Physics Informed Differentiable Solvers for Learning Parametric Solution Manifolds in Heterogeneous Physical Systems

TL;DR

The reformulating of a Physics-Informed Neural Network as a differentiable solver that learns the continuous solution manifold for steady-state Darcy flow yields accurate, mass-conserving flow solutions and supports efficient uncertainty quantification, providing a general methodology for physics-constrained data-driven modeling of heterogeneous systems.

Abstract

Learning the full family of solutions to parameterized partial differential equations (PDEs) is a central challenge to our ability to model the behavior of heterogeneous systems, with a variety of fundamental and application-oriented implications in fields such as hydrogeology where system properties exhibit significant (and often uncertain) spatial heterogeneity. We address this by reformulating a Physics-Informed Neural Network (PINN) 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 re-training for each new parameter instance. Its versatility 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 via automatic differentiation. 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.

Figures (16)

• Figure 1: Schematic depiction of the parameterized PINN framework serving as a differentiable solver. The network $\mathcal{N}$ takes both spatial coordinates $\mathbf{x}$ and a generic instance of parameter vector $\boldsymbol{\lambda}$ as input to approximate the solution field $\hat{u}(\mathbf{x}, \boldsymbol{\lambda};\mathbf{\theta})$. This architecture forms the basis for learning the entire solution manifold.
• Figure 2: Schematic depiction of the parameterized PINN framework relying on an autoencoder-based hydraulic ($K$) parameterization. (Left) Autoencoder architecture: a CNN encoder maps reference realizations of $K$ to a 2D latent vector $\boldsymbol{\lambda}$. (Center) Learned two-dimensional (2D) latent space distribution $p(\boldsymbol{\lambda})$, which defines the sampling space for $\boldsymbol{\lambda})$ for PINN training. (Top-Right) Parameterized PINN solver, which takes spatial coordinates and latent parameters as input. The differentiable INR decoder $\mathcal{D}_{\phi_2}$ is embedded into the computation of the physics-informed loss. (Bottom-Right) Sketch of the PirateNet architecture (see Supporting Information \ref{['app:piratenet']}).
• Figure 3: Training dynamics of Scenario 1. Evolution of (a) the physics-informed loss $\mathcal{J}(\theta)$, (b) the PDE residual component $\mathcal{J}_{PDE}(\theta)$, and (c) the boundary condition residual component $\mathcal{J}_{BC}(\theta)$. (d) Evolution of the trainable $\alpha_i$ ($i = 1, 2$) parameters associated with the adaptive skip connections. (e) Convergence of the mean influx and outflux, demonstrating global mass conservation. Shaded regions in panels (a)-(e) correspond to the initial warm-up (transfer learning) phase where the network is trained on the mean of the latent parameter distribution before training on the full sampled latent space. Filled circles indicate the starting point for each iteration of the training phase (see Section \ref{['sec:training_strategy']}). (f) Violin plots showing the distribution of $\text{Rel. } L^2_{\hat{h}}$ (light blue) and $\text{Rel. } L^2_{\hat{\mathbf{v}}}$ (salmon) errors over all 1024 $\boldsymbol{\lambda}$ samples mean (solid black line), median (dotted black line), and 25th/75th percentiles (dashed black lines) are also identified.
• Figure 4: Training dynamics of Scenario 2. Evolution of (a) the total physics-informed loss $\mathcal{J}(\theta)$, (b) the PDE residual component $\mathcal{J}_{PDE}(\theta)$, and (c) the boundary condition residual component $\mathcal{J}_{BC}(\theta)$. (d) Evolution of the trainable $\alpha_i$ ($i = 1, 2$) parameters associated with the adaptive skip connections. (e) Convergence of the mean influx and outflux, the two reported plots display results with different vertical axis scale. The narrow shaded region in panels (a)-(e) corresponds to the initial warm-up phase where the network is trained on the mean of the latent parameter distribution before training on the full sampled latent space. Filled circles in panels (a)-(c) indicate the starting point for each iteration of the training phase (see section \ref{['sec:training_strategy']})
• Figure 5: Performance evaluation of the parameterized PINN trained on the Scenario 1. (a) Contour plot of the relative $L^2$ norm error for model-based hydraulic heads ($\text{Rel. } L^2_{\hat{h}}$) across the $\boldsymbol{\lambda}=(\lambda_1, \lambda_2)$ parameter space, (b) Relative $L^2$ norm error for the velocity field ($\text{Rel. } L^2_{\hat{\mathbf{v}}}$).