Self-adaptive loss balanced Physics-informed neural networks for the incompressible Navier-Stokes equations

TL;DR

This paper introduces a self-adaptive loss weighting method for Physics-informed neural networks (PINNs) that improves the accuracy and robustness in solving complex incompressible Navier-Stokes equations by dynamically adjusting loss term weights during training.

Contribution

The paper proposes a novel self-adaptive loss function approach using Gaussian probabilistic models and maximum likelihood estimation to automatically balance multiple loss terms in PINNs.

Findings

Enhanced accuracy in simulating complex flows

Robustness to initialization of noise parameters

Applicable to various fluid flow problems

Abstract

There have been several efforts to Physics-informed neural networks (PINNs) in the solution of the incompressible Navier-Stokes fluid. The loss function in PINNs is a weighted sum of multiple terms, including the mismatch in the observed velocity and pressure data, the boundary and initial constraints, as well as the residuals of the Navier-Stokes equations. In this paper, we observe that the weighted combination of competitive multiple loss functions plays a significant role in training PINNs effectively. We establish Gaussian probabilistic models to define the loss terms, where the noise collection describes the weight parameter for each loss term. We propose a self-adaptive loss function method, which automatically assigns the weights of losses by updating the noise parameters in each epoch based on the maximum likelihood estimation. Subsequently, we employ the self-adaptive loss balanced Physics-informed neural networks (lbPINNs) to solve the incompressible Navier-Stokes equations,\hspace{-1pt} including\hspace{-1pt} two-dimensional\hspace{-1pt} steady Kovasznay flow, two-dimensional unsteady cylinder wake, and three-dimensional unsteady Beltrami flow. Our results suggest that the accuracy of PINNs for effectively simulating complex incompressible flows is improved by adaptively appropriate weights in the loss terms. The outstanding adaptability of lbPINNs is not irrelevant to the initialization choice of noise parameters, which illustrates the robustness. The proposed method can also be employed in other problems where PINNs apply besides fluid problems.