Complementary, Not Cumulative: Interaction Effects in Physics-Informed Neural Networks for Navier-Stokes Vortex Shedding

Physics-Informed Neural Networks (PINNs) have shown promise in solving incompressible Navier-Stokes equations, yet existing approaches are predominantly designed for single-flow settings. When extended to multi-flow scenarios, these methods face three key challenges: (1) difficulty in simultaneously capturing both shared physical principles and flow-specific characteristics, (2) susceptibility to inter-task negative transfer that degrades prediction accuracy, and (3) unstable training dynamics caused by disparate loss magnitudes across heterogeneous flow regimes. To address these limitations, we propose UniPINN, a unified multi-flow PINN framework that integrates three complementary components: a shared-specialized architecture that disentangles universal physical laws from flow-specific features, a cross-flow attention mechanism that selectively reinforces relevant patterns while suppressing task-irrelevant interference, and a dynamic weight allocation strategy that adaptively balances loss contributions to stabilize multi-objective optimization. Extensive experiments on three canonical flows demonstrate that UniPINN effectively unifies multi-flow learning, achieving superior prediction accuracy and balanced performance across heterogeneous regimes while successfully mitigating negative transfer. The source code of this paper will be released on https://github.com/Event-AHU/OpenFusion

This work establishes rigorous first-of-its-kind upper bounds on the generalization error for the method of approximating solutions to the (d+1)-dimensional incompressible Navier-Stokes equations by training depth-2 neural networks trained via the unsupervised Physics-Informed Neural Network (PINN) framework. This is achieved by bounding the Rademacher complexity of the PINN risk. For appropriately weight bounded net classes our derived generalization bounds do not explicitly depend on the network width and our framework characterizes the generalization gap in terms of the fluid's kinematic viscosity and loss regularization parameters. In particular, the resulting sample complexity bounds are dimension-independent. Our generalization bounds suggest using novel activation functions for solving fluid dynamics. We provide empirical validation of the suggested activation functions and the corresponding bounds on a PINN setup solving the Taylor-Green vortex benchmark.

Physics-informed neural networks (PINNs) provide a mesh-free approach for solving differential equations, but their performance can depend strongly on loss weighting, collocation placement, and optimization strategy. This study develops an adaptive PINN framework for the Blasius boundary-layer equation using gradient-norm-based adaptive loss weighting, nonuniform and residual-based collocation, and sequential Adam--L-BFGS optimization. In the representative run using the architecture $[1,100,100,1]$, the model predicts $f''(0)=0.3320762918$, compared with the high-accuracy benchmark $0.332057336215$, giving an absolute error of $1.896\times10^{-5}$. The final weighted loss is $6.789\times10^{-8}$, and the predicted stream-function, velocity, and shear profiles agree closely with an independent numerical boundary-value solution. A separate full-training architecture study shows that the two-hidden-layer model achieves the smallest wall-shear error among the four tested architectures, $1.629\times10^{-6}$, whereas the deepest network attains the smallest weighted objective but a substantially larger wall-shear error. Compared with the previously reported PINN value $f''(0)=0.33165$, the representative run reduces the wall-shear error by approximately a factor of $21.5$. The results show that the combined adaptive training framework can achieve high accuracy for the Blasius problem and that weighted loss alone is insufficient for identifying the most physically accurate PINN. Because the adaptive components are applied jointly, their individual contributions cannot be isolated from the present results and would require a controlled ablation study for separate assessment.