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

Devesh Shah

cs.LGAug 20, 2026

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

Authors: Devesh Shah

Organizations: Independent Researcher

Abstract

Physics-informed neural networks (PINNs) embed governing partial differential equations directly into the training loss, offering a promising alternative to costly CFD solvers for unsteady flows. Yet the growing list of techniques proposed to improve PINN training is typically validated one at a time, leaving open whether these techniques actually compose. We study this question in depth on the DFG/Schafer-Turek unsteady cylinder wake benchmark. In isolation, nearly every technique performs no better than an untreated baseline. However, combining periodic (SIREN) activations with causal weighting unlocks a previously inaccessible regime, reconstructing velocity and pressure fields to within 4.1% average relative L2 error against an OpenFOAM reference solution. Adding further techniques instead causes catastrophic performance degradation, demonstrating that individually effective PINN interventions can interact nonlinearly and that more elaborate training recipes are not necessarily better.

Figures & tables

Figure 1: $U_{y}$ velocity field snapshots from our OpenFOAM ground-truth dataset. Top: $Re=20$ , steady laminar wake, essentially unchanged across a 4-second span. Bottom: $Re=100$ , periodic Kármán vortex street, visibly evolving within a fraction of one shedding period.

Re	Physics Loss	Data Supervision	Ux	Uy	p
20	✓	$\times$	1.1% ∗	2.1% ∗	2.8% ∗
100	✓	$\times$	68.3%	86.2%	118.4%
100	✓	✓	19.8%	70.5%	36.1%
100	$\times$	✓	21.6%	108.3%	35.5%

Table 1: Relative L2 error against OpenFOAM for all baseline experiments. ∗ Note Re=20 results are reported at steady state (

t\geq 2.0

s); all other rows report error over the full trajectory.

Individual techniques
Configuration	Ux	Uy	p	St pred	Amp. ratio	Corr.
Baseline	19.8%	70.5%	36.1%	0.013	0.005	0.005
Fourier Features	19.4%	70.5%	35.2%	0.025	0.007	0.003
Causal Weighting	19.2%	70.5%	35.4%	0.013	0.007	0.005
SIREN Activation	19.5%	70.5%	35.3%	0.025	0.010	0.007
Hard Boundary Conditions	22.2%	72.4%	45.0%	0.025	0.005	$-$ 0.002

Table 2: Relative L2 error, predicted Strouhal number, wake-probe amplitude ratio, and wake-probe correlation (each averaged over

x/D=2,4,6

) for various experiments. True St=0.300 for all rows.

Configuration	Ux	Uy	p	St pred	Amp. ratio	Corr.
SIREN + Causal Weighting (reference)	7.8%	17.1%	16.7%	0.300	0.833	0.988
+ more epochs, trained to convergence	4.4%	10.8%	10.6%	0.300	0.901	0.996
+ deeper model, trained to convergence	3.0%	4.8%	4.6%	0.300	0.983	1.000
+ L-BFGS fine-tuned (on deeper model)	22.4%	80.3%	43.0%	0.013	0.041	0.003

Table 3: Understanding capacity and training budget limitations of the best configuration.

Figure 2: For various configurations we compare (a) Pressure drop

\Delta p(t)

and (b, c) streamwise wake velocity profiles

U_{x}(y)

t=5.0

s and

t=5.4

s against the ground truth.

Figure 3: (a) Training loss for SIREN+causal weighting with and without Self-Adaptive Weights. (b) Self-adaptive loss weights over training: all four escalate to the clamp ceiling in a staggered order, data first, physics last.

Figure 4: Field comparison at $t=5.0$ s for various experimental configurations. Note that SIREN + Fourier Features together collapse to a near-uniform field.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Parameter	Value	Units
Channel length $L$	2.2	m
Channel height $H$	0.41	m
Cylinder centre $(c_{x},c_{y})$	(0.20, 0.20)	m
Cylinder radius $R$	0.05	m
Cylinder diameter $D$	0.10	m
Domain extent ( $z$ )	0.01	m (pseudo-2D)

Appendix

Table 4: Computational geometry. All dimensions are in metres.

Parameter	Value	Units / notes
Meshing tool	Gmsh 4.x	—
Element type	Hexahedral (quad-extruded)	—
Total cells	50,184	—
Far-field element size $l_{c,far}$	0.030	m
Wake refinement size $l_{c,wake}$	0.003	m
Near-cylinder size $l_{c,near}$	0.001	m

Appendix

Table 5: Mesh generation parameters. The mesh is shared by both

Re=20

and

Re=100

simulations.

Boundary	Patch type	Velocity ( $U$ )	Pressure ( $p$ )
Inlet ( $x=0$ )	patch	Parabolic profile ( equation 9 )	zeroGradient
Outlet ( $x=2.2$ )	patch	zeroGradient	fixedValue $=0$
Top/bottom walls	wall	noSlip	zeroGradient
Cylinder surface	wall	noSlip	zeroGradient
Front/back faces	empty	empty (2-D)	empty (2-D)

Appendix

Table 6: Boundary condition specification for all patches.

Parameter	Value
Solver	icoFoam (transient laminar NS)
Time scheme	Euler (1st-order)
Divergence scheme	Gauss linearUpwind
Laplacian scheme	Gauss linear corrected
Pressure solver	GAMG + Gauss–Seidel
Velocity solver	smoothSolver + Gauss–Seidel

Appendix

Table 7: Solver and numerical scheme configuration applied to both simulations.

Parameter	Value	Units / notes
Reynolds number $Re$	20	—
Peak inlet velocity $U_{m}$	0.30	m/s
Mean inlet velocity $U_{mean}$	0.20	m/s $=\tfrac{2}{3}U_{m}$
Time step $\Delta t$	0.001	s
Simulation end time	10.0	s
Write interval	0.1	s (every 100 steps)

Appendix

Table 8: Time integration and output parameters for the

Re=20

(steady) simulation.

Parameter	Value	Units / notes
Reynolds number $Re$	100	—
Peak inlet velocity $U_{m}$	1.50	m/s
Mean inlet velocity $U_{mean}$	1.00	m/s $=\tfrac{2}{3}U_{m}$
Time step $\Delta t$	0.0002	s
Simulation end time	8.0	s (ST benchmark spec)
Write interval	0.1	s (every 500 steps)

Appendix

Table 9: Time integration and output parameters for the

Re=100

(unsteady) simulation.

Figure 5: Full field comparison for the baseline configuration: true (OpenFOAM), predicted (PINN), and absolute error for $U_{x}$ , $U_{y}$ , and $p$ at $t=5.0$ s and $t=5.4$ s. The baseline fails to reconstruct any wake oscillation, collapsing to a smoothed, near-steady near-field response.

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

Dengdi Sun, Jie Chen, Xiao Wang, Jin Tang

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

Physics-Informed ML Multi-Task Learning Computational Fluid Dynamics Multi-Task Optimization

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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

Abstract

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