Generalization Bounds for Physics-Informed Neural Networks for the Incompressible Navier-Stokes Equations

Sebastien Andre-Sloan; Dibyakanti Kumar; Alejandro F Frangi; Anirbit Mukherjee

doi:10.48550/arXiv.2603.23072

Generalization Bounds for Physics-Informed Neural Networks for the Incompressible Navier-Stokes Equations

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Abstract

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.

Original language	English
Publisher	arXiv
DOIs	https://doi.org/10.48550/arXiv.2603.23072
Publication status	Submitted - 24 Mar 2026

Keywords

cs.LG
math.AP
math.NA

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10.48550/arXiv.2603.23072Licence: CC BY

2603.23072v1

MCAIF: Centre for AI Fundamentals
Kaski, S. (PI), Alvarez, M. (Researcher), Pan, W. (Researcher), Mu, T. (Researcher), Rivasplata, O. (PI), Sun, M. (PI), Mukherjee, A. (PI), Caprio, M. (PI), Sonee, A. (Researcher), Leroy, A. (Researcher), Wang, J. (Researcher), Lee, J. (Researcher), Parakkal Unni, M. (Researcher), Sloman, S. (Researcher), Menary, S. (Researcher), Quilter, T. (Researcher), Hosseinzadeh, A. (PGR student), Mousa, A. (PGR student), Glover, E. (PGR student), Das, A. (PGR student), Dursun, F. (PGR student), Zhu, H. (PGR student), Abdi, H. (PGR student), Dandago, K. (PGR student), Piriyajitakonkij, M. (PGR student), Rachman, R. (PGR student), Shi, X. (PGR student), Keany, T. (PGR student), Liu, X. (PGR student), Jiang, Y. (PGR student), Wan, Z. (PGR student), Harrison, M. (Support team), Hartford, J. (PI), Kangin, D. (Researcher), Harikumar, H. (PI), Dubey, M. (PI), Parakkal Unni, M. (PI), Dash, S. P. (PGR student), Mi, X. (PGR student), Barlas, Y. (PGR student), Osho, T. (Support team), Tariq, M. (Support team), Zhang, X. (Researcher), Nzoyem, R. D. (Researcher), Hudson, T. (Researcher) & Casado Tellechea, I. (Researcher)
1/10/21 → 30/09/26
Project: Research

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title = "Generalization Bounds for Physics-Informed Neural Networks for the Incompressible Navier-Stokes Equations",

abstract = "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.",

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author = "Sebastien Andre-Sloan and Dibyakanti Kumar and Frangi, \{Alejandro F\} and Anirbit Mukherjee",

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language = "English",

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T1 - Generalization Bounds for Physics-Informed Neural Networks for the Incompressible Navier-Stokes Equations

AU - Andre-Sloan, Sebastien

AU - Kumar, Dibyakanti

AU - Frangi, Alejandro F

AU - Mukherjee, Anirbit

PY - 2026/3/24

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N2 - 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.

AB - 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.

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KW - math.AP

KW - math.NA

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M3 - Preprint

BT - Generalization Bounds for Physics-Informed Neural Networks for the Incompressible Navier-Stokes Equations

PB - arXiv

ER -

Generalization Bounds for Physics-Informed Neural Networks for the Incompressible Navier-Stokes Equations

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