[Paper Review] Learning Parameters and Constitutive Relationships with Physics Informed Deep Neural Networks

Alexandre M. Tartakovsky, Carlos Ortiz Marrero+3 more|arXiv (Cornell University)|Aug 10, 2018

Model Reduction and Neural NetworksPhysics and Astronomy19 references76 citations

TL;DR

Physics-informed deep neural networks are used to estimate unknown parameters and constitutive relationships in PDEs, using both data and governing equations, even when some measurements are missing.

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.

Motivation & Objective

• Motivate solving parameter and constitutive closure problems in PDEs with limited data.
• Develop a physics-informed DNN framework that jointly learns states and unknown functions.
• Demonstrate learning of a space-dependent diffusion coefficient in a linear diffusion equation.
• Demonstrate learning of a state-dependent constitutive relationship in a nonlinear diffusion equation.
• Compare performance against traditional MAP-based approaches and assess robustness to noise.

Proposed method

• Define two DNNs for the state u(x) and the unknown constitutive relation K(x,u).
• Construct auxiliary DNNs by substituting the DNNs into the PDE and boundary conditions and compute derivatives via automatic differentiation.
• Train the networks by minimizing a loss that combines data misfit, boundary condition residuals, and PDE residuals at collocation points.
• Use a quasi-Newton L-BFGS-B optimizer with Xavier initialization for training.
• Evaluate performance on linear diffusion with K(x) and nonlinear diffusion with K(u), including noise robustness.

Experimental results

Research questions

• RQ1 Can a physics-informed DNN accurately learn an unknown space-dependent diffusion coefficient K(x) from partial measurements of K and u?
• RQ2 Can the method recover an unknown nonlinear constitutive relation K(u) using only measurements of u when K is completely unknown?
• RQ3 How does incorporating PDE constraints affect estimation accuracy compared to traditional MAP methods?
• RQ4 What is the sensitivity of the method to initialization, collocation point placement, and measurement sparsity?

Key findings

• For linear diffusion with K(x) and partial measurements, relative L2 errors were about 0.5% for u and 1.7% for K.
• Enforcing PDE constraints reduces the error in K and u by approximately 50%.
• K measurements are more informative than u measurements for reducing error, and increasing K data improves both K and u estimates.
• In the nonlinear diffusion case with K(u) unknown and no direct K measurements, the method accurately estimates K(u) from state measurements alone.
• The approach remains accurate under 1% observation noise, with relative errors of 7.4e-4 for u and 6.4e-3 for K, in the tested scenario.