Sensitivity analysis using Physics-informed neural networks

TL;DR

This work addresses local sensitivity analysis for PDE solutions by augmenting Physics-informed neural networks (PINN) with SA-PINN, a derivative-regularized loss that targets a neighborhood around a nominal parameter $\hat{\mu}$. The neural network receives space, time, and parameter inputs, but collocation points are kept in space-time only, while automatic differentiation yields $\partial u/\partial \mu$ from the augmented loss. The key contributions are the SA-PINN formulation, its ability to compute sensitivities for multiple parameters, and demonstrated accuracy in problems with moving boundaries or sharp gradients across 1D and 2D domains including porous-media two-phase flows. The results show SA-PINN often matches or exceeds parametric PINN performance with substantially lower computational cost, making it practical for local sensitivity analysis and design-oriented studies in complex PDE systems. The approach has potential impact for uncertainty quantification and optimization tasks where parameter perturbations are small and localized.

Abstract

The goal of this paper is to provide a simple approach to perform local sensitivity analysis using Physics-informed neural networks (PINN). The main idea lies in adding a new term in the loss function that regularizes the solution in a small neighborhood near the nominal value of the parameter of interest. The added term represents the derivative of the loss function with respect to the parameter of interest. The result of this modification is a solution to the problem along with the derivative of the solution with respect to the parameter of interest (the sensitivity). We call the new technique SA-PNN which stands for sensitivity analysis in PINN. The effectiveness of the technique is shown using four examples: the first one is a simple one-dimensional advection-diffusion problem to show the methodology, the second is a two-dimensional Poisson's problem with nine parameters of interest, and the third and fourth examples are one and two-dimensional transient two-phase flow in porous media problem.

Figures (19)

• Figure 1: Diagram explaining the methodology of SA-PINN. The solution of the PDE is approximated by a neural network that takes an input of space $\mathbf{X}$, time $t$, and the parameter of interest $\mu$. Automatic differentiation is used to obtain the residual of the PDE. An extra step is performed to obtain the derivative of the residual with respect to the parameter of interest. The total loss function is composed of the PDE residual and boundary conditions term, as well as the derivative of the residual and boundary conditions term with respect to the parameter of interest.
• Figure 2: Loss function contribution for the 1D diffusion-advection example vs. the number of iterations.
• Figure 3: Solution $u$ at $\epsilon=0.1$ using PINN and SA-PINN along with the analytical solution of the 1D advection-diffusion problem.
• Figure 4: $\frac{\partial u}{\partial \epsilon}$ at $\epsilon=0.1$ using PINN and SA-PINN along with the finite difference solution of the 1D advection-diffusion problem.
• Figure 5: Loss term $loss_f$ for different $\epsilon$ values using PINN and SA-PINN of the 1D advection-diffusion problem.