MF-PINN: Model Fusion for Multi-Task Physics-Informed Neural Networks Solving the Navier–Stokes Equations

Navier–Stokes equations at different Reynolds numbers exhibit substantial differences, so their solution models must capture both the shared physical structures across flow regimes and the specific local flow patterns of each regime. A conventional single shared multi-task model is easily affected by task interference. Model fusion can reuse information from multiple task models, but for the complex characteristics of the Navier–Stokes equations, direct parameter fusion may introduce conflicting updates and weaken physical consistency.To address this issue, we propose MF-PINN (Model-Fusion Physics-Informed Neural Network), a framework that combines task-vector fusion with physics-informed task adaptation. MF-PINN extracts task-specific updates from a commonly pretrained PINN and fuses them into a shared Base model. It then keeps the Base model fixed and uses lightweight task-specific Adapters to recover local flow characteristics under different flow regimes. The model is trained using the Navier–Stokes equations, boundary conditions, and real flow-field data, while a stream-function representation enforces incompressibility by construction.Experiments on six Reynolds-number tasks from PDEBench show that MF-PINN achieves lower and more stable physics-informed test losses across tasks than independent PINNs, shared multi-task PINNs, parameterized PINNs, and general model-fusion methods. It also requires substantially fewer task-specific parameters than storing a complete model for each task. These results demonstrate that task-vector fusion and physics-informed adaptation can effectively balance cross-task knowledge sharing and task-specific prediction accuracy.