Quantifying local and global mass balance errors in physics-informed neural networks.

Area of Science:

• Computational hydrogeology
• Numerical methods for PDEs
• Machine learning in geosciences

Background:

• Physics-informed neural networks (PINN) are increasingly used to solve partial differential equations (PDEs).
• PINN incorporates physical laws, like mass balance, into the loss function for soft enforcement.
• The accuracy of PINN in satisfying these physical constraints, particularly mass balance, requires thorough investigation.

Purpose of the Study:

• To investigate the mass balance errors of Physics-informed neural networks (PINN) when solving 1D saturated groundwater flow equations.
• To compare PINN's mass balance performance against traditional numerical methods and analytical solutions.
• To evaluate the impact of training data and hyperparameter tuning on PINN's accuracy and mass balance compliance.

Main Methods:

• Solving 1D saturated groundwater flow (diffusion) equations using PINN for homogeneous and heterogeneous media.
• Evaluating local and global mass balance errors in PINN solutions.
• Comparing PINN results with a two-point finite volume method and analytical solutions.
• Assessing PINN accuracy with and without hydraulic head training data.

Main Results:

• PINN exhibited significant local and global mass balance errors compared to the finite volume method.
• Hyperparameter tuning (collocation points, data, network architecture, epochs, learning rate) did not substantially reduce errors.
• PINN's accuracy and mass balance compliance were not significantly improved by incorporating hydraulic head data.

Conclusions:

• Significant mass balance errors limit the practical utility of PINN in applications demanding strict adherence to physical laws.
• PINN's current implementation may not be suitable for groundwater flow modeling where mass conservation is critical.
• Further research is needed to improve the mass balance fidelity of PINN for scientific applications.