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

Quantifying parametric uncertainty in partial differential equations is a central challenge to our ability to model the behavior of heterogeneous systems. This challenge is relevant to a variety of fundamental and application-oriented implications where system properties exhibit significant (and often uncertain) spatial heterogeneity. We address this by reformulating a physics-informed neural network as a differentiable solver that learns the continuous solution manifold for steady-state Darcy flow. Our framework requires only a single training run, circumventing the need for costly retraining for each new parameter instance. The approach is demonstrated through two representations of spatially heterogeneous hydraulic conductivity fields: a direct analytical form and a novel data-driven formulation resting on an autoencoder to create a low-dimensional latent encoding. A key innovation is the integration of the differentiable decoder into the physics-informed loss function, enabling on-the-fly reconstruction of complex conductivity fields. The approach yields accurate, mass-conserving flow solutions and supports efficient uncertainty quantification, providing a general methodology for physics-constrained data-driven modeling of heterogeneous systems.