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DiscretizationNet: A Machine-Learning based solver for Navier-Stokes Equations using Finite Volume Discretization
Rishikesh Ranade, Chris Hill, Jay Pathak
TL;DR
Highly nonlinear, coupled PDEs require accurate, stable, and rapidly convergent solution methods. DiscretizationNet addresses this need with a data-free generative CNN encoder-decoder that embeds discretization schemes and iterative training, and it is validated on three-dimensional steady incompressible Navier-Stokes cases. The solver shows good agreement with ANSYS Fluent across lid-driven cavity, cylinder-flow, and conjugate-heat-transfer cases.
Problem
Highly nonlinear, multidimensional, stiff, coupled PDEs make accurate, stable, and rapidly convergent neural-network training difficult.
Method
DiscretizationNet is a data-free generative CNN encoder-decoder that embeds finite-volume discretization in the computational graph and iteratively replaces inputs with generated solutions during training.
Results
The solver achieved good agreement with ANSYS Fluent on three-dimensional steady incompressible Navier-Stokes cases: lid-driven cavity, laminar cylinder flow, and conjugate heat transfer.
Takeaways & Limitations
DiscretizationNet combines numerical discretization and iterative solution behavior with generative networks for data-free Navier-Stokes solving.
Takeaways & Limitations
At convergence, the model requires actual PDE solutions as inputs and therefore cannot directly infer solutions for other geometries and boundary conditions.
Abstract
from arXiv · showhide
Over the last few decades, existing Partial Differential Equation (PDE) solvers have demonstrated a tremendous success in solving complex, non-linear PDEs. Although accurate, these PDE solvers are computationally costly. With the advances in Machine Learning (ML) technologies, there has been a significant increase in the research of using ML to solve PDEs. The goal of this work is to develop an ML-based PDE solver, that couples important characteristics of existing PDE solvers with ML technologies. The two solver characteristics that have been adopted in this work are: 1) the use of discretization-based schemes to approximate spatio-temporal partial derivatives and 2) the use of iterative algorithms to solve linearized PDEs in their discrete form. In the presence of highly non-linear, coupled PDE solutions, these strategies can be very important in achieving good accuracy, better stability and faster convergence. Our ML-solver, DiscretizationNet, employs a generative CNN-based encoder-decoder model with PDE variables as both input and output features. During training, the discretization schemes are implemented inside the computational graph to enable faster GPU computation of PDE residuals, which are used to update network weights that result into converged solutions. A novel iterative capability is implemented during the network training to improve the stability and convergence of the ML-solver. The ML-Solver is demonstrated to solve the steady, incompressible Navier-Stokes equations in 3-D for several cases such as, lid-driven cavity, flow past a cylinder and conjugate heat transfer.
1. Introduction
Solving highly nonlinear, coupled PDEs requires methods that preserve accuracy, stability, and convergence while handling difficult solution spaces. This work introduces DiscretizationNet, which combines finite-volume solver characteristics with generative neural networks for data-free PDE solution generation.
- 1. Introduction: Physics-based deep-learning methods include data-driven models that depend on simulation or experimental data and data-free models constrained by governing PDEs.The passage contrasts their reliance on data fidelity and generalizability.
- 1. Introduction: Highly nonlinear, multidimensional, stiff, coupled PDEs make gradient computation and network training central to accuracy, stability, and convergence.Navier-Stokes solutions can contain sharp local gradients across broad computational domains.
- 1. Introduction: Traditional PDE solvers discretize equations and iteratively solve their linearized forms using methods such as finite volume, finite element, and finite difference.Discretization provides higher-order derivative approximations and can add artificial dissipation for stability.
- 1. Introduction: DiscretizationNet couples finite-volume numerical schemes with a generative network to solve stiff, coupled PDEs without training data.The schemes are implemented inside the computational graph for vectorized GPU operations, while the encoder-decoder solves equations iteratively.
2. Solution Methodology
DiscretizationNet solves steady, incompressible Navier–Stokes equations with a generative encoder-decoder whose PDE residuals are computed using finite-volume schemes inside the computational graph. Iterative training and latent-space inference support convergence and generalization across geometries and boundary conditions.
- 2.1. DiscretizationNet Architecture: DiscretizationNet uses a generative CNN encoder-decoder with flow variables, boundary encoding, and geometry level sets as inputs while minimizing PDE residuals.The network also learns the solutions in an encoded latent space.
- 2.2. Training Mechanics: The iterative training procedure replaces inputs with newly generated solutions whenever residuals fall by an order of magnitude, improving stability and convergence from partial to fully converged solutions.At convergence, the network effectively behaves as a conditional autoencoder.
- 2.3. Geometry and Boundary Encoders: Geometry and boundary information are encoded separately, with a boundary encoder needed for spatially or temporally varying conditions and custom encoding sufficient for constant conditions.The geometry encoder represents level sets in a lower-dimensional latent space.
- 2.4. Loss Formulation: Finite-volume residual computation treats voxels as control-volume centers, converts volume integrals to face-based surface terms, and evaluates convective and diffusive fluxes with specified numerical schemes.The schemes are implemented inside the computational graph for fast GPU computation, while alternative discretizations remain possible.
- 2.4. Loss Formulation: Boundary voxels enforce constraints through flux computation, including stair-step discretization for unstructured boundaries such as cells adjacent to cylinder walls.The formulation uses ghost, boundary, and neighboring interior values to impose boundary conditions.
- 2.5. Inference for other geometry and boundary conditions: Because the converged network requires actual PDE solutions as inputs, the proposed inference algorithm iteratively steers a solution latent vector toward the training latent space for new geometries and boundary conditions.The fixed geometry and boundary encodings provide constraints during this iterative inference process.
3. Results
DiscretizationNet was validated against ANSYS Fluent for three-dimensional steady incompressible flow problems, including lid-driven cavity flow, flow past a cylinder, and conjugate heat transfer. Across these cases, it produced convergent solutions with close agreement, while iterative training improved convergence and stability.
- 3.1. Lid-driven cavity flow: Relative errors were less than 1% at each centerline point for lid-driven cavity velocity-magnitude comparisons with ANSYS Fluent.The comparisons covered centerlines along the x, y, and z directions.
- 3.1. Lid-driven cavity flow: The iterative procedure reached an absolute residual of 1e−3 faster than training without iteration, despite transient residual peaks after input replacement.Both configurations used fewer than 3x10^4 epochs, with about 1 second required per epoch.
- 3.1. Lid-driven cavity flow: DiscretizationNet produced convergent steady solutions at Re = 5000, beyond the range where other PDE solvers converge easily, and captured Reynolds-number-dependent vortical structures.The streamline comparisons considered Re = 5000 alongside lower Reynolds numbers.
- 3.2. Laminar flow past a cylinder: For flow past a cylinder at Re = 10, 20, and 40, predicted velocity and pressure fields agreed greatly with ANSYS Fluent and captured upstream and downstream flow structures.Line-plot differences were attributed partly to stair-step discretization near the curved cylinder surface.
- 3.3. Conjugate heat transfer: For conjugate heat transfer, normalized temperature and velocity profiles were within 1% of ANSYS Fluent solutions in both fluid and solid domains.The comparison used a 10 m long solid cube and included coupled fluid–solid behavior.
- 3.3. Conjugate heat transfer: The solver also matched ANSYS Fluent for a 13 m solid cube not seen during training, while predicting a lower peak normalized solid-domain temperature than for the smaller cube.The lower peak was attributed to differences in surface area.
4. Conclusion
DiscretizationNet combines finite-volume discretization, residual-based training, and iterative generative inference to solve steady incompressible Navier–Stokes equations without training data. Validation across three 3-D cases showed agreement with ANSYS Fluent, while the authors identify extensions for unsteady problems and improved boundary accuracy.
- 4. Conclusion: DiscretizationNet solves steady incompressible Navier–Stokes equations without training data by generating solutions while learning them during training.It computes PDE residuals with finite-volume discretization and updates network weights using the residual L-2 norm.
- 4. Conclusion: The generative CNN encoder-decoder is conditioned on geometry and boundary conditions and uses iterative replacement of inputs with outputs to improve stability and convergence.A latent-space inference algorithm iteratively generates solutions using trained model weights.
- 4. Conclusion: Agreement with ANSYS Fluent was observed for 3-D lid-driven cavity, flow past a cylinder, and conjugate heat-transfer cases, with stable convergence within 3x10^4 training epochs.The comparison used contour and line plots across all three validation cases.
- 4. Conclusion: The framework can be extended to unsteady problems, while cut-cell unstructured-grid discretization is proposed to improve boundary accuracy over stair-step discretization.The authors also identify applications to other complex-physics PDEs and computationally inexpensive low-dimensional models.