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Learning solutions of parametric Navier-Stokes with physics-informed neural networks

M. Naderibeni, M. J. T. Reinders, L. Wu, D. M. J. Tax

TL;DR

The paper addresses learning parametric Navier–Stokes solutions across a range of Reynolds numbers by treating $Re$ as an input to Physics-Informed Neural Networks (PINNs) and training on CFD-generated solutions to enable interpolation to unseen parameter values. The approach integrates NSE residuals $f,g,h$ into the loss alongside labeled data, encouraging physically consistent predictions. Compared with unconstrained neural networks, the PINN preserves mass and momentum and yields improved gradient fields such as vorticity, albeit with some trade-offs in velocity/pressure accuracy for seen cases. This framework offers fast, physics-informed flow predictions for new parameter instances, with implications for real-time design and parametric CFD tasks.

Abstract

We leverage Physics-Informed Neural Networks (PINNs) to learn solution functions of parametric Navier-Stokes Equations (NSE). Our proposed approach results in a feasible optimization problem setup that bypasses PINNs' limitations in converging to solutions of highly nonlinear parametric-PDEs like NSE. We consider the parameter(s) of interest as inputs of PINNs along with spatio-temporal coordinates, and train PINNs on generated numerical solutions of parametric-PDES for instances of the parameters. We perform experiments on the classical 2D flow past cylinder problem aiming to learn velocities and pressure functions over a range of Reynolds numbers as parameter of interest. Provision of training data from generated numerical simulations allows for interpolation of the solution functions for a range of parameters. Therefore, we compare PINNs with unconstrained conventional Neural Networks (NN) on this problem setup to investigate the effectiveness of considering the PDEs regularization in the loss function. We show that our proposed approach results in optimizing PINN models that learn the solution functions while making sure that flow predictions are in line with conservational laws of mass and momentum. Our results show that PINN results in accurate prediction of gradients compared to NN model, this is clearly visible in predicted vorticity fields given that none of these models were trained on vorticity labels.

Learning solutions of parametric Navier-Stokes with physics-informed neural networks

TL;DR

The paper addresses learning parametric Navier–Stokes solutions across a range of Reynolds numbers by treating as an input to Physics-Informed Neural Networks (PINNs) and training on CFD-generated solutions to enable interpolation to unseen parameter values. The approach integrates NSE residuals into the loss alongside labeled data, encouraging physically consistent predictions. Compared with unconstrained neural networks, the PINN preserves mass and momentum and yields improved gradient fields such as vorticity, albeit with some trade-offs in velocity/pressure accuracy for seen cases. This framework offers fast, physics-informed flow predictions for new parameter instances, with implications for real-time design and parametric CFD tasks.

Abstract

We leverage Physics-Informed Neural Networks (PINNs) to learn solution functions of parametric Navier-Stokes Equations (NSE). Our proposed approach results in a feasible optimization problem setup that bypasses PINNs' limitations in converging to solutions of highly nonlinear parametric-PDEs like NSE. We consider the parameter(s) of interest as inputs of PINNs along with spatio-temporal coordinates, and train PINNs on generated numerical solutions of parametric-PDES for instances of the parameters. We perform experiments on the classical 2D flow past cylinder problem aiming to learn velocities and pressure functions over a range of Reynolds numbers as parameter of interest. Provision of training data from generated numerical simulations allows for interpolation of the solution functions for a range of parameters. Therefore, we compare PINNs with unconstrained conventional Neural Networks (NN) on this problem setup to investigate the effectiveness of considering the PDEs regularization in the loss function. We show that our proposed approach results in optimizing PINN models that learn the solution functions while making sure that flow predictions are in line with conservational laws of mass and momentum. Our results show that PINN results in accurate prediction of gradients compared to NN model, this is clearly visible in predicted vorticity fields given that none of these models were trained on vorticity labels.
Paper Structure (7 sections, 7 equations, 10 figures)

This paper contains 7 sections, 7 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Physics-Informed Neural Networks
  3. Experiment setup; 2D Navier-Stokes Equations
  4. Results and Discussions
  5. Conclusions
  6. CFD Solver and computational grid.
  7. Lift force computations.

Figures (10)

  • Figure 1: A snapshot of generated data for flow past cylinder problem for (a) the x component of velocity, (b) the y component of velocity, (c) pressure and (d) the vorticity (the curl of velocity field), generated for a flow with $Re=100$. Figure (e) demonstrates the evolution of a snapshot of vorticity with changing Re number.
  • Figure 2: PINN architecture for solving Navier-Stokes Equation.
  • Figure 3: Strategies for sampling data points required in training PINN for the flow past cylinder problem. Residual points with (a) a uniform distribution, (b) a uniform distribution that is refined around the cylinder. (c) Training points sampled from initial and boundary conditions. (green : inlet, red: outlet, black: cylinder wall, blue: points on top and bottom). (d) Distribution of points along the Re parameter. (blue: residual points, orange: points from two available solutions)
  • Figure 4: A comparison on the performance of the PINNs and the fully connected Neural Networks on predicting the velocity and pressure fields of the flow past cylinder problem for a range of Re numbers. Models are trained on velocity and pressure data generated from numerical simulation of the flow at: $\mathrm{Re = [100, 200, 333.\bar{3}, 400, 500]}$ (specified with filled square markers). The test set includes data from $\mathrm{Re = [66.\bar{6}, 142.8, 250, 444.\bar{4}, 666.\bar{6}]}$ as well (specified with empty circle markers).
  • Figure 5: A comparison on the performance of PINNs and the NN models on (a) satisfying PDEs residuals, (b) vorticity field predictions for different Re numbers. Models are trained on velocity and pressure data generated from numerical simulation of the flow at: $\mathrm{Re = [100, 200, 333.\bar{3}, 400, 500]}$ (specified with filled square markers). Test set includes data from $\mathrm{Re} = [66.\bar{6}, 142.8, 250, 444.\bar{4}, 666.\bar{6}]$ as well (specified with empty circle markers).
  • ...and 5 more figures