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Bridging Computational Fluid Dynamics Algorithm and Physics-Informed Learning: SIMPLE-PINN for Incompressible Navier-Stokes Equations

This paper introduces SIMPLE-PINN, a novel Physics-Informed Neural Network framework that integrates velocity and pressure correction terms inspired by the SIMPLE algorithm to overcome training instability and accelerate convergence, thereby achieving highly accurate, fully data-free solutions for complex incompressible Navier-Stokes problems at high Reynolds numbers and over long time horizons.

Original authors: Chang Wei, Yuchen Fan, Chin Chun Ooi, Jian Cheng Wong, Heyang Wang, Pao-Hsiung Chiu

Published 2026-03-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Chang Wei, Yuchen Fan, Chin Chun Ooi, Jian Cheng Wong, Heyang Wang, Pao-Hsiung Chiu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to teach a super-smart robot how to predict how water flows through pipes, around bridges, or even inside a jet engine. This is the job of Computational Fluid Dynamics (CFD).

Traditionally, we solve these problems using "brute force" math: we chop the space into millions of tiny Lego bricks (a grid) and calculate the flow in each brick. It's accurate, but it's slow and requires massive supercomputers.

Enter PINNs (Physics-Informed Neural Networks). These are like a different kind of robot. Instead of using a grid, they are "learners." You give them the rules of physics (like Newton's laws) and a few boundary conditions, and they try to guess the entire flow pattern all at once. It's like asking a student to solve a whole math problem by understanding the concept rather than doing step-by-step arithmetic.

The Problem:
While PINNs are cool, they are notoriously bad at solving fluid problems, especially when the fluid moves fast or gets turbulent.

The Analogy: Imagine trying to balance a broom on your hand. If you just try to keep it upright (minimizing error), it wobbles and falls. In fluid dynamics, the "broom" is the relationship between speed (velocity) and pressure. If the speed goes up, the pressure must go down, and vice versa. Standard PINNs often get this relationship wrong, causing the simulation to crash or produce nonsense results. They try to learn the speed and pressure separately, but they forget to talk to each other.

The Solution: SIMPLE-PINN
The authors of this paper built a new framework called SIMPLE-PINN. To understand it, let's look at the name: SIMPLE stands for Semi-Implicit Method for Pressure-Linked Equations. It's a famous, old-school algorithm used by traditional CFD experts for decades to fix the speed-pressure balance.

The authors asked: "What if we could teach the neural network to use this old-school 'balancing trick'?"

Here is how they did it, using simple metaphors:

1. The "Correction" Mechanism (The Coach)

In the old-school method, if the speed and pressure get out of sync, the computer calculates a "correction" and nudges them back into place.

The Metaphor: Imagine a dance partner (Speed) and a lead (Pressure). If they step on each other's toes, a coach (the SIMPLE algorithm) steps in, says, "Whoa, you're out of sync! Speed, step back a bit; Pressure, lean forward," and they correct their move.
The Innovation: Standard PINNs don't have this coach. They just guess and hope. SIMPLE-PINN builds the coach directly into the robot's brain. Every time the robot makes a guess, it calculates a "correction loss" (a penalty score) based on how well the speed and pressure are dancing together. If they are out of sync, the penalty is huge, forcing the robot to fix the relationship immediately.

2. The "Hybrid" Strategy (The Swiss Army Knife)

Fluids often flow around weird shapes, like a curved airplane wing or a jagged rock.

The Problem: The "brute force" math (Finite Volume Method) works great in open spaces but gets confused near jagged edges. The "learning" method (Automatic Differentiation) is flexible but can be imprecise.
The Solution: SIMPLE-PINN is a Swiss Army Knife.
- In the open, safe zones of the fluid, it uses the robust, grid-like math (Finite Volume) because it's stable.
- Right next to the jagged edges or walls, it switches to the flexible "learning" math (Automatic Differentiation) to handle the curves perfectly.
- It seamlessly blends the two, getting the best of both worlds.

3. The "Time Travel" Trick (No Time Windows)

Simulating fluid over a long time (like watching a vortex form over 100 seconds) is usually a nightmare for AI. Usually, you have to break time into tiny chunks (windows), solve one, then use that result to start the next. If you make a tiny mistake in the first window, the error piles up like a snowball rolling down a hill, ruining the whole simulation.

The Breakthrough: SIMPLE-PINN can look at the entire timeline at once without breaking it into chunks. It's like reading a whole book in one go rather than reading one page, closing the book, and trying to remember the plot for the next page. This allows it to simulate complex, long-term events (like a cylinder shedding vortices for 100 seconds) without the errors piling up.

Why This Matters (The Results)

The paper shows that this new "Coach" and "Swiss Army Knife" approach is a game-changer:

Speed: It solved a notoriously difficult high-speed flow problem (Reynolds number 20,000) in just 448 seconds. That's less than 8 minutes! Other methods either failed or took hours.
Accuracy: It perfectly predicted the swirling "vortex streets" behind a cylinder for a long time, something previous AI models struggled to do.
Versatility: It worked on everything from simple boxes to complex airplane wings and even heat-driven fluid instabilities.

In a Nutshell

SIMPLE-PINN is like taking a brilliant but clumsy student (the Neural Network) and giving them a veteran coach (the SIMPLE algorithm) and a multi-tool (the Hybrid strategy). The coach ensures they don't make fundamental mistakes in how they balance forces, and the multi-tool lets them handle any shape or obstacle. The result is a solver that is fast, stable, and accurate enough to tackle the most difficult fluid dynamics problems in engineering and science.

1. Problem Statement

Physics-Informed Neural Networks (PINNs) have emerged as a promising tool for solving Partial Differential Equations (PDEs) without large labeled datasets. However, when applied to incompressible Navier-Stokes (N-S) equations, particularly at high Reynolds numbers (Re) or in complex geometries, standard PINNs face significant challenges:

Training Instability & Slow Convergence: The strong nonlinearity of convection terms and the tight coupling between velocity and pressure create a highly non-convex optimization landscape.
Velocity-Pressure Decoupling: Standard PINNs often fail to strictly enforce the continuity equation ( $\nabla \cdot \mathbf{u} = 0$ ) globally, leading to non-physical solutions where mass conservation is violated at unsampled points.
Geometric Limitations: Traditional Finite Volume Method (FVM) discretizations struggle with irregular boundaries (e.g., airfoils, complex obstacles) due to stencil dependencies, while pure Automatic Differentiation (AD) can be computationally expensive and less stable for stiff gradients.
Long-term Prediction: Existing methods often require time-windowing (dividing time into small segments) to simulate unsteady flows, leading to error accumulation.

2. Methodology: The SIMPLE-PINN Framework

The authors propose SIMPLE-PINN, a novel framework that bridges classical numerical algorithms (specifically the SIMPLE algorithm) with PINNs. The core innovation is the derivation of velocity and pressure correction loss terms inspired by the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE).

Key Methodological Components:

Correction Loss Function ( $L_{RC}$ ):
- Instead of relying solely on PDE residuals, the framework introduces explicit correction terms derived from the momentum and continuity residuals.
- Derivation: The authors discretize the N-S equations using a simplified FVM. By analyzing the residuals at iteration $n$ and assuming physical constraints are satisfied at $n+1$ , they derive expressions for pressure ( $p$ ) and velocity ( $u, v$ ) corrections.
- Approximation for NNs: To make these corrections compatible with neural networks (which cannot access future iteration variables $n+1$ $n + 1$ or complex neighbor stencils directly), the authors employ:
  1. Taylor Expansion & Extrapolation: To approximate neighbor contributions and next-iteration values (e.g., $u^{n+1} \approx 2u^n - u^{n-1}$ ).
  2. Relaxation Factors: $\alpha_p, \alpha_u, \alpha_v$ are introduced to stabilize the correction updates.
- The correction loss is formulated as an $L_1$ norm penalty enforcing that the update from iteration $n$ to $n+1$ follows the derived correction physics.
Hybrid Differentiation Strategy (AD + FVM):
- To handle complex geometries (e.g., airfoils, square cylinders), the framework uses a hybrid approach:
  - Interior Points: Use simplified FVM discretization for efficiency and conservation properties.
  - Near-Boundary Points: Use Automatic Differentiation (AD) when FVM stencils fall outside the domain. This preserves geometric flexibility while maintaining the robustness of FVM in the bulk flow.
Architecture:
- Uses a Multi-Layer Perceptron (MLP) with frequency annealing mapping for inputs.
- Shared hidden layers followed by specific output branches for velocity ( $u, v$ ) and pressure ( $p$ ).
- Trained using Adam optimizer with a warmup cosine decay learning rate.

3. Key Contributions

Algorithmic Integration: Successfully transforms the classical SIMPLE algorithm's velocity-pressure coupling mechanism into a differentiable loss function suitable for neural network training, overcoming the "black box" limitation of standard PINNs.
Data-Free High-Re Solution: Achieves the first fully data-free PINN solution for lid-driven cavity flow at Re = 20,000, a regime where most PINNs fail or require massive labeled data.
Long-Term Unsteady Prediction: Demonstrates the ability to simulate unsteady vortex shedding (Kármán vortex street) over a continuous time domain ( $t=0$ to $100$) without time-windowing, a significant advancement over existing PINN methods that suffer from error accumulation.
Multiphysics Capability: Successfully solves the Rayleigh-Taylor instability (Ra = $10^6$ ) involving coupled momentum and energy equations, capturing complex thermal-fluid interactions.
Geometric Flexibility: Validates the hybrid AD/FVM strategy on complex geometries including NACA0012 airfoils and arrays of square cylinders, showing superior accuracy in resolving steep gradients near sharp corners compared to pure AD or pure FVM approaches.

4. Experimental Results

The framework was validated on six benchmark problems:

Benchmark Case	Key Achievement	Performance Metrics
Lid-Driven Cavity (Re=20k)	First PINN solution at this Re.	448s training time; Relative $L_2$ error $\approx 3.65 \times 10^{-2}$ (velocity). Outperforms FFV-PINN which failed to converge.
Wavy Channel (Re=100)	Complex geometry handling.	Relative $L_2$ error $\approx 3.91 \times 10^{-3}$ ; Training time 219s.
NACA0012 Airfoil	Curved boundaries & pressure gradients.	Hybrid AD/FVM reduced velocity error by >1 order of magnitude compared to non-AD. Captured suction peaks accurately.
3 Square Cylinders	Multiple obstacles & sharp corners.	Successfully captured complex vortex interactions and wake interference. Training time 335s (open) / 451s (channel).
Unsteady Cylinder (Re=100)	Long-term prediction ( $t=0-100$ ).	Captured the full evolution of the Kármán vortex street without time-windowing. MSE $\approx 10^{-3}$ .
Rayleigh-Taylor (Ra= $10^6$ )	Multiphysics (Thermal-Fluid).	Continuous time simulation ( $t=0-6$ ) without decomposition. 9x more efficient than SOAP-based PINN (which used time-windowing).

Comparison: SIMPLE-PINN consistently outperformed state-of-the-art variants (e.g., JAXPI, PirateNet, SOAP, FFV-PINN) in terms of convergence speed, accuracy at high Re, and stability in long-term unsteady simulations.

5. Significance and Impact

Paradigm Shift: The paper moves beyond simply embedding PDE residuals into loss functions. It demonstrates that classical numerical insights (like pressure-velocity coupling corrections) can be mathematically reformulated as neural network loss terms to drastically improve stability and convergence.
Engineering Applicability: By solving high-Re flows and complex geometries efficiently, SIMPLE-PINN bridges the gap between academic PINN research and practical engineering applications in aerospace, civil, and mechanical engineering.
Scalability: The ability to simulate long-duration unsteady flows without time-windowing opens new possibilities for PINNs in transient fluid dynamics, potentially reducing the computational overhead associated with traditional time-marching schemes.
Future Potential: The authors note that the underlying principle of "constraint correction" could be extended to other coupled physical systems (e.g., solid mechanics, electromagnetism) where conservation laws and divergence-free conditions are critical.

In conclusion, SIMPLE-PINN represents a significant step forward in making Physics-Informed Learning a robust, competitive, and reliable solver for complex, nonlinear, and high-Reynolds number fluid dynamics problems.