Solving and Learning Navier-Stokes Dynamics with the Neural Basis Method
Authors: Yuhe Wang, Zhu Wang, Min Wang
Organizations: Institute for Scientific Computation, Texas A&M University, College Station, TX 77843, USA · Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA · Department of Mathematics, University of Houston, TX 77204, USA
Physics-informed machine learning has become a major direction in predictive modeling of fluid dynamics, combining governing physics with learned representations. Yet prevailing formulations neither guarantee that small PDE-residual losses imply small solution errors nor scale effectively with increasing problem size. Here we present the neural basis method (NBM) as a new approach for Navier--Stokes dynamics. NBM is built on a ``representation-then-projection'' principle: fixed neural bases define the representation space, while projection of the governing equations determines the expansion coefficients. The same bases define common coordinates for the parametric solution manifold, recasting operator learning as a parameter-to-coefficient mapping and enabling scalability by removing explicit spatial dependence. For incompressible flow, NBM yields a velocity-only formulation whose residual provides an a posteriori velocity-error estimate, establishing a certified, data-free training objective for coefficient learning. Theoretical analysis and numerical experiments demonstrate the accuracy, efficiency, generalization, and scalability of NBM, with substantial improvements over baselines.
Figures & tables
Fig. 1: NBM formulation for incompressible Newtonian fluid dynamics. a, Neural representation. Scalar neural bases are generated by fixed-parameter, dual-layer residual networks. Divergence-free velocity bases are then constructed via rotated gradient in 2D, or from curls of vector-valued neural potentials in 3D. Pressure field is represented using scalar neural bases. b, Neural discretization and projection. Residuals are evaluated on “virtual” control-volumes and assembled into a linearized least-squares system to determine the velocity and pressure coefficients. c, Velocity-only decomposition. The full momentum residual consists of velocity-dependent and pressure-gradient contributions, rfull=ru+r∇p , where ru=Auα−b and r∇p=Gpβ . The velocity-dependent residual is decomposed into components parallel ( r∥ ) and orthogonal ( r⊥ ) to the discrete pressure-gradient space Gh , such that ru=r∥+r⊥ . With incompressibility satisfied by the divergence-free velocity representation, applying the Leray projection eliminates the component parallel to the pressure-gradient space, leaving r⊥=P⊥Wru and yielding the equivalent velocity-only least-squares formulation: minα,βrfullW2=minαP⊥WruW2 . Pressure can be recovered a posteriori. d, Learning in velocity-coefficient space. NBO maps parameter-time inputs to velocity coefficients.
Fig. 2: Kovasznay flow and Taylor–Green vortex benchmarks with analytical reference solutions. a–h, Kovasznay flow. a,b, Speed field with velocity quivers and pressure field, followed by absolute error fields w.r.t. the analytical solution for NBM, OpenFOAM, and PINN. c, Relative errors. d, Energy spectrum. e, Residual vs. Nb ( Nc=50×50 ). f, Relative errors vs. Nb ( Nc=50×50 ). g, Relative errors vs. Nc ( Nb=500 ). h, Relative error of E(k) vs. Nc ( Nb=500 ). i–p, Taylor–Green vortex. i,j, Speed field with velocity quivers and pressure field, followed by absolute error fields w.r.t. the analytical solution for NBM, OpenFOAM, and PINN. k, Relative errors. l, Energy spectrum. m, Residual εrel vs. Nb , ( Nc=50×50 ). n, Relative errors vs. Nb ( Nc=50×50 ). o, Relative errors vs. Nc ( Nb=500 ). p, Relative error of E(k) vs. Nc ) ( Nb=500 ). Note: Kovasznay flow: Re=500 . Taylor–Green vortex: Re=1000 ; results are at t=0.01 . See further details in Methods and Supplementary Information sections.
Fig. 3: Lid-driven cavity and shear-layer flow benchmarks. a–h, Lid-driven cavity. a,b, Speed field with velocity quivers and pressure field, followed by OpenFOAM reference, NBM solution, absolute difference between NBM and OpenFOAM, and absolute difference between PINN and OpenFOAM. c, Relative differences using OpenFOAM as reference. d, Energy spectrum. e, Residual vs. Nb ( Nc=100×100 ). f, Relative differences vs. Nb ( Nc=100×100 ). g, Relative differences vs. Nc ( Nb=1000 ). h, Relative difference of E(k) vs. Nc ( Nb=800 ). i–p, Shear-layer flow. i,j, Speed field with velocity quivers and pressure field, followed by OpenFOAM reference, NBM solution, absolute difference between NBM and OpenFOAM, and absolute difference between PINN and OpenFOAM. k, Relative differences using OpenFOAM as reference. l, Energy spectrum. m, Residual vs. Nb ( Nc=100×100 ). n, Relative differences vs. Nb ( Nc=100×100 ). o, Relative differences vs. Nc ( Nb=1000 ). p, Relative difference of E(k) vs. Nc ( Nb=800 ). Note: Lid-driven cavity: Re=1000 . Shear-layer flow: Re∼104 ; results are at t=30 . See further details in Methods and Supplementary Information sections.
Fig. 4: Residual-certified neural basis operator learning for parametric Kolmogorov flow. a, Representative speed fields with velocity quivers at t=3 for NBM reference, NBO-RC, and NBO-DD. b, Corresponding results of DeepO-PI-CT, DeepO-PI-DT, and DeepO-DD. In a,b , erel: relative L2 velocity error. c, Kinetic-energy spectra E(k) of NBM reference, OpenFOAM, and learned operators. The inset shows relative L2 errors of the high-pass-filtered velocity field for k>60 , using OpenFOAM as reference. d, Normalized training-loss histories. e, Normalized relative L2 velocity error vs. normalized square root of the training loss with linear fit for NBO-RC. f, Relative errors of the velocity field and energy spectrum, with training and inference timing. Training speedups are relative to DeepO-PI-DT, while inference speedups are relative to OpenFOAM; parenthetical values compare NBO with DeepO. g, Test relative L2 velocity-error distributions under random train–test splits of ζi,j=(χi,tj) , from 80:20 to 0.1:99.9 . h, Relative L2 velocity-error distributions for in-distribution (ID) and out-of-distribution (OOD) magnitude values. Note: See Methods and Supplementary Information Sections for further details.
Fig. 5: Residual-certified neural basis operator learning for parametric Taylor–Couette flow. a, Representative meridional-speed fields with velocity quivers at t=0.2 . Left to right: NBM reference, NBO-RC, NBO-DD, DeepO-PI-DT, and DeepO-DD; erel: relative L2 velocity error. b, Meridional kinetic-energy spectra E(k) of NBM reference, OpenFOAM, and learned operators. The inset shows relative L2 errors of the high-pass-filtered meridional velocity for k>60 , using OpenFOAM as reference. c, Normalized training-loss histories. d, Normalized relative L2 velocity error vs. normalized square root of training loss with linear fit for NBO-RC. e, Relative errors of meridional velocity and its energy spectrum, with training and inference timing and corresponding speedups. Training speedups are relative to DeepO-PI-DT, while inference speedups are relative to OpenFOAM; parenthetical values compare NBO with DeepO. f, Test relative L2 meridional velocity-error distributions under random train–test splits of ζi,j=(Ωi,tj) , from 80:20 to 0.1:99.9 . g, Relative L2 meridional velocity-error distributions for in-distribution (ID) and out-of-distribution (OOD) rotation rates. Note: See Methods and Supplementary Information Section for further details.
Fig. 6: Mechanistic probe of residual-certified neural basis operator learning. a, Directional quadratic forms of the sample-averaged residual Gram GR,mean and the velocity-energy metric E , evaluated along 50 random unit directions q in the neural basis coefficient space. b, Evolution of the 95 th-percentile relative discrepancy ΔGN(v)=∥(H−HGN)v∥2/∥HGNv∥2 between the full Hessian H and its Gauss–Newton component HGN , evaluated over 20 random unit directions v in the space of trainable network parameters. Insets compare the final directional curvatures vTHv and vTHGNv . c,d, Velocity-energy curvature vs. loss-induced Gauss–Newton (GN) curvature for Kolmogorov flow ( c ) and Taylor–Couette flow ( d ). In each panel, the left evaluates RC-GN and MSE-reference-GN curvatures using the NBO-RC network Jacobian, whereas the right plot evaluates MSE-GN and RC-reference-GN curvatures using the NBO-DD network Jacobian. e, Normalized local quadratic basins in selected stiff and weak network-parameter directions for Kolmogorov flow (left) and Taylor–Couette flow (right), comparing the RC, velocity-energy, and MSE-reference metrics. f, Rank-percentile correspondence for directions in the upper half of velocity-energy curvature for Kolmogorov flow (left) and Taylor–Couette flow (right), comparing RC with velocity energy for NBO-RC and velocity energy with MSE for NBO-DD. Note: “corr. (lin/rank)” reports the Pearson linear correlation and Spearman rank correlation, respectively. See Methods and Supplementary Information Sections for the Hessian decomposition, metric construction, and probe procedure.
Notation
Definition / Meaning
Remarks
Physical and geometric quantities
Ω,∂Ω
Spatial domain and its boundary
Ω⊂Rd .
d
Spatial dimension
d=1 , 2 , or 3 .
x
Spatial coordinate
x=(x,y) in 2D Cartesian.
x,y
Cartesian spatial coordinates
r,θ,z
Radial/azimuthal/axial coordinates
For axisymmetric Taylor–Couette flow.
Table S1: Notation used in the main text and Supplementary Information.
Benchmark
Re
Nb
NBM Nc
OpenFOAM mesh
Kovasznay
500
500
50×50
50×50
Taylor–Green
1000
500
50×50
50×50
Lid-driven cavity
1000
1000
100×100
100×100
Shear-layer
O(104)
1000
100×100
100×100
Table S2: NBM/OpenFOAM configurations for the single-instance benchmarks.
Benchmark
Hidden layers
Width
Activation
Kovasznay
5
96
Tanh
Taylor–Green
5
96
Tanh
Lid-driven cavity
6
256
SiLU
Shear-layer
8
256
SiLU
Table S3: PINN configurations for the single-instance benchmarks.
Benchmark
Nb
NBM Nc
Latent dimension Nz
Kolmogorov
1000
50×50
300
Taylor–Couette
1000
30×180
300
Table S4: NBO configurations for the parametric benchmarks.
Configuration
Input
Parameter–time module
Kolmogorov DeepO-PI-CT
(χ,t)
Two 4×128 MLPs
Kolmogorov DeepO-PI-DT
(χ,t)
One 4×128 MLP
Kolmogorov DeepO-DD
(χ,t)
One 4×128 MLP
Taylor–Couette DeepO-PI-DT
(Ωi,t)
One 4×128 MLP
Taylor–Couette DeepO-DD
(Ωi,t)
One 4×128 MLP
Table S5: DeepO configurations for the parametric benchmarks.
Neural PDE solvers use trainable models to approximate solutions or solution operators for partial differential equations, often with physical constraints.
Physics-governed models are increasingly paired with machine learning for accelerated predictions, yet most "physics--informed" formulations treat the governing equations as a penalty loss whose scale and meaning are set by heuristic balancing. This blurs operator structure, thereby confounding solution approximation error with governing-equation enforcement error and making the solving and learning progress hard to interpret and control. Here we introduce the Neural Basis Method, a projection-based formulation that couples a predefined, physics-conforming neural basis space with an operator-induced residual metric to obtain a well-conditioned deterministic minimization. Stability and reliability then hinge on this metric: the residual is not merely an optimization objective but a computable certificate tied to approximation and enforcement, remaining stable under basis enrichment and yielding reduced coordinates that are learnable across parametric instances. We use advective multiscale Darcian dynamics as a concrete demonstration of this broader point. Our method produce accurate and robust solutions in single solves and enable fast and effective parametric inference with operator learning.
Yuhe Wang, Min Wang
Institute for Scientific Computation, Texas A&M University, College Station, TX 77843, USA · Department of Mathematics, University of Houston, Houston, TX 77204, USA
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.
Sebastien Andre-Sloan, Dibyakanti Kumar, Alejandro F Frangi +1
Department of Computer Science, The University of Manchester
We present a two-part neural framework for forced three-dimensional incompressible Navier--Stokes flow. PartI develops the computational forcing system. A physics-informed neural model generates structured external-force trajectories, candidates are optimized through differentiable PDE rollouts or PPO-Clip, and selected forcings are frozen and checked by independent fixed-force replay. PartII provides the mathematical certification layer. It separates neural candidate discovery from continuum analysis, derives integrated reciprocal-vorticity criteria that imply Riccati-type growth and finite-time loss of smooth continuation, develops a validated computational-to-continuum transfer strategy, and establishes a conditional positive-probability closure for a nondegenerate neural output law. The proof is complete at the continuum level.