Fengbo: a Clifford Neural Operator pipeline for 3D PDEs in Computational Fluid Dynamics
Alberto Pepe Mattia Montanari Joan Lasenby
We introduce Fengbo, a pipeline entirely in Clifford Algebra to solve 3D partial differential equations (PDEs) specifically for computational fluid dynamics (CFD). Fengbo is an architecture composed of only 3D convolutional and Fourier Neural Operator (FNO) layers, all working in 3D Clifford Algebra. It models the PDE solution problem as an interpretable mapping from the geometry to the physics of the problem. Despite having just few layers, Fengbo achieves competitive accuracy, superior to 5 out of 6 proposed models reported in \cite{li2024geometry} for the $\emph{ShapeNet Car}$ dataset, and it does so with only 42 million trainable parameters, at a reduced computational complexity compared to graph-based methods, and estimating jointly pressure \emph{and} velocity fields. In addition, the output of each layer in Fengbo can be clearly visualised as objects and physical quantities in 3D space, making it a whitebox model.  
By leveraging Clifford Algebra and establishing a direct mapping from the geometry to the physics of the PDEs, Fengbo provides an efficient, geometry- and physics-aware approach to solving complex PDEs.
Clifford Algebra, Neural Operator, PDE Modelling, Geometric Machine Learning
Fengbo is a Clifford Algebra pipeline for solving 3D PDEs in CFD. With 42M parameters, it achieves competitive accuracy, outperforms five out of six models and enables 3D visualization, offering an efficient and interpretable geometry-aware solution.