Physics-Informed Neural Networks for Multiphysics Data Assimilation with Application to Subsurface Transport (1912.02968v1)

Abstract: 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.

• The paper introduces a PINN framework that embeds Darcy’s law and advection–dispersion equations into neural network training to estimate hydraulic conductivity, head, and concentration fields.
• The study demonstrates that incorporating physical constraints into deep learning models significantly reduces error and variability compared to standard data-driven approaches.
• The multiphysics PINN approach leverages sparse measurements and sequential training to enhance predictive accuracy, highlighting its potential scalability for real-world subsurface applications.

Physics-Informed Neural Networks for Multiphysics Data Assimilation with Application to Subsurface Transport

This paper presents a sophisticated machine learning framework tailored to tackle the challenges of data assimilation in subsurface transport problems, specifically through Physics-Informed Neural Networks (PINNs). The authors introduce a unique application of PINNs to simultaneously estimate space-dependent hydraulic conductivity, hydraulic head, and concentration fields using sparse measurements. The proposed methodology is underscored by the integration of physical laws—Darcy's law and the advection–dispersion equations—into the training of deep neural networks (DNNs).

The paper articulates the advantages of PINNs over traditional data-driven models, particularly in scenarios where measurement data is sparse, a common characteristic of subsurface environments. Here, the PINN framework enhances parameter estimation by embedding the governing physics directly into the neural network training process. This is achieved through a loss function that includes residual terms from the differential equations governing subsurface flow and transport, additionally weighted by the measurement errors.

Numerical simulations demonstrate the improved accuracy of PINNs in assimilating multiphysical data to estimate hydraulic properties compared to standard DNNs. The paper presents a comprehensive analysis of various scenarios, including the effect of measurement sparsity, neural network architectural choices, and initialization methods on estimation accuracy. Results underscore the significant reduction in error and variability when physical constraints are enforced, particularly in settings where direct measurements are limited.

Further exploration involves comparing the performance of purely data-driven DNNs, PINNs constrained by Darcy's law (PINN-Darcy), and a multiphysics PINN approach (MPINN) that combines both flow and transport physics. With the strategic inclusion of concentration data and a sequential training approach, MPINN showcases enhanced estimation performance, highlighting the potential for indirect measurements to bolster predictive accuracy.

The research speculates on the future potential and scalability of PINNs in real-world applications, emphasizing the need for optimized training algorithms and computational resources to handle large-scale problems. The application of automatic differentiation and neural network architectures specific to the correlation characteristics of the parameter fields is particularly noted for their role in improving the model's adaptability and precision.

This paper contributes substantively to the fields of computational modeling and subsurface hydrology, offering a robust tool that bridges the gap between data availability and the predictive power of environmental models. The PINN framework is well-positioned to be a transformative approach in environmental engineering, building upon recent advancements in machine learning and automatic differentiation technologies. As such, it paves the way for more accurate, physics-consistent predictions in various applications beyond subsurface transport, potentially extending to other domains involving partial differential equations and complex systems.