Source-linked AI summary
Physics-informed neural networks for solving forward and inverse flow problems via the Boltzmann-BGK formulation
Qin Lou, Xuhui Meng, George Em Karniadakis
TL;DR
The paper addresses whether PINNs can solve Boltzmann-BGK flow problems across continuum and rarefied regimes, including inverse problems with unknown boundary conditions. It introduces PINN-BGK with separate equilibrium and non-equilibrium approximations and reports accurate field inference from sparse interior velocity data, while identifying reduced accuracy near boundaries for line-distributed measurements.
Problem
Existing PINN flow studies focused on continuum regimes, while rarefied inverse problems are difficult because accurate boundary conditions may be unavailable.
Method
PINN-BGK uses separate neural networks for equilibrium and non-equilibrium distribution functions and enforces the Boltzmann-BGK equation with appropriate conditions or interior measurement mismatches.
Results
The PINN-BGK accurately solves forward multiscale flows and infers rarefied velocity fields from limited interior measurements without explicit boundary conditions, with relative L2 errors of 2.04%, 2.04%, and 1.44% for Kn = 1, 5, and 10 in micro Couette flow.
Takeaways & Limitations
PINN-BGK extends physics-informed neural-network flow modeling to continuum and rarefied Boltzmann-BGK problems, including inverse settings with unknown boundary conditions.
Takeaways & Limitations
In micro cavity flow, uniformly distributed measurements along five lines were less accurate than measurements used in the previous case, especially near boundaries.
Abstract
from arXiv · showhide
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.
1. Introduction
The paper addresses the lack of PINN methods for rarefied and transitional flows governed by the Boltzmann-BGK equation, especially when boundary conditions are unavailable or difficult to obtain.
- The Boltzmann-BGK equation is used to describe flows spanning rarefied and continuum regimes in applications including porous media, vacuum technology, and microfluidics.
- PINNs offer an alternative for solving PDEs with sparse data and can address both forward and inverse problems without discretizing differential operators.
- Existing PINN flow studies considered continuum flows governed by Navier-Stokes or Euler equations, not rarefied or transitional flows.
- Prior deep convolutional neural networks solved forward Boltzmann-BGK problems across Knudsen numbers from 10^-3 to 10, but did not address inverse problems.
- The paper asks whether velocity fields can be predicted from limited scattered interior velocity measurements when boundary conditions are unknown.
2. Methodology
The methodology discretizes the Boltzmann-BGK model in velocity space and uses separate neural networks for equilibrium and non-equilibrium distributions, with physics and condition residuals defining training losses.
- 2.1. From Boltzmann-BGK equation to discrete velocity Boltzmann-BGK model: The Boltzmann-BGK model describes a particle distribution function relaxing toward a Maxwellian equilibrium distribution over relaxation time τ = µ/p.
- 2.1. From Boltzmann-BGK equation to discrete velocity Boltzmann-BGK model: The continuous particle velocity is approximated using a discrete velocity model based on a Hermite-polynomial expansion truncated at order N.
- 2.1. From Boltzmann-BGK equation to discrete velocity Boltzmann-BGK model: Different discrete velocity models are selected for flows with different Knudsen numbers.
- 2.2. Physics-informed neural networks for solving DVB model: PINN-BGK uses separate neural networks to approximate equilibrium and non-equilibrium distribution components, plus a network encoding the equation and initial or boundary conditions.
- 2.2. Physics-informed neural networks for solving DVB model: Forward training minimizes governing-equation, initial-condition, and boundary-condition losses, whereas inverse training combines equation residuals with mismatch to scattered interior velocity measurements.
- 2.2. Physics-informed neural networks for solving DVB model: Continuous flows use kinetic constraints based on prescribed velocity, while rarefied flows use diffuse-reflection boundary conditions for gas-surface interactions.
3. Forward problems
PINN-BGK models continuous and rarefied forward flows by combining neural approximations of equilibrium and non-equilibrium distributions with Boltzmann-BGK residual and boundary-condition constraints. Across benchmark flows, scaling the non-equilibrium output improves Kovasznay accuracy, transfer learning accelerates optimization, and predictions agree with reference or exact solutions.
- Kovasznay flow: Scaling the non-equilibrium distribution lowers Kovasznay training loss by about three orders and residuals by about two orders compared with unscaled output.The scaled case also produces predictions much better aligned with exact solutions.
- Kovasznay flow: Non-equilibrium-distribution errors are about one order greater than velocity errors, attributed to roundoff in computing the non-equilibrium distribution.
- Kovasznay flow: Transfer learning retains comparable accuracy while providing up to a three-fold speedup over standard Adam plus L-BFGS-B training.With random initialization, L-BFGS-B can converge to a local optimum; transfer learning achieves about two orders better vertical-velocity accuracy at Re = 40 and 60.
- Taylor-Green flow: Taylor-Green velocity predictions agree well with exact solutions at t = 1, 5, and 10, with relative L2 errors reported for velocity and pressure.
- Cavity and micro-Couette flows: Cavity-flow velocity profiles agree well with a lattice Boltzmann reference, while micro-Couette predictions capture velocity slip with relative L2 errors of 2.1%, 1.12%, and 1.32%.The reported micro-Couette errors correspond to Kn = 0.01, 0.05, and 0.09, respectively.
4. Inverse problems
The PINN-BGK reconstructs rarefied microflow velocity fields from sparse interior velocity measurements without explicit boundary conditions, across Couette and cavity configurations. Accuracy depends on physics-informed constraints and measurement placement or density.
- Micro Couette flows: 2.04%, 2.04%, and 1.44% relative L2 errors were obtained for micro Couette flow at Kn = 1, 5, and 10, respectively.Predictions agreed well with linearized Boltzmann reference solutions using only partial scattered interior velocity measurements.
- Micro Couette flows: PINN-BGK predicted micro Couette upper-boundary slip velocities in good agreement with reference solutions for Kn ranging from 0.01 to 10.These inferred boundary velocities can provide boundary conditions for numerical Boltzmann-BGK solvers.
- Micro cavity flows: PINN-BGK predictions agreed well with DUGKS reference solutions for micro cavity flow at Kn = 0.1, 1, and 10 despite unknown velocity boundary conditions.The cavity tests used interior velocity measurements and evaluated profiles across the cavity center.
- Micro cavity flows: Less than 5% relative L2 error was achieved by PINN-BGK in all random-data cavity cases, compared with more than 15% for unconstrained regression.The comparison used DUGKS solutions as references and indicates that enforcing the governing equation improves reconstruction accuracy, especially near boundaries.
- Micro cavity flows: Increasing cavity measurements from 50 samples along 5 lines to 100 samples along 10 lines improved accuracy, particularly after adding measurements near the vortex zone.Predictions with line-distributed measurements were less accurate near boundaries than those with randomly distributed data, while additional samples improved results.
5. Summary
The paper develops PINN-BGK for forward and inverse multiscale flow problems governed by the Boltzmann-BGK equation. It reports accurate forward solutions across continuum and rarefied regimes, accurate inverse velocity reconstruction from sparse interior data without boundary information, and three-fold training speedups through transfer learning.
- 5. Summary: PINN-BGK uses three sub-networks to approximate equilibrium and non-equilibrium distributions and encode the Boltzmann-BGK equation with boundary or initial conditions.Training minimizes governing-equation residuals and mismatches with supplied boundary or initial conditions.
- 5. Summary: PINN-BGK accurately approximated forward Boltzmann-BGK solutions for continuous and rarefied flows with Knudsen numbers from 0.01 to 5.The tested forward problems included Taylor-Green, cavity, and other benchmark flows given boundary or initial conditions.
- 5. Summary: PINN-BGK reconstructed micro Couette and cavity velocity fields with good accuracy from limited scattered interior velocity observations and no boundary-condition information for Knudsen numbers from 0.01 to 10.The paper identifies this setting as relevant to multiscale flows where accurate boundary conditions are difficult to obtain.
- 5. Summary: Three-fold speedup was obtained with transfer learning compared with standard training using Adam plus L-BFGS-B for the considered two-dimensional flow problems.The speedup result is reported for the training process used in the study.