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PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain
Han Gao, Luning Sun, Jian-Xun Wang
TL;DR
CNN-based physics-informed learning is efficient for large-scale PDE fields but is limited by irregular, non-uniform geometries. PhyGeoNet uses a geometry-adaptive architecture to learn parametric PDE solutions without labeled data, achieving convergence more than an order of magnitude faster than FC-NN PINN and higher accuracy under equal training budgets.
Problem
Classic CNNs are not directly applicable to irregular domains with non-uniform grids, while traditional and FC-NN approaches face efficiency or scalability challenges.
Method
PhyGeoNet uses physics-constrained CNN learning with elliptic geometry transformation to leverage classic CNN backbones for data-free parametric PDE solution learning on irregular domains.
Results
Convergence is more than an order of magnitude faster than PINN with FC-NN formulation, and accuracy is much higher under the same total training budget.
Takeaways & Limitations
The proposed architecture enables direct use of classic CNN backbones for physics-informed learning of PDEs on irregular domains without labeled data.
Takeaways & Limitations
The method addresses the challenge that classic CNNs are not directly applicable when Euclidean-distance convolution filters lose invariance on non-uniform meshes.
Abstract
from arXiv · showhide
Recently, the advent of deep learning has spurred interest in the development of physics-informed neural networks (PINN) for efficiently solving partial differential equations (PDEs), particularly in a parametric setting. Among all different classes of deep neural networks, the convolutional neural network (CNN) has attracted increasing attention in the scientific machine learning community, since the parameter-sharing feature in CNN enables efficient learning for problems with large-scale spatiotemporal fields. However, one of the biggest challenges is that CNN only can handle regular geometries with image-like format (i.e., rectangular domains with uniform grids). In this paper, we propose a novel physics-constrained CNN learning architecture, aiming to learn solutions of parametric PDEs on irregular domains without any labeled data. In order to leverage powerful classic CNN backbones, elliptic coordinate mapping is introduced to enable coordinate transforms between the irregular physical domain and regular reference domain. The proposed method has been assessed by solving a number of PDEs on irregular domains, including heat equations and steady Navier-Stokes equations with parameterized boundary conditions and varying geometries. Moreover, the proposed method has also been compared against the state-of-the-art PINN with fully-connected neural network (FC-NN) formulation. The numerical results demonstrate the effectiveness of the proposed approach and exhibit notable superiority over the FC-NN based PINN in terms of efficiency and accuracy.
1. Introduction
The paper addresses efficient, data-free learning of parametric PDE solutions, focusing on the mismatch between CNN requirements and irregular geometries. It proposes PhyGeoNet to extend classic CNN backbones to irregular domains.
- Traditional PDE solvers require substantial computation for complex systems and repeated simulations in optimization, inverse problems, and uncertainty quantification.
- CNNs offer efficient large-scale field learning through parameter sharing, but existing physics-informed CNN studies primarily assume image-like regular domains.
- Physics-informed learning trains neural networks by minimizing PDE-residual violations, enabling deterministic solution of specified PDEs without labeled data.
- Irregular-domain parametric inputs such as geometry, boundary conditions, and equation parameters remain comparatively underexplored in physics-informed surrogate modeling.
- PhyGeoNet enables data-free parametric PDE learning on irregular geometries, directly leveraging uniform Cartesian-grid CNN architectures and hard boundary-condition enforcement.
- The paper identifies CNN-based learning of parametric Navier–Stokes equations on complex geometries without labeled data as a first attempt to its authors’ knowledge.
2. Methodology
The methodology formulates physics-constrained CNN learning through PDE residuals and addresses irregular geometries by encoding elliptic coordinate transformation into a geometry-adaptive CNN framework.
- Physics-constrained learning computes PDE residuals from neural-network-approximated solutions and uses them to construct a PDE-based loss function.
- Classic CNN derivatives are computed with finite-difference convolution operations, which fail on irregular domains with non-uniform grids.
- PhyGeoNet introduces a geometry-adaptive CNN framework that encodes elliptic coordinate transformation to handle non-uniform grids and irregular geometries.
2.1. Physics-constrained learning with classic convolutional neural network
The classic CNN formulation maps spatial coordinates and PDE parameters to solution fields and trains using physics-based residuals, but rasterization introduces major irregular-domain limitations.
- The parametric PDE objective is to learn steady-state solution fields from spatial coordinates and parameters, supporting efficient online inference after offline training.
- CNNs generate entire solution fields end-to-end, whereas FC-NNs produce pointwise outputs and impose considerable training burdens on large-scale problems.
- Physics-constrained CNN training minimizes PDE residuals without solving the governing equations during optimization, while boundary conditions may be enforced softly or architecturally.
- Classic CNNs require uniform grid points in rectangular domains, making their spatially invariant convolution filters unsuitable for irregular non-uniform meshes.
- Rasterized representations are not differentiable with respect to input design parameters, limiting their use for geometry optimization and design.
2.2. Physics-Informed Geometry-Adaptive Convolutional Neural Network
PhyGeoNet adapts CNN-based physics-informed learning to non-uniform grids and irregular geometries by mapping physical solution fields to a rectangular reference domain. It reformulates the PDE constraints there and strictly enforces boundary conditions through padding.
- PhyGeoNet enables CNN-based physics-informed learning for problems with non-uniform grids and irregular geometries.
- Coordinate transformation between physical and reference domains: Coordinate transformations map irregular physical domains to rectangular reference domains, allowing a classic uniform-grid CNN backbone to be used.The deterministic geometry transformation can be precomputed as part of the network architecture.
- Coordinate transformation between physical and reference domains: The forward map G: Ωr 7→Ωp locates physical-domain collocation points corresponding to uniform reference-domain mesh grids.The forward map is obtained numerically by solving elliptic equations with specified boundary conditions.
- Reformulate physics-constrained learning on reference domain: Jacobians of G convert differential operators from the physical domain to the reference domain, where finite-difference CNN filters implement derivatives.Higher derivatives are obtained by applying the finite-difference filters multiple times instead of using the chain rule.
- Hard enforcement of boundary conditions: Hard boundary-condition enforcement strictly satisfies both Dirichlet and Neumann conditions at the discrete level through padding.Constant padding imposes Dirichlet conditions, while Neumann padding is derived from internal-node solutions using finite differences.
- Physics-informed CNN architecture in reference domain: Separate sub-CNNs represent each state variable on the reference domain, decoupling trainable filters across variables with different scales.The architecture uses physical parameters and coordinates as input channels and adapts a classic CNN with three hidden convolution layers.
3. Numerical Results
The numerical experiments evaluate PhyGeoNet on irregular-domain heat, Navier–Stokes, and other parameterized PDEs without labeled data, using finite-volume simulations as benchmarks. Across the reported cases, predictions capture benchmark solution fields, while accuracy varies with Reynolds number, geometry, and testing-point proximity.
- Experimental scope: The experiments cover non-parametric heat and Navier–Stokes equations on arbitrary irregular shapes, followed by parameterized boundary conditions, varying geometries, and spatially varying source terms.All parameterized solutions are learned without data, with finite-volume simulations used for comparison.
- Heat equation: The heat-equation case maps the irregular physical domain to a regular reference domain through elliptic transformation, enabling PDE-constrained CNN learning.The physical domain uses a boundary-fitted mesh, whereas optimization and the PDE loss are defined on the mapped regular domain.
- Heat equation: 0.098 relative prediction error was obtained for the non-parametric heat equation against the finite-volume benchmark, while accurately capturing the temperature distribution.The benchmark imposes T = 0 on the upper boundary and T = 1 on the remaining sides.
- Navier–Stokes equations: PhyGeoNet captures irregular-domain flow features reasonably well across different Reynolds numbers in the laminar regime, including boundary layers and pressure variations.The reported error increases slightly as Reynolds number rises because convection becomes more dominant.
- Parameterized geometries: For varying vascular geometries, predictions generally agree with CFD benchmarks and reproduce geometry-dependent boundary layers, pressure regions, and outlet acceleration without training data.Performance deteriorates as stenosis increases, and errors rise when testing points move farther from training input points.
- Parameterized problems: 0.0687 mean relative error for training samples and 0.0703 for testing samples were reported, supporting PhyGeoNet as a surrogate for high-dimensional problems.The corresponding variances were 0.0025 for training and 0.0041 for testing samples.
4. Discussion
The discussion compares PhyGeoNet with FC-NN PINN formulations and examines constrained data-driven learning on irregular domains. PhyGeoNet converges faster, achieves higher accuracy under matched budgets, and better satisfies physical constraints, while limitations remain for dynamic systems, highly complex geometries, and unlabeled turbulent Navier–Stokes problems.
- Comparison with PINN: FC-NN PINNs may not scale efficiently to complex, large-scale problems, motivating CNN-based formulations for improved training efficiency.The comparison evaluates convergence cost and predictive accuracy under both fully trained and equal-budget scenarios.
- Comparison with PINN: Both methods capture the general flow pattern, but PhyGeoNet agrees better with CFD in developing velocity regions and near-inlet pressure distributions.PINN shows notable velocity discrepancies and fails to accurately predict pressure near the inlet.
- Comparison with PINN: PhyGeoNet needs about 10 times fewer iterations and roughly 15 minutes instead of 2.5 hours to converge, with higher predictive accuracy than PINN.The paper attributes this efficiency to convolutional updates of the entire field rather than pointwise training.
- Constrained data-driven learning: In a turbulent irregular-domain data-driven test at Re = 50000, enforcing divergence-free structure improves mass conservation but introduces slightly larger oscillations.The constrained and purely data-driven solutions reach the same training-error level.
- Limitations and future perspectives: The framework currently targets parametrized steady-state PDEs, struggles with geometries exceeding five C0-continuous edges, and has not solved unlabeled turbulent Navier–Stokes equations.The authors discuss temporal architectures, domain decomposition, and turbulence closure or subgrid-scale models as possible future developments.
5. Conclusion
The conclusion presents PhyGeoNet as a physics-informed CNN for parameterized PDEs on irregular domains without labeled data. Elliptic mapping enables classic CNN backbones, and comparisons report faster convergence and higher accuracy than FC-NN PINNs.
- Conclusion: PhyGeoNet solves parameterized PDEs on irregular domains without relying on labeled data.The method is described as a physics-informed geometry-adaptive CNN.
- Conclusion: Elliptic mapping transforms irregular physical domains to regular reference domains, enabling classic CNN backbones on non-rectangular geometries and non-uniform grids.Because the mapping is obtained numerically, the method can address geometries that are not parameterizable.
- Conclusion: The method is demonstrated on nonparametric and parametric heat and Navier–Stokes equations across irregular domains.The evaluation includes varying geometries and parameterized settings.
- Conclusion: PhyGeoNet converges more than an order of magnitude faster than FC-NN PINN and is much more accurate when the total training budget is fixed.The conclusion identifies fast convergence and matched-budget accuracy as the principal comparison outcomes.
Appendix A. Proof of forward elliptic transformation
The appendix proves derivative relationships for the forward elliptic transformation and clarifies how the Jacobian is handled during and after mapping construction.
- Forward elliptic transformation: Combining the transformed derivative identities establishes the required relationships between physical and reference coordinates.The proof multiplies and sums the relevant identities to reach the final result.
- Forward elliptic transformation: The forward map’s derivatives are obtained from the mapping equations and substituted into the transformed differential relations.The proof uses the stated equations to derive the coordinate derivatives required by the transformation.
- Forward elliptic transformation: The Jacobian J is variable while mappings are built, but becomes a constant non-zero point-wise value after the mapping is established.Its partial derivatives are calculated during construction using the specified equations.
Appendix B. Convolution filter for derivatives on reference domain
The appendix defines convolution filters for approximating derivatives on the regular reference domain. It uses high-order central differences internally and one-sided differences near boundaries to avoid padding artifacts.
- Derivative filters: Internal reference-domain nodes use fourth-order central differences represented as convolution filters.Figure B.25 illustrates the finite-difference convolution filter for the differential operator.
- Derivative filters: Boundary and near-boundary nodes use one-sided upwind or downwind differences to avoid ghost-cell padding artifacts.Third-order one-sided formulas are given for the lower and left boundaries.
Appendix C. Compuational hyperparameter setting for PhyGeoNet
Table C.4 presents the hyperparameter settings used for PhyGeoNet across the numerical experiments.
- Table C.4 lists PhyGeoNet hyperparameter settings for the numerical experiments.
- The settings are organized by numerical experiment.
- The table concerns the computational configuration of PhyGeoNet.
Appendix D. Convergence History of PhyGeoNet Training
Figure D.26 shows convergence histories for PhyGeoNet training across heat-equation, Navier–Stokes, and Poisson-equation experiments. The figure distinguishes parametric and non-parametric cases where indicated and includes continuity and momentum histories for Navier–Stokes.
- The heat-equation cases include non-parametric and parametric experiments.
- The displayed scales include 10^1 through 10^5 and 0 through 2.
- The Navier–Stokes histories include continuity, x-momentum, and y-momentum.
- The Navier–Stokes cases include non-parametric and parametric experiments.
- The figure also includes a parametric Poisson-equation case.
- Figure D.26 presents convergence histories of PhyGeoNet training.