UniPINN: A Unified PINN Framework for Multi-task Learning of Diverse Navier-Stokes Equations

Dengdi Sun; Jie Chen; Xiao Wang; Jin Tang

cs.CVMar 11, 2026

UniPINN: A Unified PINN Framework for Multi-task Learning of Diverse Navier-Stokes Equations

Authors: Dengdi Sun, Jie Chen, Xiao Wang, Jin Tang

Organizations: Key Laboratory of Intelligent Computing & Signal Processing (ICSP), Ministry of Education, School of Artificial Intelligence, Anhui University, Hefei 230601, China · Anhui Provincial Key Laboratory of Multimodal Cognitive Computing, School of Computer Science and Technology, Anhui University, Hefei 230601, China

Abstract

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

Figures & tables

Figure 1: Single-task vs. unified multi-flow learning. (a) Conventional single-task PINNs train independent networks for each flow type, incurring parameter redundancy and neglecting the shared Navier-Stokes structure across flows. (b) UniPINN unifies multiple flow types under a shared backbone with task-specific heads and cross-flow attention, enabling joint learning and knowledge transfer while preserving flow-specific fidelity.

Figure 2: UniPINN architecture. It consists of three components: (1) Task-shared backbone network: extracts common physical features from spatiotemporal coordinates and task embeddings for all flow types. (2) Task-specific feature extraction: the task-aware multi-task attention selectively reinforces the shared representation; the task-specific dedicated layers decode the enhanced features into stream function and pressure, with velocity obtained via automatic differentiation. (3) Dynamic weight allocation: adaptively reweights equation, boundary, and data loss terms across flows to ensure stable joint optimization.

Method	Lid-Driven	Pipe Flow	Couette flow
Linear Regression [ 41 ]	6.61e-02	5.79e-01	6.61e-02
GPR [ 42 ]	2.51e-02	3.97e-01	4.95e-01
DNN [ 43 ]	3.25e-02	2.80e-01	6.83e-02
LAAF-PINN [ 44 ]	-	2.26e-01	-
KIH-PINN [ 45 ]	2.49e-02	2.41e-01	3.39e-02
AL-PINN [ 46 ]	3.12e-02	1.91e-01	4.73e-02

Table 1: Comparative Study Results: Performance Comparison (MSE) Across Different Methods.

Figure 3: Predicted flow field visualization (streamline plot). The model accurately captures characteristic physical structures: (a) Lid-driven cavity primary vortex; (b) Pipe flow parabolic profile; (c) Couette flow linear profile.

Attention	Input Norm.	DWA	Lid-Driven	Pipe Flow	Couette flow
✓	✓	✓	1.27e-02	1.25e-01	1.27e-02
✓	✓	$\times$	1.83e-02	2.17e-01	1.18e-01
✓	$\times$	✓	2.57e-02	1.65e-01	2.40e-02
$\times$	✓	✓	8.59e-02	2.62e-01	1.62e-02
$\times$	$\times$	✓	1.22e-01	3.51e-01	6.39e-02
$\times$	$\times$	$\times$	1.42e-01	3.78e-01	9.21e-02

Table 2: Ablation study results: Component analysis and performance comparison (MSE).

Figure 4: Training convergence analysis (loss curves) during multi-flow PINN training. (a) Unweighted Total Loss; (b) Equation Loss (Navier-Stokes residuals); (c) Boundary Condition Loss; (d) Data Loss. All loss components and the total loss show stable convergence under the DWA strategy.

Transfer Scenario	Setting	Final MSE	Loss Reduction	Initial Loss
Source: Pipe & Lid Target: Couette	From scratch	0.1226	Baseline	3004
	Frozen Backbone	0.0236	80.7%	1.51
	Fine-tuned	0.0216	82.4%	1.08
Source: Pipe & Couette Target: Lid-Driven	From scratch	6.2407	Baseline	97.7
	Frozen Backbone	0.0268	99.96%	4.84
	Fine-tuned	0.0214	99.97%	1.76

Table 3: Cross-task transfer learning performance using pre-trained shared backbone.

Target Task	Baseline	Source: Lid	Source: Pipe	Source: Couette
Lid-Driven	0.087	—	0.135 (+55%)	0.158 (+81%)
Pipe	0.235	0.297 (+26%)	—	0.339 (+44%)
Couette	0.049	0.056 (+14%)	0.068 (+39%)	—

Table 4: Negative transfer analysis when reusing task-specific attention features. Values indicate Final MSE and Relative Increase over Baseline.

Explore similar work

MUSA-PINN: Multi-scale Weak-form Physics-Informed Neural Networks for Fluid Flow in Complex Geometries
Mar 9, 2026Weizheng Zhang, Xunjie Xie, Hao Pan +4Neural PDE Solvers PDE Solving
Shandong University, Qingdao, China · Tsinghua University, Beijing, China · Institute of Engineering Thermophysics, Chinese Academy of Sciences, Beijing, China
Complementary, Not Cumulative: Interaction Effects in Physics-Informed Neural Networks for Navier-Stokes Vortex Shedding
Aug 20, 2026Devesh ShahPhysics-Informed ML Neural Network Training Dynamics
Independent Researcher
Multiscale Physics-Informed Neural Network for Complex Fluid Flows with Long-Range Dependencies
Apr 7, 2026Prashant Kumar, Rajesh RanjanNeural PDE Solvers Physics-Informed ML
Department of Aerospace Engineering, Indian Institute of Technology Kanpur, Kanpur, India 208016

Mar 9, 2026J/K move · Enter open · S save

MUSA-PINN: Multi-scale Weak-form Physics-Informed Neural Networks for Fluid Flow in Complex Geometries

Weizheng Zhang, Xunjie Xie, Hao Pan, Xiaowei Duan+3

Shandong University, Qingdao, China · Tsinghua University, Beijing, China · Institute of Engineering Thermophysics, Chinese Academy of Sciences, Beijing, China

Neural PDE Solvers PDE Solving Physics-Informed ML Computational Fluid Dynamics

Open paper AskPDF arXiv

While Physics-Informed Neural Networks (PINNs) offer a mesh-free approach to solving fluid-flow PDEs, standard point-wise residual minimization suffers from convergence pathologies in topologically complex domains like Triply Periodic Minimal Surfaces (TPMS). The locality bias of point-wise constraints fails to propagate global information through tortuous channels, causing unstable gradients and conservation violations. To address this, we propose the Multi-scale Weak-form PINN (MUSA-PINN), which reformulates Navier-Stokes equation constraints as integral conservation laws over hierarchical spherical control volumes. We enforce continuity and momentum conservation via flux-balance residuals on control surfaces. Our method utilizes a three-scale subdomain strategy-comprising large volumes for long-range coupling, skeleton-aware meso-scale volumes aligned with transport pathways, and small volumes for local refinement-alongside a two-stage training schedule prioritizing continuity. Experiments on steady incompressible flow in TPMS geometries show MUSA-PINN outperforms state-of-the-art baselines, reducing relative errors by up to 93% and preserving mass conservation.

Abstract

Explore similar work

MUSA-PINN: Multi-scale Weak-form Physics-Informed Neural Networks for Fluid Flow in Complex Geometries

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

Multiscale Physics-Informed Neural Network for Complex Fluid Flows with Long-Range Dependencies