Learning Parameters and Constitutive Relationships with Physics Informed Deep Neural Networks

TL;DR

This work tackles the challenge of learning unknown constitutive relationships and parameters in PDEs from limited data by introducing physics-informed deep neural networks. It jointly learns the state $u(\boldsymbol{x})$ and an unknown constitutive relation $K(\boldsymbol{x},u)$ by enforcing the governing operator $\\mathcal{L}[u,K(\boldsymbol{x},u)]=0$ and boundary conditions through automatic differentiation, using data and collocation points. The method demonstrates strong performance on linear diffusion with a space-dependent $K(\boldsymbol{x})$—outperforming MAP with relative errors around $\\varepsilon_u \\approx 0.5\%$ and $\\varepsilon_K \\approx 1.7\%$—and extends to nonlinear diffusion with unknown $K(u)$ learned from $u$ data alone, achieving accurate $K(u)$ recovery even with noise. These results indicate a data-efficient, PDE-constrained framework for discovering unknown physics, with potential for robust, scalable applications in porous media and other complex systems.

Abstract

We present a physics informed deep neural network (DNN) method for estimating parameters and unknown physics (constitutive relationships) in partial differential equation (PDE) models. We use PDEs in addition to measurements to train DNNs to approximate unknown parameters and constitutive relationships as well as states. The proposed approach increases the accuracy of DNN approximations of partially known functions when a limited number of measurements is available and allows for training DNNs when no direct measurements of the functions of interest are available. We employ physics informed DNNs to estimate the unknown space-dependent diffusion coefficient in a linear diffusion equation and an unknown constitutive relationship in a non-linear diffusion equation. For the parameter estimation problem, we assume that partial measurements of the coefficient and states are available and demonstrate that under these conditions, the proposed method is more accurate than state-of-the-art methods. For the non-linear diffusion PDE model with a fully unknown constitutive relationship (i.e., no measurements of constitutive relationship are available), the physics informed DNN method can accurately estimate the non-linear constitutive relationship based on state measurements only. Finally, we demonstrate that the proposed method remains accurate in the presence of measurement noise.

Figures (11)

• Figure 1: Reference $K$ (left) and $u$ (right) fields.
• Figure 2: Estimated (a) $\hat{K}(\mathbf{x})$ and (b) $\hat{u}(\mathbf{x})$ fields. The red dots indicate the observation locations. Absolute point errors in (c) $\hat{K}(\mathbf{x})$ and (d) $\hat{u}(\mathbf{x})$. $N_K = 250$, $N_u = 100$, and $N_c = 1024$.
• Figure 3: Mean and standard deviation of $\hat{K}$ and $\hat{u}$ obtained with 11 different network initializations using Xavier's initialization scheme as a function of $N=N_K=N_u$. The $u$ and $K$ measurement locations are fixed, and $N_c = 1024$ collocation points are used.
• Figure 4: (a)-(b): Mean and standard deviation of the estimated $u$ and $K$ relative errors as a function of the number of collocation points $N_c$. For a given $N_c$, the DNNs are trained 11 times for different configurations of collocation points to compute the mean and variance of the relative errors. The $u$ and $K$ measurement locations are fixed, and $N=N_K=N_u=20$. Shaded area width is equal to two standard deviations. (c)-(d): Mean $K$ and $u$ relative errors as a function of $N_c$ and $N$.
• Figure 5: (a) and (b): Mean relative $L_2$ error of the predicted $K$ and $u$ as a function of the number of $u$ observations. The number of $K$ observations is 20. Shaded area width is equal to two standard deviations computed for 11 different configurations of $u$ observations. (c) and (d): Mean relative $L_2$ error of the predicted $K$ and $u$ as a function of the number of $u$ and $K$ observations.