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

arXiv: https://arxiv.org/abs/2607.06587 · HTML: https://mlsanity.com/paper/2607.06587/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 61 references that are in the map.
- First posted on arXiv 5 Jul 2026.

## Builds on (most cited first, 17 of 17)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Lookahead Optimizer: k steps forward, 1 step back](https://mlsanity.com/paper/1907.08610/) (2019)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Deep Learning Methods for Reynolds-Averaged Navier-Stokes Simulations of Airfoil Flows](https://mlsanity.com/paper/1810.08217/) (2018)
- [Prediction of Aerodynamic Flow Fields Using Convolutional Neural Networks](https://mlsanity.com/paper/1905.13166/) (2019)
- [Physics-Informed Diffusion Models](https://mlsanity.com/paper/2403.14404/) (2024)
- [Deep learning of free boundary and Stefan problems](https://mlsanity.com/paper/2006.05311/) (2020)
- [Aero-Nef: Neural Fields for Rapid Aircraft Aerodynamics Simulations](https://mlsanity.com/paper/2407.19916/) (2024)
- [X-MeshGraphNet: Scalable Multi-Scale Graph Neural Networks for Physics Simulation](https://mlsanity.com/paper/2411.17164/) (2024)
- [Machine Learning-augmented Predictive Modeling of Turbulent Separated Flows over Airfoils](https://mlsanity.com/paper/1608.03990/) (2016)
- [U-net architectures for fast prediction of incompressible laminar flows](https://mlsanity.com/paper/1910.13532/) (2019)
- [NeuralFoil: An Airfoil Aerodynamics Analysis Tool Using Physics-Informed Machine Learning](https://mlsanity.com/paper/2503.16323/) (2025)
- [GoRINNs: Godunov-Riemann Informed Neural Networks for Learning Hyperbolic Conservation Laws](https://mlsanity.com/paper/2410.22193/) (2024)
- [Generative prediction of flow fields around an obstacle using the diffusion model](https://mlsanity.com/paper/2407.00735/) (2024)
- [Adaptive Interface-PINNs (AdaI-PINNs): An Efficient Physics-informed Neural Networks Framework for Interface Problems](https://mlsanity.com/paper/2406.04626/) (2024)
- [Accelerated Airfoil Design Using Neural Network Approaches](https://mlsanity.com/paper/2503.24052/) (2025)

## Abstract (arXiv)

We present CoFINN (Conservation Flux Informed Neural Networks), a physics-informed deep learning framework for predicting compressible flow fields governed by conservation laws. Unlike conventional data-driven convolutional neural networks (CNNs), which optimize only pixel-wise similarity metrics, CoFINN embeds finite-volume conservation physics directly into the training process. Unlike classical physics-informed methods which enforce differential-equation residuals at collocation points through automatic differentiation, CoFINN adopts a finite-volume perspective consistent with modern CFD methodology. CoFINN interprets CNN output fields as structured computational grids, where each pixel represents a finite-volume cell, and enforces conservation consistency through sophisticated numerical flux calculations. The framework is evaluated on transonic flow prediction around airfoils at (M=0.7, Re=6 * 10^6), including challenging conditions involving shock waves and high angles of attack. Results show that CoFINN improves aerodynamic force prediction accuracy, reducing drag prediction error by up to 34% at extreme angles of attack and by approximately 15% on average across the test set. Improvements are particularly significant in limited-data regimes, demonstrating that the conservation-based loss acts as an effective physical regularizer. The proposed approach maintains the computational efficiency advantages of CNN surrogates while significantly improving physical consistency and conservation behavior. The framework is architecture-agnostic and extensible to broader classes of conservation-law-governed physical systems.