DPGNN: Differentiable Physics- and Geometry-Assisted Neural Network	

Alberto Pepe		
Cambridge University/Huawei

We propose DPGNN, a differentiable physics- and geometry-aided pipeline for the solution of 2D partial differential equations describing the turbulent flow of a fluid around obstacles. We build upon the work of \cite{brach2023}, that proposes a hybrid framework through a differentiable solver-in-the-loop strategy called differentiable physics-assisted neural network (DPNN). While the DPNN outperforms other supervised learning approaches by keeping the network solutions sound from a physical perspective, it lacks explicit geometrical information about the problem. We include such geometrical information by working in two-dimensional (2D) Geometric Algebra, $G(2,0,0)$, replacing the ResNet of the DPNN with a Clifford ResNet and by recasting the inputs in multivector form, with scalar part encoding information about the obstacles and vector part encoding information about the fluid velocity. DPGNN, which modifies DPNN minimally, achieves lower minima during training, it shows comparable performances to its geometry agnostic counterpart at a fraction of the number of its trainable parameters and it is more robust when estimating the fluid flow for large $\Delta t$.