A Point-Cloud Deep Learning Framework for Prediction of Fluid Flow Fields on Irregular Geometries

Ali Kashefi; Davis Rempe; Leonidas J. Guibas

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A Point-Cloud Deep Learning Framework for Prediction of Fluid Flow Fields on Irregular Geometries

Ali Kashefi, Davis Rempe, Leonidas J. Guibas

arXiv:2010.09469v2Oct 15, 2020cs.LGphysics.flu-dyn

TL;DR

Predicting CFD fields on irregular geometries is difficult because neural networks handle Cartesian grids more readily than unstructured, geometry-dependent data. This paper uses a PointNet-based point-cloud framework to predict flow fields, achieving hundreds-fold speedups while preserving CFD-data accuracy and conserving mass and momentum.

Problem
Neural-network CFD surrogates need effective representations for geometry-dependent flow fields, but connecting scattered data on unavoidable unstructured grids remains challenging.
Method
The framework treats CFD grid vertices as point clouds and uses a PointNet-based network to map spatial coordinates to velocity and pressure fields.
Results
Predictions on unseen cylinder geometries were hundreds of times faster than a traditional CFD solver while conserving mass and momentum with excellent to reasonable accuracy.
Takeaways & Limitations
The framework preserves CFD-data accuracy, captures small geometry changes, maintains curved-surface smoothness, and supports non-uniform point distributions for design optimization.
Takeaways & Limitations
Current physics-informed neural network versions remain limited to fixed geometries, motivating future integration with the proposed framework.

Abstract

from arXiv · show

We present a novel deep learning framework for flow field predictions in irregular domains when the solution is a function of the geometry of either the domain or objects inside the domain. Grid vertices in a computational fluid dynamics (CFD) domain are viewed as point clouds and used as inputs to a neural network based on the PointNet architecture, which learns an end-to-end mapping between spatial positions and CFD quantities. Using our approach, (i) the network inherits desirable features of unstructured meshes (e.g., fine and coarse point spacing near the object surface and in the far field, respectively), which minimizes network training cost; (ii) object geometry is accurately represented through vertices located on object boundaries, which maintains boundary smoothness and allows the network to detect small changes between geometries; and (iii) no data interpolation is utilized for creating training data; thus accuracy of the CFD data is preserved. None of these features are achievable by extant methods based on projecting scattered CFD data into Cartesian grids and then using regular convolutional neural networks. Incompressible laminar steady flow past a cylinder with various shapes for its cross section is considered. The mass and momentum of predicted fields are conserved. We test the generalizability of our network by predicting the flow around multiple objects as well as an airfoil, even though only single objects and no airfoils are observed during training. The network predicts the flow fields hundreds of times faster than our conventional CFD solver, while maintaining excellent to reasonable accuracy.

I. INTRODUCTION AND MOTIVATION · II. PROBLEM FORMULATION AND METHODOLOGY · A. Governing equations of fluid dynamics

The paper motivates replacing costly CFD iterations with a PointNet-based point-cloud framework that preserves irregular geometry and CFD data while mapping spatial positions to flow quantities. It formulates incompressible viscous flow around arbitrary cylinder cross sections and evaluates generalization to unseen geometries, including an airfoil.

I. INTRODUCTION AND MOTIVATION: CFD-based design optimization remains computationally expensive because flow-field analysis requires many iterations.Machine learning is motivated primarily as a way to reduce CFD computational costs.
I. INTRODUCTION AND MOTIVATION: Cartesian-grid pixelation can reduce CFD-data accuracy through interpolation or extrapolation and miss small geometry changes without costly super-resolved inputs.The paper identifies these as central limitations of CNN-based representations for irregular CFD geometries.
I. INTRODUCTION AND MOTIVATION: The proposed framework treats CFD grid vertices as a point cloud and uses PointNet segmentation with mean squared error to map vertex positions to velocity and pressure.This representation jointly captures spatial geometry and flow quantities without requiring Cartesian-grid projection.
I. INTRODUCTION AND MOTIVATION: The framework is evaluated on incompressible flow past cylinders with different cross sections and on an airfoil absent from training.The benchmark tests both varying cylinder geometries and generalization to a standard airfoil.
A. Governing equations of fluid dynamics: The governing Navier–Stokes and continuity equations express momentum and mass conservation for incompressible viscous Newtonian flow.Velocity, pressure, density, viscosity, body force, and boundary quantities define the formulation.
A. Governing equations of fluid dynamics: The CFD domain is a 38 m × 32 m rectangle containing an arbitrary rigid cylinder cross section centered at (8 m, 16 m), with no-slip object boundaries and prescribed inflow conditions.The setup also specifies free-stream flow at the inflow, bottom, and top boundaries and an open outflow condition.
A. Governing equations of fluid dynamics: With density, viscosity, and free-stream velocity fixed, the object-dependent length scale is the only parameter varied to change Reynolds number.Unstructured finite-volume meshes are generated with refinement near objects and in wakes before numerical solutions are obtained.
A. Governing equations of fluid dynamics: The framework also applies when flow predictions depend on variations in outer-boundary geometry, beyond the inner-boundary cylinder cases considered here.This extends the stated applicability from external flows around objects to domains whose outer boundaries vary.

B. Data generation

The data set covers 2,595 automated CFD geometries formed from seven cylinder cross-section shapes, rotations, and size variations. Training focuses on a task-specific neural-network subdomain around the cylinder and wake, with normalized velocity and pressure outputs.

Geometry and data-set construction: The data set includes seven cross-section shapes—circle, square, triangle, rectangle, ellipse, pentagon, and hexagon—augmented by cylinder rotations and cross-section size changes.These variations generate the training data set.
Flow conditions: The Reynolds number range is 20.0 to 84.0, with the characteristic length defined from the cross-section dimensions.Length scale L is a for several regular shapes and b for rectangles, ellipses, and triangles.
Geometry and data-set construction: 2,595 different geometries require automated mathematical geometry definition, unstructured-grid generation, and CFD computation of velocity and pressure fields.The procedure is implemented using C++ code because manual generation is impractical.
Neural-network domain: The neural-network domain can be a subdomain of the CFD domain, emphasizing the cylinder and wake while excluding unimportant far-field data.The framework permits selecting any VNN such that VNN ⊂ V.
Data normalization: Velocity and pressure outputs are nondimensionalized and scaled to [0, 1] to accelerate convergence and equalize their contributions to network-parameter determination.Input spatial coordinates remain in the physical domain because their normalization did not significantly improve training.

C. Neural network architecture

The framework uses a PointNet segmentation network to map an unstructured point cloud of CFD-domain vertices to pointwise CFD quantities while preserving permutation invariance. Shared MLPs and max pooling form a global feature that is combined with per-point features, while geometry is implicitly encoded by the point-set null space.

Point-cloud regression: The network formulates CFD prediction as f(X) = Y, mapping N spatial points X = {x_i ∈ R^d} to corresponding outputs Y = {y_i ∈ R^nCFD}.Each output y_i corresponds to the input point x_i.
Permutation-invariant processing: PointNet addresses unstructured inputs by applying a shared point-processing function followed by symmetric max pooling, producing outputs invariant to input-point ordering.Permutation invariance is required because the output should not depend on how input points are ordered.
Network architecture: The bottom network concatenates each 64-dimensional intermediate feature with the global feature, forming N features of dimension 1088 before a final shared MLP regresses CFD quantities for every point.This segmentation architecture produces an individual output corresponding to each input point.
Network architecture: The top network uses shared MLPs with sizes (64, 64) and (64, 128, 1024), followed by max pooling to produce a 1024-dimensional global feature.Batch normalization separates MLP layers to improve convergence during training.
Implicit geometry representation: The point set implicitly defines cylinder cross-section geometry through its null space, with the cylinder occupying regions where no points exist.The network is expected to learn the relationship between this implicit geometry representation and the flow solution during training.

D. Training

Training uses mean squared error with Adam optimization, scaled velocity and pressure variables, and randomly split CFD data. A grid search identifies a 1024-dimensional global feature with batch size 25 as optimal by test loss, while predictions improve substantially across training epochs.

Training: Mean squared error is used as the loss function for training.The paper describes mean squared error as a reliable norm for deep-learning applications in computational mechanics.
Training: Ground-truth velocity and pressure fields come from the CFD solver, while predicted fields are scaled back into physical space.The scaled predictions are denoted by ˜u, ˜v, and ˜p.
Training: The Adam optimizer uses learning rate α = 5 × 10−4, β1 = 0.9, β2 = 0.999, and ˆϵ = 10−6 with batch size 256.The dataset contains 2595 generated data, split randomly into training (80%), validation (10%), and test (10%) sets.
Training: After 10 epochs, network predictions are inaccurate, whereas after 100 epochs their overall flow-field structure is correctly configured.At 100 epochs, the homogeneous velocity Dirichlet boundary condition on the object surface is not yet satisfied.

III. RESULTS AND DISCUSSION · A. General analysis

The network accurately predicts velocity and pressure fields across varied object geometries and point distributions, including flow separation, without apparent overfitting to a particular object class. Errors concentrate near solid-wall boundaries, while prediction is substantially faster than CFD simulation.

A. General analysis: The network successfully predicts flow separation across test examples with varied object classes, sizes, orientations, and length scales.The examples also use varying spatial distribution densities, demonstrating the network’s flexibility and generality.
A. General analysis: The largest averaged pointwise error occurs for the x-component of velocity, while the smallest occurs for pressure.The x-component is more complicated in this regime, whereas no pressure boundary condition is imposed in the domain of interest.
A. General analysis: Pointwise errors do not indicate overfitting to a particular object class, and maximum loss does not occur in grids with the highest spatial-density variation.The minimum or maximum loss is not associated with a specific object.
A. General analysis: Maximum pointwise error occurs at object edges where zero-velocity no-slip conditions are applied.The difficulty is attributed to the velocity jump from wall surfaces into the active space and differing point distributions across objects at the boundary.
A. General analysis: 34% lower maximum pointwise error for ˜u results when viscous zero-velocity conditions are replaced with extrapolated inviscid-flow data.This indicates that training is more straightforward for inviscid flows under the tested setup.
A. General analysis: Averaged pointwise error roughly doubles when output data are scaled to [−1, 1] instead of [0, 1].This comparison is based on numerical experiments for the test set.
A. General analysis: Approximately 1846× speedup is achieved for predicting 259 unseen test cases, with average network prediction time of approximately 6 seconds.The corresponding CFD simulations take approximately 11071 seconds; the speedup depends strongly on solver efficiency and CPU/GPU types.

B. Investigation of conservation of mass and momentum

The study evaluates conservation of mass and momentum in predicted flow fields, extending beyond pointwise L2 error to assess all predicted spaces. Residuals are computed by post-processing network predictions on finite-volume meshes for 259 unseen data.

Assessment approach: Conservation analysis complements pointwise L2 error by assessing mass and momentum conservation across all predicted spaces.The authors identify this as a more precise performance analysis than evaluating each predicted field individually.
Residual definitions: Momentum residuals measure Navier–Stokes equation errors in the x and y directions, while continuity residuals measure incompressible-flow mass-conservation error.Residual magnitudes are represented using absolute values.
Residual computation: A first-order finite-volume method numerically approximates spatial derivatives for the conservation residuals.The network input consists of finite-volume grids with available cell connectivity, although the network itself does not use that connectivity.
Results: The predicted velocity and pressure fields are returned to the finite-volume mesh for post-processing analysis on 259 unseen data.The conservation results are reported in Table IV using the International Unit System.

C. Physical interpretation

The network’s intermediate representations reflect physical structure: critical points identify geometry-defining boundary points, while derivative-based residuals assess momentum and continuity in predicted fields. These residuals remain close to zero without imposing the governing equations as loss functions.

Physical interpretation: Critical points are input points whose latent features contribute maximum values to at least one dimension of the global 1024-dimensional feature.They provide a way to inspect features learned by PointNet during inference.
Physical interpretation: All points on bluff-body cylinder boundaries are critical, indicating that the network passes boundary geometry into its latent global feature.This reflects the dependence of flow fields on object geometry.
Physical interpretation: Momentum and continuity residuals are defined for interior points using derivatives of predicted CFD quantities with respect to input coordinates.TensorFlow auto-differentiation computes derivatives after predicting (ũ, ṽ, p̃) at each point.
Physical interpretation: Residuals for momentum and continuity are relatively close to zero at both critical and non-critical points.Average, maximum, and minimum residual values are reported in Table V.
Physical interpretation: The network does not impose the momentum and continuity equations as loss functions.The residuals are evaluated to interpret predictions rather than directly constrained during training.

D. Neural network generalizability

The study evaluates whether the neural network generalizes to unseen domain-geometry categories by predicting velocity and pressure fields around multiple cylinders with different cross sections and an airfoil, none included in training.

Generalizability tests: The generalizability tests target velocity and pressure fields around multiple cylinders with different cross sections and an airfoil absent from the training data.The authors state that high accuracy is not expected for these unseen geometries, but the investigation may reveal network behavior and guide performance improvements.

1. Prediction of flow around multiple objects

The network predicts steady, laminar velocity and pressure fields around multiple objects despite training only on single-object data. It accounts for interactions among bodies and achieves reasonable accuracy under the training flow regime.

Multiple-object prediction: The network predicts velocity and pressure fields around multiple objects without seeing multi-object data during training.Test cases use object spacings that keep the flow steady and laminar.
Object interactions: Different pressure and velocity values in front of two cylinders show that the network accounts for both bodies rather than treating them independently.The first cylinder experiences higher pressure in front of it than the second cylinder.
Generalization: The network also handles three cylinders with elliptical, circular, and rectangular cross sections, a case not performed in cited CNN studies.Those CNN studies were limited by domains too small to accommodate several reasonably separated bodies.
Mechanism and conditions: A 1024-dimensional global latent code extracts point-cloud geometry, while nonlinear functions represent the governing-equation solution as a function of geometric features.This mechanism supports prediction across different numbers of bodies when the flow remains within the trained regime.

2. Prediction of flow around an airfoil

The PointNet-based network predicts velocity and pressure fields around an unseen NACA 0028 airfoil at multiple angles of attack, achieving reasonable accuracy despite no airfoils in training. Errors are lowest at 0°, concentrated near the airfoil front or surface, and increase with rotation because unseen curvature patterns can be misinterpreted.

Generalization to an unseen airfoil: The network predicts velocity and pressure fields around NACA 0028 at angles of attack of 0°, 30°, and −30° despite never seeing airfoils during training.The airfoil’s length scale and associated Reynolds number were within the training-data range.
Accuracy and error analysis: The predictions achieve only a reasonable level of accuracy, with the minimum pointwise L2 error at 0° compared with 30° and −30°.For every angle, the pointwise L2 error of ũ exceeds the corresponding error of ṽ.
Accuracy and error analysis: Unseen upper- and lower-surface curvature can cause the network to treat the airfoil like a circle or ellipse, producing rear-region maximum errors.This curvature mismatch explains the rotation-dependent pressure errors.
Geometric representation: The point cloud accurately represents NACA 0028 through its null space and accommodates high point density near the airfoil surface, which traditional CNNs cannot capture.The network had not encountered this near-surface point organization during training.

E. Potentials for turbulent flow predictions

The framework may extend to turbulent-flow prediction, but thorough machine-learning experiments are still required. Such applications would require more input points for finer resolved meshes, while the network’s compatibility with unstructured data is advantageous for boundary layers and complex geometries.

Scope and validation: Turbulent-flow predictions require a thorough set of machine-learning experiments because the article primarily focuses on laminar steady flows.The authors identify turbulent-flow capability as an assessment requiring additional experiments.
Resolution requirements: Resolved turbulent solutions generally require finer meshes than corresponding laminar flows, making increased N essential to capture relevant grid vertices.The passage explicitly links finer turbulent-flow meshes to more input points.
Unstructured-grid compatibility: Because efficient unstructured grids are key for turbulent boundary layers and complex geometries, the network’s design for connecting unstructured data supports this application.The approach is described as connecting unstructured data to the neural network for training and prediction.

IV. CONCLUSIONS AND FUTURE DIRECTIONS

The paper introduces a PointNet-based framework for predicting velocity, pressure, and other fields in complicated domains while preserving CFD-data accuracy and geometric fidelity. It demonstrates strong performance on varied cylinder geometries and identifies unsteady, three-dimensional, and physics-informed extensions.

Framework: The proposed framework uses a PointNet-based segmentation architecture to predict velocity, pressure, or other fields in complicated domains.It is designed for geometries where Cartesian-grid-based approaches are unsuitable.
Framework: The framework preserves CFD-data accuracy, captures small geometric changes, maintains curved-surface smoothness, and permits non-uniform point distributions.These properties support efficient design optimization without introducing artificial geometric roughness.
Evaluation: For incompressible laminar steady flow past cylinders with varied cross sections, unseen-data L2 norms of predicted velocity and pressure fields demonstrated excellent performance.The dataset varied cross-section rotation, shape, and size while keeping fluid density and viscosity fixed.
Broader applicability: The framework can support computational-mechanics applications beyond CFD, including replacing a Poisson solver within projection schemes for pressure prediction.The proposed use predicts pressure or intermediate pressure fields from the divergence of intermediate velocity.
Future directions: Future work includes unsteady-flow prediction, including regimes where physical or geometric changes produce unsteady behavior, using temporal or spatio-temporal point-cloud representations.The paper notes that two-dimensional circular-cylinder flow becomes unsteady when Reynolds number approximately exceeds 40.0.
Future directions: Future studies will extend the method to three-dimensional objects and merge it with PINNs for semi-supervised prediction in realistic, complicated, and dynamic geometries.The three-dimensional methodology would use grid points located on the object surface, while current PINN versions are limited to fixed geometries.