HPST: A Hybrid Physics-Spectral-Threshold Framework for Fluid Flow Analysis with Theorem Proving and Graph Neural Networks

We present HPST, a unified framework that inte- grates symbolic theorem proving, statistical physical analysis, and graph neural networks (GNNs) for ro-bust and interpretable modeling of fluid flows. HPST combines an associative-commutative (AC) matching rewriting engine to verify algebraic identities, a data- driven module to compute physical invariants (e.g. ,Bernoulli’s principle and adaptive thresholds), and a GNN surrogate based on EdgeConv layers that learns velocity fields from scattered point clouds. Exten- sive experiments on a synthetic cylinder wake dataset (40,000 points) demonstrate that HPST successfully proves three fundamental theorems (commutativity, associativity, distributivity) and that the optimized GNN achieves a coefficient of determination up to R2 = 0.208 with an average R2 = 0.164 ± 0.03 over mul- tiple runs, while also reducing the mean absolute di-vergence to 0.27—a measure of physical consistency.Comparison with baseline models (k-nearest neigh- bors, linear regression) shows that HPST offers com- petitive accuracy while providing interpretability and a layer of mathematical verification. The framework’s modular design and robust performance make it di-rectly applicable to industrial scenarios such as aero-dynamic shape optimization, automotive drag prediction, and wind-farm layout planning. All experi-ments were conducted on a Kaggle environment with an NVIDIA P100 GPU (16 GB RAM), and the complete source code is publicly available.