PI-GANO: Physics-Informed Neural Operator

• PI-GANO is a mesh-free neural operator that integrates Navier–Stokes and Darcy–Forchheimer equations to model steady-state flows.
• It uses a unified physics-informed loss function to learn from variable geometries, boundary conditions, and material parameters without retraining.
• The approach delivers rapid inference—reducing computational time by up to three orders of magnitude compared to traditional CFD methods.

PI-GANO (Physics-Informed Geometry-Aware Neural Operator) is a mesh-free neural operator architecture designed to model steady-state flows simultaneously through and around porous structures of arbitrary geometry, boundary condition, and material parameter, without retraining for each new configuration. It systematically integrates the governing physics of the Navier–Stokes and Darcy–Forchheimer equations within a unified loss formulation, supports variable boundary conditions and material properties, and achieves rapid, geometry-agnostic inference suitable for design acceleration in engineering and environmental flow scenarios (Ciceri et al., 15 Feb 2026).

1. Problem Definition and Motivation

Many problems in engineering, environmental, or atmospheric science feature bodies with both permeable (porous) interiors and free-fluid exteriors—for example, windbreaks, catalysts, or submerged gabions. Standard computational fluid dynamics (CFD) utilizes the Navier–Stokes equations in fluid regions (Ωₚ) and the Darcy–Forchheimer extension in porous regions (Ω_f), with appropriate coupling on the fluid–porous interface (Γ). Classical approaches demand meshing bespoke to each geometry and repeated PDE solves whenever boundary conditions or material parameters (e.g., Darcy D, Forchheimer F, viscosity μ) change.

PI-GANO circumvents these bottlenecks by learning the mapping

$${ \{\partial \Omega_p,\ D,\ F,\ \mu,\ \mathrm{BC}\}\ \longrightarrow\ \{\mathbf{u}(x),\ p(x)\} }$$

in a single, end-to-end physics-informed neural operator. This architecture generalizes to unseen geometries, boundary conditions, and parameter settings without retraining (Ciceri et al., 15 Feb 2026).

2. Mathematical and Physical Model

2.1. Governing Equations

PI-GANO partitions the domain Ω into:

• $\Omega_p$: porous region
• $\Omega_f$: free-fluid region
• Γ = ∂Ωₚ: interface
• ∂Ωₚ, ∂Ω_f: exterior boundaries

On $\Omega_f$, it enforces the incompressible steady Navier–Stokes equations: $$\nabla\cdot \mathbf{u} = 0, \quad \rho\,(\mathbf{u}\cdot\nabla)\mathbf{u} = -\nabla p + \mu\,\nabla^2\mathbf{u}$$ On $\Omega_p$, the Darcy–Forchheimer extension applies: $$\rho(\mathbf{u}\cdot\nabla)\mathbf{u} = -\nabla p + \mu \nabla^2 \mathbf{u} - (\mu D + \tfrac{1}{2} \rho F |\mathbf{u}|)\mathbf{u}$$ Here, $D$ and $F$ are calculated from the porosity $\phi$ and particle diameter $\Omega_p$0 via Kozeny–Carman relations: $\Omega_p$1 Boundary and interface conditions require continuity of velocity and normal stress at the interface Γ, and prescribed Dirichlet/Neumann conditions at the domain boundary ($\Omega_p$2).

2.2. Unified Physics-Informed Loss

Collocation points $\Omega_p$3 throughout Ω are annotated with an indicator function

$\Omega_p$4

At each point, the residual for the full steady PDE is: $\Omega_p$5 The incompressibility residual is: $\Omega_p$6 The total loss minimized by PI-GANO is: $\Omega_p$7 with

$\Omega_p$8

Here, $\Omega_p$9 hyperparameters control the enforcement strength, and $\Omega_f$0 denotes the number of various collocation/data points (Ciceri et al., 15 Feb 2026).

3. Architecture and Training

3.1. Input Encoding and Latent Representations

PI-GANO extends the PointNet backbone of PIPN, forming a full neural operator through:

• Per-point features: coordinates $\Omega_f$1, signed-distance to interface $\Omega_f$2, porous-region indicator $\Omega_f$3, and boundary-type vector $\Omega_f$4 (e.g., inlet, outlet, wall).
• Global parameter inputs: inlet velocity $\Omega_f$5, Darcy $\Omega_f$6, Forchheimer $\Omega_f$7 (with absent BCs set to zero).

A shared MLP encoder $\Omega_f$8 produces a per-point embedding, with max-pooling aggregation yielding a global geometry latent $\Omega_f$9. Boundary/parameter information is embedded by a second MLP $\Omega_f$0 acting on boundary samples, generating a latent $\Omega_f$1.

For each query (or collocation) point $\Omega_f$2, the triple $\Omega_f$3 per-point encoding, $\Omega_f$4, $\Omega_f$5 $\Omega_f$6 is fed to a shared decoder MLP $\Omega_f$7 which outputs $\Omega_f$8 and $\Omega_f$9. This shared decoder design reduces parameter count and achieves better field consistency than separate output branches (Ciceri et al., 15 Feb 2026).

3.2. Training Protocol

• Optimizer: Adam with initial learning rate $$\nabla\cdot \mathbf{u} = 0, \quad \rho\,(\mathbf{u}\cdot\nabla)\mathbf{u} = -\nabla p + \mu\,\nabla^2\mathbf{u}$$0; exponential decay factor $$\nabla\cdot \mathbf{u} = 0, \quad \rho\,(\mathbf{u}\cdot\nabla)\mathbf{u} = -\nabla p + \mu\,\nabla^2\mathbf{u}$$1 set per task.
• Activation: SiLU; $$\nabla\cdot \mathbf{u} = 0, \quad \rho\,(\mathbf{u}\cdot\nabla)\mathbf{u} = -\nabla p + \mu\,\nabla^2\mathbf{u}$$2 for manufactured solutions only.
• Dropout: $$\nabla\cdot \mathbf{u} = 0, \quad \rho\,(\mathbf{u}\cdot\nabla)\mathbf{u} = -\nabla p + \mu\,\nabla^2\mathbf{u}$$3 on last two layers (disabled for manufactured solution).
• Epochs: $$\nabla\cdot \mathbf{u} = 0, \quad \rho\,(\mathbf{u}\cdot\nabla)\mathbf{u} = -\nabla p + \mu\,\nabla^2\mathbf{u}$$4.
• Loss weights: see Table 1.

Experiment	$$\nabla\cdot \mathbf{u} = 0, \quad \rho\,(\mathbf{u}\cdot\nabla)\mathbf{u} = -\nabla p + \mu\,\nabla^2\mathbf{u}$$5	$$\nabla\cdot \mathbf{u} = 0, \quad \rho\,(\mathbf{u}\cdot\nabla)\mathbf{u} = -\nabla p + \mu\,\nabla^2\mathbf{u}$$6	$$\nabla\cdot \mathbf{u} = 0, \quad \rho\,(\mathbf{u}\cdot\nabla)\mathbf{u} = -\nabla p + \mu\,\nabla^2\mathbf{u}$$7	$$\nabla\cdot \mathbf{u} = 0, \quad \rho\,(\mathbf{u}\cdot\nabla)\mathbf{u} = -\nabla p + \mu\,\nabla^2\mathbf{u}$$8 (lr–decay)	#int	#bnd	#obs
MMS	1	1	—	0.9995	667	168	—
Duct (fixed)	1	1	100	0.9990	1000	200	500
Windbreak (3D)	1	10	1	0.9990	2000	1000	1000

Explanation of terms: MMS—manufactured solutions; Duct—2D; Windbreak—3D scenarios. #int/#bnd/#obs: counts of interior, boundary, and observation points.

4. Dataset Generation and Test Cases

4.1. CFD Dataset Creation

Simulation data is generated with OpenFOAM's simpleFoam solver. Domains include both 2D ducts with porous obstacles and 3D windbreak environments (trees, buildings). Boundary conditions and meshes are precisely matched to the requirements of steady, laminar, incompressible, Newtonian flow, with accurate convergence.

• 2D shapes: star, octagon, ellipse; composite geometries held out for generalization tests.
• 3D: Extracted from 3D tree species (oak, pine, cypress, willow, acacia, eucalyptus) and house models, with porosity estimated by projected leaf count.

Features per-point include velocity $$\nabla\cdot \mathbf{u} = 0, \quad \rho\,(\mathbf{u}\cdot\nabla)\mathbf{u} = -\nabla p + \mu\,\nabla^2\mathbf{u}$$9, pressure $\Omega_p$0, region indicator $\Omega_p$1, porosity $\Omega_p$2, SDF, and boundary-type. Uniform sampling with increased density near the interface ensures balanced loss enforcement.

4.2. Data Preprocessing

Point features are standardized (binary fields to [0,1], $\Omega_p$3 via $\Omega_p$4-score). PDE coefficients are scaled accordingly.

• Splits: $\Omega_p$5 train, $\Omega_p$6 validation, $\Omega_p$7 test for all cases.

5. Performance and Generalization Results

5.1. Method of Manufactured Solutions

Verification with manufactured 2D fields,

$\Omega_p$8

demonstrates PI-GANO can reproduce known analytical solutions to within MAEs of order $\Omega_p$9 on both training and unseen geometries.

5.2. 2D Fixed-BC, Variable-BC, and 3D Cases

Case 2 (2D, fixed-BC duct, PIPN):

• Test MAE: $$\rho(\mathbf{u}\cdot\nabla)\mathbf{u} = -\nabla p + \mu \nabla^2 \mathbf{u} - (\mu D + \tfrac{1}{2} \rho F |\mathbf{u}|)\mathbf{u}$$0, $$\rho(\mathbf{u}\cdot\nabla)\mathbf{u} = -\nabla p + \mu \nabla^2 \mathbf{u} - (\mu D + \tfrac{1}{2} \rho F |\mathbf{u}|)\mathbf{u}$$1, $$\rho(\mathbf{u}\cdot\nabla)\mathbf{u} = -\nabla p + \mu \nabla^2 \mathbf{u} - (\mu D + \tfrac{1}{2} \rho F |\mathbf{u}|)\mathbf{u}$$2
• Inference: $$\rho(\mathbf{u}\cdot\nabla)\mathbf{u} = -\nabla p + \mu \nabla^2 \mathbf{u} - (\mu D + \tfrac{1}{2} \rho F |\mathbf{u}|)\mathbf{u}$$3 s per case, OpenFOAM baseline $$\rho(\mathbf{u}\cdot\nabla)\mathbf{u} = -\nabla p + \mu \nabla^2 \mathbf{u} - (\mu D + \tfrac{1}{2} \rho F |\mathbf{u}|)\mathbf{u}$$4 s

Case 3 (2D, variable-BC, PI-GANO):

• Test MAE: $$\rho(\mathbf{u}\cdot\nabla)\mathbf{u} = -\nabla p + \mu \nabla^2 \mathbf{u} - (\mu D + \tfrac{1}{2} \rho F |\mathbf{u}|)\mathbf{u}$$5, $$\rho(\mathbf{u}\cdot\nabla)\mathbf{u} = -\nabla p + \mu \nabla^2 \mathbf{u} - (\mu D + \tfrac{1}{2} \rho F |\mathbf{u}|)\mathbf{u}$$6, $$\rho(\mathbf{u}\cdot\nabla)\mathbf{u} = -\nabla p + \mu \nabla^2 \mathbf{u} - (\mu D + \tfrac{1}{2} \rho F |\mathbf{u}|)\mathbf{u}$$7
• Unseen BC/geometry: $$\rho(\mathbf{u}\cdot\nabla)\mathbf{u} = -\nabla p + \mu \nabla^2 \mathbf{u} - (\mu D + \tfrac{1}{2} \rho F |\mathbf{u}|)\mathbf{u}$$8, $$\rho(\mathbf{u}\cdot\nabla)\mathbf{u} = -\nabla p + \mu \nabla^2 \mathbf{u} - (\mu D + \tfrac{1}{2} \rho F |\mathbf{u}|)\mathbf{u}$$9, $D$0
• Inference: $D$1 s, OpenFOAM baseline $D$2 s

Case 4 (3D windbreak, PI-GANO):

• Global MAE: $D$3, $D$4, $D$5, $D$6
• Inference: $D$7 s, OpenFOAM baseline $D$8 s

In all cases, maximum errors concentrate near sharp corners, region interfaces, and steep-gradient zones; the streamwise velocity $D$9 is particularly affected.

6. Comparative Analysis and Limitations

6.1. Comparative Assessment

PI-GANO, in contrast to PINNs and PIPN:

• Handles novel geometry, boundary condition, and material-parameter combinations in a single forward pass, by virtue of its operator structure.
• Achieves test and generalization errors below those of PIPN when boundary/parameter variations are present.
• Reduces inference latency by one to three orders of magnitude relative to full CFD simulations.
• Provides mesh-free prediction and does not require re-meshing or solver reruns for new cases.

6.2. Limitations and Prospective Advancements

Dominant sources of error remain at flow–porous interfaces and in regions with steep gradients. Remedies discussed include:

• Employing PointNet++ for enhanced capture of fine-scale geometry,
• Adoption of spectral (FNO-style) kernels,
• Adaptive collocation-point sampling targeting high-error zones,
• Incorporating additional PDE physics, such as $F$0–$F$1 turbulence modeling and automatic multi-objective loss weighting.

These enhancements are suggested to further improve the robustness and accuracy of the PI-GANO operator (Ciceri et al., 15 Feb 2026).

7. Conclusion

PI-GANO establishes a systematic, physics-informed, mesh-free neural operator framework for steady-state flows through and around porous geometries. It enables real-time, accurate inference across unseen spatial configurations, boundary settings, and material properties, potentially accelerating parametric and geometric design processes without repeated training or meshing. It unifies geometric awareness, PDE constraints, and operator learning within a single model and demonstrates generalization performance and computational speed superior to conventional CFD approaches and prior physics-informed neural architectures (Ciceri et al., 15 Feb 2026).