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Physics-Informed Neural Networks for Multiphysics Data Assimilation with Application to Subsurface Transport
QiZhi He, David Brajas-Solano, Guzel Tartakovsky, Alexandre M. Tartakovsky
TL;DR
Unknown parameters in heterogeneous, data-sparse natural systems make subsurface parameter and state estimation challenging. The paper presents physics-informed neural networks for multiphysics data and model assimilation, reporting improved accuracy over data-driven approaches and increased conductivity-estimation accuracy when additional variables are jointly inverted.
Problem
Heterogeneity and data sparsity leave parameters of natural systems often unknown, complicating parameter and state estimation.
Method
The paper presents physics-informed neural networks for multiphysics data and model assimilation, jointly training neural networks with physical-model constraints for subsurface transport estimation.
Results
Physics-informed methods improve the accuracy of data-driven estimates, while MPINN improves conductivity-estimation accuracy over PINN-Darcy.
Takeaways & Limitations
Joint physics-informed assimilation supports more accurate multiphysics parameter and state estimation than data-driven estimation, with additional variables improving parameter-estimation accuracy.
Takeaways & Limitations
Applying physics-informed DNNs to large-scale problems requires addressing the increased computational cost of training caused by physical-model constraints.
Abstract
from arXiv · showhide
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.
1. Introduction
Subsurface transport data assimilation is difficult because measurements are sparse, porous media are heterogeneous, and inverse problems are nonlinear. The paper extends physics-informed neural networks to jointly estimate conductivity, hydraulic head, and concentration, reporting improved sparse-data accuracy over data-only DNNs.
- Sparse measurements and heterogeneous porous media make accurate subsurface transport modeling and parameter estimation challenging.
- Data assimilation updates model parameters and system states using measurements together with governing equations.
- Subsurface flow and transport inverse problems are highly nonlinear, with non-Gaussian states and parameters that challenge direct and Bayesian estimation methods.
- The study extends PINN-based parameter estimation to assimilate hydraulic conductivity, hydraulic head, and solute concentration measurements jointly.
- PINNs significantly improve parameter and state estimation accuracy over standard data-only DNNs when data are sparse.
2. Physics-informed neural networks
The PINN framework augments neural-network data fitting with governing-equation and boundary-condition residuals. This physics-informed regularization enables estimation from sparse measurements while introducing computational and optimization costs.
- PINNs enforce physics in neural-network regression, enabling assimilation of different types of data.
- Fully connected feed-forward DNNs approximate differentiable spatial parameters and PDE state variables.
- The physics-informed loss combines measurement mismatch with governing-PDE and boundary-condition mismatch penalties.
- Physics residuals are imposed at finite collocation points, and the authors demonstrate solution convergence as their number increases.
- Automatic differentiation computes spatial derivatives needed for PDE and Neumann boundary-condition residuals.
- Physics-informed losses act as regularization because the number of network parameters can greatly exceed the number of measurements.
3. PINN for data assimilation and parameter estimation in subsurface transport problems
The proposed multiphysics PINN uses separate neural networks for conductivity, head, and concentration in a steady-state subsurface transport model. Numerical comparisons show improved accuracy over data-driven training, while larger applications remain computationally constrained.
- The objective is to learn conductivity, hydraulic head, and concentration fields from sparse measurements in a subsurface transport domain.
- Three DNNs represent K(x), h(x), and C(x), with spatial derivatives computed by automatic differentiation.
- MPINN jointly trains all three fields with Darcy, advection–dispersion, and boundary-condition constraints, while PINN-Darcy uses only Darcy constraints.
- Adding physics constraints complicates the non-convex loss landscape and makes minimization more challenging.
- Sequential training first pretrains conductivity and head with PINN-Darcy, then retrains them with concentration using MPINN.
4. Numerical example 1: periodic conductivity field
The numerical example evaluates data-driven, PINN-Darcy, and MPINN approaches for estimating periodic conductivity, hydraulic head, and concentration fields from synthetic measurements. Accuracy improves with more measurements, while physics constraints and joint inversion improve sparse-data estimation, especially for concentration.
- Numerical setup: The DNNs use two-dimensional spatial inputs and one scalar output for each of conductivity K, hydraulic head h, or concentration C.The study varies hidden-layer depth and neuron count to examine architecture effects.
- Data-driven DNNs: For all variables and network sizes, mean error and standard deviation decrease as the number of measurements increases.With more measurements, DNN size has little effect; with sparse measurements, network size matters substantially.
- Data-driven DNNs: For highly nonlinear K and C fields, smaller DNNs outperform larger ones when measurements are sparse, whereas deeper networks require more data.The smooth hydraulic-head field can be accurately approximated with relatively few measurements regardless of DNN size.
- PINN-Darcy and MPINN: Physics constraints act as a regularizer for sparse-data training while preserving solutions that satisfy the governing equations.Unlike L1 or L2 regularization, the physics-informed solutions satisfy the governing equations in general.
- PINN-Darcy and MPINN: PINN-Darcy accuracy improves with measurements and residual points, and physics-informed models can use larger DNNs with slightly smaller approximation error.The effect of enforcing the Darcy equation is especially strong for sparse data.
- PINN-Darcy and MPINN: MPINN improves K and h estimation over PINN-Darcy when concentration measurements are included, showing benefits from jointly inverting additional variables.The concentration-field relative error falls from more than 0.2 with a data-driven DNN to less than 0.05 with MPINN.
- PINN-Darcy and MPINN: Sequential training matches simultaneous training for K and h but performs better for C, particularly for medium and deep DNNs with few measurements.The results indicate that sequential training helps minimize the loss function without introducing additional errors or instabilities.
5. Numerical example 2: lognormal conductivity fields with different correlation lengths
The study examines how conductivity-field correlation length and DNN size affect approximation accuracy, then evaluates physics-informed data assimilation with sparse measurements. Smoother fields require smaller networks, while combining measurements with Darcy and advection–dispersion constraints improves estimation, especially for sparse data and short correlation lengths.
- The optimal optimizer and training effort depend on field smoothness: Adam replaces inefficient L-BFGS-B for large measurement sets, requiring fewer iterations for larger correlation lengths.
- DNN approximation error initially decreases as network width increases because larger networks have greater representation ability.
- Smoother conductivity fields with larger correlation lengths require smaller DNNs, while oversized networks can overfit.
- The optimal DNN size follows an approximately power-law dependence on the modeled field’s correlation length, but also depends on architecture and activation choices.
- Physics constraints improve conductivity approximation, with the largest reduction from hydraulic-head measurements and the Darcy constraint, followed by concentration data and the advection–dispersion constraint.
- For λ = 0.2 and NK = 20, relative parameter-estimation error falls from more than 180% with data-driven DNNs to 66.8% with PINN-Darcy and 57.7% with MPINN.
- MPINN improves hydraulic-head and concentration estimates and captures local concentration-plume features using sparse direct and indirect measurements with PDE constraints.
6. Conclusion
The study presents physics-informed DNNs as a unified framework for assimilating sparse, multiphysics data in subsurface transport. Compared with data-driven DNNs, physics-informed methods improve estimation accuracy and reduce prediction uncertainty, while network size and training cost remain important constraints.
- Physics-informed framework: The framework jointly trains DNNs for unknown parameters and states using multiphysics measurements and governing-equation constraints.The approach models conductivity, hydraulic head, and concentration fields and minimizes a loss combining measurement errors with physical-model residuals.
- Compared approaches: MPINN combines conductivity, head, and concentration measurements with Darcy and advection–dispersion equations, whereas PINN-Darcy uses conductivity, head, and Darcy constraints.The study compares these physics-informed approaches with a pure data-driven DNN trained only on data.
- Accuracy and uncertainty: MPINN improves PINN-Darcy conductivity estimates, while both physics-informed methods improve data-driven conductivity estimation and reduce prediction uncertainty, especially with limited direct measurements.The results support physics-informed methods for sparse and indirect subsurface observations.
- Network size: Small networks can have poor representability and large networks can overfit, with an optimal size showing power-law dependence on the modeled K-field correlation length.For small measurement sets, the optimal-size DNN outperforms both larger and smaller networks.
- Network size: Physics constraints and added measurements reduce dependence on DNN size once the network is sufficiently representative.This indicates that adequate network capacity remains necessary even when additional physical information is available.
- Computational considerations: Nonlinear PDE constraints increase training cost, and large-scale applications require multi-GPU computing and scalable training algorithms.The authors also identify training algorithms and DNN hyperparameters as topics requiring further study.