Physics-informed differentiable solvers for learning parametric solution manifolds in heterogeneous physical systems.

Area of Science:

• Computational fluid dynamics
• Machine learning in geoscience
• Partial differential equations

Background:

• Parametric uncertainty quantification is crucial for modeling heterogeneous systems.
• Spatial heterogeneity in system properties presents significant modeling challenges.
• Existing methods often require extensive retraining for new parameter instances.

Purpose of the Study:

• To develop a novel physics-informed neural network (PINN) framework for efficient parametric uncertainty quantification.
• To reformulate PINNs as differentiable solvers capable of learning continuous solution manifolds.
• To enable single-run training for steady-state Darcy flow problems with heterogeneous parameters.

Main Methods:

• Reformulating a physics-informed neural network as a differentiable solver.
• Utilizing a single training run to learn the continuous solution manifold.
• Employing autoencoders for low-dimensional latent encoding of hydraulic conductivity fields.
• Integrating a differentiable decoder into the physics-informed loss function for on-the-fly field reconstruction.

Main Results:

• Accurate and mass-conserving solutions for steady-state Darcy flow were achieved.
• Efficient uncertainty quantification was demonstrated for heterogeneous systems.
• The framework circumvents the need for repeated retraining for different parameter instances.
• Successful reconstruction of complex conductivity fields was enabled through the integrated decoder.

Conclusions:

• The proposed differentiable PINN solver offers a general methodology for physics-constrained data-driven modeling of heterogeneous systems.
• This approach significantly enhances the efficiency of uncertainty quantification in complex subsurface flow simulations.
• The integration of autoencoders and differentiable decoders provides a powerful tool for handling spatially varying parameters.