Source-linked AI summary
Physics-informed PointNet: A deep learning solver for steady-state incompressible flows and thermal fields on multiple sets of irregular geometries
Ali Kashefi, Tapan Mukerji
TL;DR
Physics-informed models face limitations in labeled-data requirements, GPU memory, training convergence, and forward-solver speed, while existing methods generally target one computational domain. PIPN combines PointNet-based geometry encoding with physics-informed losses to solve across multiple irregular domains and generalize to unseen geometries, achieving low errors in forward and inverse tests.
Problem
Existing physics-informed models generally solve PDEs for only one computational domain per training procedure, while supervised approaches require plentiful labeled measurements.
Method
PIPN combines a PointNet-based neural network for geometric features and unordered point clouds with a physics-informed loss built from governing-equation residuals and problem constraints.
Results
PIPN solved forward problems across varied geometries with 0.1% velocity and 1.3% pressure average relative pointwise errors, and inverse problems on new geometries with errors below 12%.
Takeaways & Limitations
The framework enables one trained model to predict solutions across multiple irregular domains, including unseen geometries from seen categories.
Takeaways & Limitations
Point-cloud distributions can impair implicit shape representation and cause higher errors for some geometries; physics-informed models also require more GPU memory and slower training than supervised models.
Abstract
from arXiv · showhide
We present a novel physics-informed deep learning framework for solving steady-state incompressible flow on multiple sets of irregular geometries by incorporating two main elements: using a point-cloud based neural network to capture geometric features of computational domains, and using the mean squared residuals of the governing partial differential equations, boundary conditions, and sparse observations as the loss function of the network to capture the physics. While the solution of the continuity and Navier-Stokes equations is a function of the geometry of the computational domain, current versions of physics-informed neural networks have no mechanism to express this functionally in their outputs, and thus are restricted to obtain the solutions only for one computational domain with each training procedure. Using the proposed framework, three new facilities become available. First, the governing equations are solvable on a set of computational domains containing irregular geometries with high variations with respect to each other but requiring training only once. Second, after training the introduced framework on the set, it is now able to predict the solutions on domains with unseen geometries from seen and unseen categories as well. The former and the latter both lead to savings in computational costs. Finally, all the advantages of the point-cloud based neural network for irregular geometries, already used for supervised learning, are transferred to the proposed physics-informed framework. The effectiveness of our framework is shown through the method of manufactured solutions and thermally-driven flow for forward and inverse problems.
1. Introduction and motivation
The paper introduces Physics-Informed PointNet (PIPN), combining PointNet-based geometric representation with physics-informed losses to solve PDEs across irregular computational domains. It addresses the lack of geometric awareness in existing physics-informed models and evaluates the framework on forward and inverse incompressible-flow problems.
- Physics-informed modeling: Physics-informed models minimize governing-equation and condition residuals, reducing labeled-data requirements for forward problems and enabling inverse discovery from sparse measurements.Forward models use PDE and boundary-condition residuals, while inverse models can use sparse interior observations when some boundary conditions are unknown.
- Motivation: Existing physics-informed neural networks lack a mechanism to capture computational-domain geometry in their outputs.Their commonly used fully connected architectures do not explicitly represent the geometric characteristics of the domain.
- Point-cloud representation: PointNet avoids pixelation artifacts, supports sensitivity to geometric changes, and permits adaptive point distributions for irregular domains.Its point-cloud input also makes spatial derivatives explicit with respect to network inputs, unlike image-based representations.
- Proposed framework: PIPN combines PointNet’s point-cloud geometry encoding with a physics-informed loss to solve PDEs on multiple irregular domains using one training procedure.PointNet represents domains without pixelation, while the physics-informed objective uses residuals of the governing equations and associated conditions.
- Capabilities: After training on one domain set, PIPN predicts solutions for unseen geometries from both seen and unseen categories.This extends the framework beyond the single-domain setting of existing physics-informed models.
- Evaluation: The framework is evaluated on manufactured-solution and thermally driven-flow cases, including pressure-boundary effects, formulation choices, body-force implementations, and generalizability.The experiments cover forward and inverse problems for incompressible flows and thermal fields.
2. Governing equations of interest
The paper considers steady two-dimensional incompressible Newtonian flow through conservation equations for mass, momentum, and energy. The computational domain is a non-simply connected fluid region formed by a square outer space and case-specific internal space.
- Governing equations: The governing equations describe conservation of mass, momentum, and energy for steady incompressible Newtonian flow in two dimensions.The variables include velocity, pressure, temperature, body force, density, viscosity, thermal conductivity, and specific heat.
- Computational domain: The target is to obtain PDE solutions over non-trivial fluid-domain geometries rather than a single simply connected region.The fluid domain V is defined as a non-simply connected space.
- Computational domain: The outer space H is square with side length L, while the internal space W is specified separately for each test case.This construction defines the geometry used for the two investigated cases.
3. Physics-informed PointNet (PIPN)
PIPN combines a PointNet-based architecture with physics-informed losses to map irregular-domain point clouds to coupled PDE solutions. It represents geometry through point-cloud features and trains on PDE, boundary-condition, and observation residuals.
- 3.1. Methodology: The PIPN skeleton combines a PointNet-based neural network with a physics-informed loss function.The network extracts geometric features from unordered point sets, while the loss enforces the governing physics.
- 3.1. Methodology: PIPN maps each domain point cloud to corresponding PDE fields, with each output depending on both the point coordinate and encoded domain geometry.The mapping is trained over multiple computational spaces and can be applied to new spaces after training.
- 3.2. Loss function: For forward problems, the loss uses mean squared PDE residuals at interior points and boundary-condition errors at boundary points; inverse problems additionally use sparse observation errors.The residuals are obtained from derivatives of network outputs with respect to point-cloud inputs, using automatic differentiation.
- 3.3. Neural network architecture: PointNet encodes each unordered point cloud through shared functions and a permutation-invariant max operator, producing a global feature of size 1024.The architecture uses a first branch for geometric encoding and a second branch for decoding the output fields.
- 3.3. Neural network architecture: The decoder concatenates the 1024-dimensional global feature with a 64-dimensional intermediate feature and predicts the desired fields using shared MLPs.The decoder MLPs have forms (512, 256, 128) and (128, nPDE), with hyperbolic tangent activations and batch normalization after fully connected layers.
- 3.3. Neural network architecture: Because the target equations involve second-order derivatives, PIPN requires an activation function with a well-defined second derivative; ReLU causes training divergence in this setting.The study restricts the current cases to two dimensions, with three output fields for velocity and pressure and four when temperature is included.
4.1. Method of manufactured solutions in non-trivial geometries
PIPN is evaluated on 26 irregular geometries using manufactured solutions, achieving successful steady-state incompressible-flow predictions while exposing sensitivity to point-cloud distributions and pressure boundary conditions. The experiments also test computational cost, alternative equation implementations, and generalization to unseen geometries.
- General analysis: The manufactured-solutions experiment evaluates PIPN on steady-state continuity and Navier–Stokes equations over 26 geometries with varying shapes, orientations, and point distributions.Exact solutions isolate errors attributable to PIPN, while nonconstant boundary conditions add difficulty.
- General analysis: PIPN successfully predicts velocity and pressure fields across the 26-geometry set, with graphical comparisons agreeing with the manufactured solutions.Average, maximum, and minimum relative pointwise L2 errors are reported in Table 2.
- General analysis: Maximum local errors occur at both boundary and interior points, with particularly high errors for rectangular and symmetrical-star geometries.Uneven point concentrations on short sides and sparse sampling on long sides impair their implicit shape representations.
- Effect of pressure boundary condition: Omitting the pressure boundary term increases average relative velocity errors by 36.484% in x, 43.890% in y, and pressure error by 4873.483%.Velocity errors remain on the order of 10^-2, whereas pressure error is approximately magnified by a factor of 10.
- Effect of pressure boundary condition: Pressure-gradient predictions preserve velocity accuracy more effectively than pressure-field accuracy because the Navier–Stokes equations use pressure gradients.Removing the pressure boundary term also reduces convergence time by 14036 seconds, approximately 4 hours.
- Conservative vs. non-conservative implementation: The conservative implementation increases velocity errors by 9.114% for u and 7.343% for v, and pressure error by 7.621%, while requiring more iterations.Its extra continuity-equation weighting makes optimization slower despite fewer automatic-differentiation operations.
- Generalizability: PIPN predicts solutions for unseen geometries, including domains with points outside seen spaces and two unseen circular cavities, with relative velocity and pressure errors below 2% in the latter case.Accurate predictions are obtained over most curved outer boundaries despite modifying boundary point density.
4.2. Natural convection in a square enclosure with a cylinder
The study evaluates PIPN on thermally driven flow in square enclosures containing varied inner-cylinder geometries, including forward and inverse settings. PIPN produces close solutions across many geometries and generalizes to unseen cylinder shapes, while pressure observations and point-cloud resolution strongly affect accuracy.
- Experimental setup: PIPN is trained jointly over 108 enclosure geometries, using point clouds with sensors distributed to measure velocity, temperature, and pressure fields.The experiment includes sensors measuring all three fields and sensors measuring only pressure and temperature.
- Inverse problem: The inverse problem reconstructs velocity, temperature, and pressure fields from boundary conditions and sparse sensor observations, including the unknown inner-cylinder temperature distribution.The problem is mathematically ill-posed and uses sparse measurements of velocity, temperature, and pressure.
- Forward and inverse evaluation: 2994 iterations and 99330 s of computation produced converged PIPN predictions for the 108-geometry set.The reported runtime is approximately 27.6 hours.
- Forward and inverse evaluation: PIPN predictions agree well with numerical solutions, although maximum local errors occur near inner-cylinder boundaries where geometric variation is greatest.The reported error concentration is associated with the inner-cylinder boundaries across the evaluated geometries.
- Observation effects: Removing pressure observations increases the average relative pressure error by approximately 7218%, from 2.89030E−2 to 2.11516.Velocity and temperature errors remain roughly unchanged because the framework preserves pressure-gradient accuracy.
- Resolution effects: Coarse point clouds, such as N=1000, considerably increase prediction errors because they represent geometry and fluid-flow fields less accurately.The effect reflects both poorer geometric-feature extraction and lower spatial resolution.
- Generalizability: PIPN generalizes to unseen circular and hexagonal inner-cylinder geometries, with the hexagonal case reaching 16.554% relative error for the x-velocity field.The authors report agreement with ground truth in both unseen-shape cases, while sharper or more varied boundaries remain challenging.
5. Summary and future studies
The study introduces PIPN to solve steady-state incompressible-flow problems across multiple geometries and tests its accuracy and generalizability on seen and unseen shapes. Results support accurate forward solutions and inverse predictions across several geometry sets, while future work targets broader applications.
- Summary: PIPN addresses the limitation that physics-informed neural networks traditionally solve PDEs on only one fixed computational domain.The framework incorporates geometry through a PointNet-based architecture.
- Summary: 0.1% average relative pointwise error for velocity and 1.3% for pressure were obtained over domains with varied geometries in the manufactured-solution test.These values use the L2 norm.
- Summary: Inverse predictions over 27 domains with unseen geometries from seen categories had average relative pointwise errors below 12%.The model was trained on 108 domains with equilateral nonagonal, octagonal, and heptagonal inner cylinders.
- Summary: For circular and hexagonal inner cylinders from unseen categories, predicted flow fields had relative errors ranging from approximately 2% to 17%.This test examined generalizability beyond the geometry categories used for inverse-problem training.
- Future studies: Future studies plan to extend PIPN to time-dependent incompressible flow and other computational-physics applications.Listed applications include compressible flows, turbulent flows, solid mechanics, and porous-media flow.
CRediT authorship contribution statement
The authors contributed across conceptualization, methodology, software, writing, visualization, project administration, and funding acquisition.
- Authorship contributions: Ali Kashefi handled conceptualization, methodology, software, writing, and visualization, while Tapan Mukerji contributed to conceptualization, writing, project administration, and funding acquisition.