ψ − flow: A Novel Physics-Constrained Architecture to Enforce Incompressibility and Boundary Conditions for Fast and Accurate Flow Predictions

                                
==Abstract==

Deeplearning models have demonstrated remarkable capabilities at producing fast predictions of complex flow fields. However, incorporating known physics is essential to ensure that physical solutions can generalize to flow regimes not used for training. In this study, a formulation that, by construction, enforces flow incompressibility and respects the invariance of physical laws across different unit systems is introduced. We demonstrate that this approach can achieve performance improvements of up to 100 times compared to purely data-driven methods, all while maintaining fidelity to other crucial physical quantities. Moreover, we show that for canonical flow test cases, such a physics-constrained model can yield accurate results even with training datasets as small as a few hundred points and neural networks containing only a handful of neurons. It is also shown, however, that physics-constrained machine learning models are not silver bullets out of the box, and require careful consideration in their application and integration with other constraints. Specifically, this study addresses how a problem that is mathematically simple may not necessarily be straightforward in machine learning terms, and discusses ongoing efforts to bridge this gap. We conclude by discussing the place of physics-constrained machine learning models within a landscape primarily dominated by physics-informed approaches, in particular in the context of real-world problems where data and computational resources are often limited