in the dataset; 3) reduced effort to convergence during optimization for a new, previously unseen scenario by transfer optimization of a pre-existing model. Hence, we noticed the inclusion of a physics-based regularization term can substantially improve the equivalent data-driven surrogate model in many substantive ways, including an order of magnitude improvement in test error when the dataset is very noisy, and a 2-3x improvement when only partial physics is included. In addition, we propose a novel transfer optimization scheme for use in such surrogate modeling scenarios and demonstrate an approximately 3x improvement in speed to convergence and an order of magnitude improvement in predictive performance over conventional Xavier initialization for training of new scenarios.
Near-wall blood flow and wall shear stress (WSS) regulate major forms of cardiovascular disease, yet they are challenging to quantify with high fidelity. Patient-specific computational and experimental measurement of WSS suffers from uncertainty, low resolution, and noise issues. Physics-informed neural networks (PINN) provide a flexible deep learning framework to integrate mathematical equations governing blood flow with measurement data. By leveraging knowledge about the governing equations (herein, Navier-Stokes), PINN overcomes the large data requirement in deep learning. In this study, it was shown how PINN could be used to improve WSS quantification in diseased arterial flows. Specifically, blood flow problems where the inlet and outlet boundary conditions were not known were solved by assimilating very few measurement points. Uncertainty in boundary conditions is a common feature in patient-specific computational fluid dynamics models. It was shown that PINN could use sparse velocity measurements away from the wall to quantify WSS with very high accuracy even without full knowledge of the boundary conditions. Examples in idealized stenosis and aneurysm models were considered demonstrating how partial knowledge about the flow physics could be combined with partial measurements to obtain accurate near-wall blood flow data. The proposed hybrid data-driven and physics-based deep learning framework has high potential in transforming high-fidelity near-wall hemodynamics modeling in cardiovascular disease.
We introduce conditional PINNs (physics informed neural networks) for estimating the solution of classes of eigenvalue problems. The concept of PINNs is expanded to learn not only the solution of one particular differential equation but the solutions to a class of problems. We demonstrate this idea by estimating the coercive field of permanent magnets which depends on the width and strength of local defects. When the neural network incorporates the physics of magnetization reversal, training can be achieved in an unsupervised way. There is no need to generate labeled training data. The presented test cases have been rigorously studied in the past. Thus, a detailed and easy comparison with analytical solutions is made. We show that a single deep neural network can learn the solution of partial differential equations for an entire class of problems.
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-Lo{e}ve 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-Lo{e}ve 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).
The troposphere is one of the atmospheric layers where most weather phenomena occur. Temperature variations in the troposphere, especially at 500 hPa, a typical level of the middle troposphere, are significant indicators of future weather changes. Numerical weather prediction is effective for temperature prediction, but its computational complexity hinders a timely response. This paper proposes a novel temperature prediction approach in framework ofphysics-informed deep learning. The new model, called PGnet, builds upon a generative neural network with a mask matrix. The mask is designed to distinguish the low-quality predicted regions generated by the first physical stage. The generative neural network takes the mask as prior for the second-stage refined predictions. A mask-loss and a jump pattern strategy are developed to train the generative neural network without accumulating errors during making time-series predictions. Experiments on ERA5 demonstrate that PGnet can generate more refined temperature predictions than the state-of-the-art.
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