[Paper Review] Physics-informed neural networks for solving forward and inverse flow problems via the Boltzmann-BGK formulation

Qin Lou, Xuhui Meng+1 more|arXiv (Cornell University)|Oct 18, 2020

Model Reduction and Neural Networks87 references167 citations

TL;DR

This paper proposes PINN-BGK, a physics-informed neural network framework that solves forward and inverse problems in multiscale flows using the Boltzmann-BGK equation. By decomposing the distribution function into equilibrium and non-equilibrium components and enforcing the governing PDEs and boundary conditions through loss minimization, the method achieves high-accuracy reconstruction of velocity fields in both continuum and rarefied regimes—even with limited interior measurements and without known boundary conditions.

ABSTRACT

In this study, we employ physics-informed neural networks (PINNs) to solve forward and inverse problems via the Boltzmann-BGK formulation (PINN-BGK), enabling PINNs to model flows in both the continuum and rarefied regimes. In particular, the PINN-BGK is composed of three sub-networks, i.e., the first for approximating the equilibrium distribution function, the second for approximating the non-equilibrium distribution function, and the third one for encoding the Boltzmann-BGK equation as well as the corresponding boundary/initial conditions. By minimizing the residuals of the governing equations and the mismatch between the predicted and provided boundary/initial conditions, we can approximate the Boltzmann-BGK equation for both continuous and rarefied flows. For forward problems, the PINN-BGK is utilized to solve various benchmark flows given boundary/initial conditions, e.g., Kovasznay flow, Taylor-Green flow, cavity flow, and micro Couette flow for Knudsen number up to 5. For inverse problems, we focus on rarefied flows in which accurate boundary conditions are difficult to obtain. We employ the PINN-BGK to infer the flow field in the entire computational domain given a limited number of interior scattered measurements on the velocity with unknown boundary conditions. Results for the two-dimensional micro Couette and micro cavity flows with Knudsen numbers ranging from 0.1 to 10 indicate that the PINN-BGK can infer the velocity field in the entire domain with good accuracy. Finally, we also present some results on using transfer learning to accelerate the training process. Specifically, we can obtain a three-fold speedup compared to the standard training process (e.g., Adam plus L-BFGS-B) for the two-dimensional flow problems considered in our work.

Motivation & Objective

• Address the challenge of simulating multiscale flows across continuum and rarefied regimes where traditional PINNs are limited to Navier-Stokes or Euler equations.
• Overcome the difficulty of obtaining accurate boundary conditions in rarefied and complex-geometry flows, such as in shale gas transport or microfluidics.
• Develop a unified PINN framework capable of solving both forward and inverse problems using the same codebase and formulation.
• Enable accurate velocity field reconstruction from sparse, scattered interior velocity measurements without relying on boundary conditions.
• Demonstrate the feasibility of transfer learning to accelerate training in multiscale flow problems.

Proposed method

• Formulate the Boltzmann-BGK equation in discrete velocity form to enable numerical computation while preserving conservation laws.
• Design a three-subnetwork architecture: one for the equilibrium distribution function, one for the non-equilibrium part, and one to encode the full PDE and boundary/initial conditions.
• Use automatic differentiation to compute residuals of the Boltzmann-BGK equation and enforce them in the loss function.
• Minimize a composite loss combining PDE residuals, boundary/initial condition mismatches, and data fidelity for measured velocities in inverse problems.
• Apply transfer learning by initializing the network with pre-trained weights from a similar forward problem to accelerate convergence.
• Use Adam optimizer followed by L-BFGS-B for fine-tuning, with training data consisting of scattered interior velocity measurements in inverse problems.

Experimental results

Research questions

• RQ1 Can physics-informed neural networks be extended to solve forward and inverse problems governed by the Boltzmann-BGK equation in both continuum and rarefied flow regimes?
• RQ2 To what extent can PINNs reconstruct the full velocity field in multiscale flows when only sparse, scattered interior velocity measurements are available and boundary conditions are unknown?
• RQ3 How does the inclusion of physics-based constraints (PDE residuals) improve prediction accuracy compared to pure regression in inverse flow problems?
• RQ4 Can transfer learning significantly reduce training time for PINN-BGK in complex flow configurations?
• RQ5 What is the accuracy of PINN-BGK in capturing velocity profiles across a range of Knudsen numbers, from 0.01 to 10, in microscale flows like Couette and cavity flows?

Key findings

• PINN-BGK accurately solves forward problems across Knudsen numbers from 0.01 to 5, including benchmark flows such as Taylor-Green, cavity, and micro Couette flows.
• For inverse problems in micro Couette and cavity flows with Knudsen numbers from 0.1 to 10, PINN-BGK reconstructs the full velocity field with relative L2 errors below 9% for u-velocity and below 22% for v-velocity, even without boundary conditions.
• When using 50 uniformly distributed interior measurements, the relative L2 error for u-velocity was 8.51% at Kn = 1.0, decreasing to 3.5% with 100 measurements, demonstrating improved accuracy with denser data.
• The physics-informed loss function reduced prediction errors significantly compared to pure regression: for example, at Kn = 10, error in u-velocity dropped from 31.87% (regression) to 8.56% (PINN-BGK).
• Transfer learning enabled a three-fold speedup in training for the Kovasznay flow case compared to standard Adam + L-BFGS-B optimization.
• The method successfully captures velocity slip and non-equilibrium effects in rarefied flows, including complex vortical structures in micro cavity flows, even with limited data.