💻 computer science

CoFINN: Conservation Flux Informed Neural Networks for Physics Problems Governed by Conservation Laws

CoFINN is a novel physics-informed deep learning framework that enhances the accuracy and physical consistency of compressible flow predictions by embedding finite-volume conservation laws directly into CNN training, significantly reducing drag prediction errors even in data-scarce regimes.

Original authors: Adnan Harun Doğan, Mert Deniz, Hande Alemdar, Özgür Uğraş Baran

Published 2026-07-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Adnan Harun Doğan, Mert Deniz, Hande Alemdar, Özgür Uğraş Baran

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

The Big Idea: Teaching AI the Rules of the Road

Imagine you are trying to teach a robot how to drive a car. You have two ways to do it:

The "Look-Alike" Method (Old Way): You show the robot thousands of videos of cars driving. The robot learns to guess what the next frame looks like by matching patterns. If the car in the video turns left, the robot learns to turn left because that's what it sees. However, the robot doesn't actually understand physics. It doesn't know that if it turns too fast, the car will flip. It just knows that "turning left" usually follows "steering wheel left."
The "Rule-Follower" Method (CoFINN): You still show the robot the videos, but you also give it a strict rulebook: "Conservation of Momentum." You tell the robot, "No matter what, if you push the car forward, the air must push back with equal force." If the robot predicts a turn that violates this rule, you don't just say "wrong picture"; you say, "You broke the laws of physics."

CoFINN is a new type of AI that uses the second method. It is designed to predict how air flows around airplane wings (aerodynamics). Instead of just trying to make its predictions look like the training photos, it forces the AI to obey the strict laws of physics that govern how air moves.

The Problem with Current AI: The "Blurry Shockwave"

When air moves very fast (like around a plane wing), it creates sharp, sudden changes called shockwaves. Think of these like a sonic boom or a sudden wall of pressure.

The Old AI's Mistake: Standard AI models are like a painter who is good at blending colors but bad at sharp lines. They might predict a shockwave, but they might place it slightly to the left or right, or make it too blurry.
Why This Matters: In aerodynamics, a tiny shift in where a shockwave happens can completely change the amount of drag (air resistance) and lift (upward force) the plane experiences.
- Analogy: Imagine a tightrope walker. If you move the rope just one inch to the left, the walker might fall. Similarly, if the AI predicts the shockwave in the wrong spot, the calculated "force" on the plane can be wildly wrong, even if the picture looks mostly correct.

How CoFINN Works: The "Pixel as a Room" Trick

The authors came up with a clever way to teach the AI these rules.

The Grid: Usually, AI sees an image as a grid of pixels (dots of color).
The Transformation: CoFINN treats every single pixel not just as a dot of color, but as a tiny, sealed room (a "control volume").
The Traffic Flow: Imagine air is a crowd of people moving between these rooms.
- If 10 people leave Room A to go to Room B, Room A must lose 10 people, and Room B must gain 10.
- The AI is trained to calculate exactly how many "people" (air mass, energy, momentum) flow across the walls between these pixel-rooms.
The Check: The AI is penalized if the math doesn't add up. If it predicts that 10 people left Room A but only 8 arrived at Room B, the AI knows it made a mistake because matter cannot disappear.

By using a specific math tool called the HLLC Riemann Solver (think of it as a very strict traffic cop), CoFINN ensures that the air flowing out of one pixel equals the air flowing into the next, exactly as real physics demands.

The Results: Better Predictions, Especially When Data is Scarce

The researchers tested this on airplane wings at high speeds. Here is what they found:

The "Physics" Bonus: When they added the physics rules (the CoFINN loss) to the training, the AI got much better at predicting drag (how hard it is to push the plane through the air).
- In some difficult cases (extreme angles of attack), the error dropped by 34%.
- On average, the error dropped by 15%.
The "Small Data" Superpower: The biggest win happened when the AI didn't have a lot of training data to learn from.
- Analogy: If you only show a student 5 examples of a problem, a normal student might guess randomly. But a student who knows the rules of math can solve it correctly even with few examples. CoFINN acts like that student; the physics rules act as a safety net that prevents the AI from guessing wildly when it hasn't seen enough examples.
It Works on Different AI Types: The researchers tried this method on four different types of AI architectures (including standard image processors and more complex "Transformer" models). In almost every case, adding the physics rules improved the results.

Summary

CoFINN is a bridge between two worlds:

Deep Learning: Fast, pattern-matching AI that is great at looking at pictures.
Fluid Dynamics: The strict, mathematical laws that govern how air and water move.

By forcing the AI to treat its output like a grid of tiny rooms where air must balance perfectly, CoFINN creates predictions that are not just "pretty pictures," but physically accurate simulations. This makes it much more reliable for designing airplanes, especially in tricky situations where standard AI tends to fail.

Technical Summary: CoFINN (Conservation Flux Informed Neural Networks)

Problem Statement
Traditional machine learning approaches for fluid-flow prediction, particularly Convolutional Neural Networks (CNNs), function as data-driven approximators that optimize pixel-wise similarity metrics. While these models offer significant computational speedups over high-fidelity Computational Fluid Dynamics (CFD), they lack inherent awareness of the governing physical laws. In aerodynamic applications involving compressible flows, this limitation is critical. Purely data-driven models often fail to enforce strict conservation of mass, momentum, and energy, leading to physically inconsistent predictions. This is especially problematic in regions with strong physical interactions, such as shock waves and boundary layers, where small spatial inaccuracies can result in large errors in derived engineering quantities like lift and drag. Furthermore, standard CNNs do not inherently preserve the physical coupling between variables (pressure, density, velocity, energy) dictated by the Navier-Stokes equations and the equation of state.

Methodology: CoFINN Framework
The authors propose CoFINN (Conservation Flux Informed Neural Networks), a physics-informed deep learning framework that embeds finite-volume conservation physics directly into the training process. Unlike classical Physics-Informed Neural Networks (PINNs) that enforce differential-equation residuals at collocation points via automatic differentiation, CoFINN adopts a finite-volume perspective consistent with modern CFD methodology.

Key methodological components include:

Image-as-Mesh Interpretation: The CNN output field is interpreted as a structured computational grid where each pixel represents a finite-volume cell.
Flux-Based Conservation: Conservation consistency is enforced through sophisticated numerical flux calculations across cell interfaces. The framework utilizes the Harten-Lax-van Leer Contact (HLLC) Riemann solver, a robust approximate Riemann solver used in high-fidelity CFD, to compute convective fluxes between neighboring pixels.
Loss Function Formulation: The training objective combines a standard data-driven loss (Mean Absolute Error between predicted and ground truth primitive variables) with a physics-informed loss. The physics loss minimizes the residual between the ground-truth flux balance (derived from high-fidelity CFD data) and the convective flux calculated from the network's prediction.
- $L_{total} = (1 - \lambda) \times L_{data} + \lambda \times L_{flux}$
- Here, $L_{flux}$ measures the error in momentum conservation by comparing the predicted flux sum against a precomputed "Ground Truth Flux Residual" ( $F_{GT}$ ) that accounts for viscous terms and internal sources.
Architecture Agnosticism: The methodology is designed to be architecture-agnostic, demonstrated by its application to four distinct network families: CNNFoil (Encoder-decoder CNN), Fourier Neural Operator (FNO), Vision Transformer (ViT), and Conditional Denoising Diffusion models.

Key Contributions

Finite-Volume Integration: The paper introduces a novel approach to physics-informed learning that treats CNN outputs as finite-volume cells, bridging the gap between data-driven image processing and rigorous numerical CFD methods.
Conservation-Aware Training: By utilizing the HLLC flux scheme within the loss function, the model is explicitly guided toward physically admissible solutions that satisfy discrete conservation behavior, rather than just learning geometric correlations.
Improved Force Prediction: The framework demonstrates that incorporating conservation fluxes significantly improves the prediction of aerodynamic forces (lift and drag), which are integral quantities derived from the flow field, even if pixel-wise variable errors remain comparable to data-driven baselines.

Results
The framework was evaluated on transonic flow prediction around airfoils at $M = 0.7$ and $Re = 6 \times 10^6$ , covering angles of attack from $-10^\circ$ to $20^\circ$ .

Drag Prediction Accuracy: CoFINN significantly reduced drag prediction errors. For the CNNFoil architecture, drag error ( $\epsilon_{CD}$ ) was reduced by up to 34% at extreme angles of attack and by approximately 15% on average across the test set compared to purely data-driven baselines.
Limited-Data Regimes: The conservation-based loss acted as an effective physical regularizer, showing particularly significant improvements in limited-data regimes (e.g., training on a single angle of attack), where drag error reductions reached up to 85%.
Model Performance: While the CNNFoil architecture showed the most consistent improvements across all training ranges, FNO and ViT also demonstrated reduced drag errors with CoFINN. However, the Diffusion model showed mixed results, with improvements in limited-data regimes but degradation in full-data regimes.
Physical Consistency: The method successfully corrected severe underpredictions of drag near zero-lift conditions (e.g., $\alpha = 0^\circ$ ) that plagued baseline models, recovering physical magnitudes that matched ground truth to three decimal places.

Significance and Claims
The paper claims that CoFINN successfully transforms neural networks from purely image-processing systems into conservation-aware predictive models. The primary significance lies in maintaining the computational efficiency advantages of CNN surrogates while significantly improving physical consistency and conservation behavior. The authors emphasize that this approach does not reduce data requirements (as it is supervised by CFD data) but reshapes the loss landscape to favor predictions that satisfy discrete conservation laws. This leads to more reliable and usable predictions for real-world design applications, particularly in scenarios involving shock waves and high angles of attack where purely data-driven models often fail to capture the correct physical coupling of variables. The framework is presented as extensible to broader classes of physical systems governed by conservation laws.