Solving and Learning Navier-Stokes Dynamics with the Neural Basis Method

Yuhe Wang; Zhu Wang; Min Wang

physics.flu-dynOct 23, 2026

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

Abstract

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, $\mathbf{r}^{\mathrm{full}}=\mathbf{r}^{\mathbf{u}}+\mathbf{r}^{\nabla p}$ , where $\mathbf{r}^{\mathbf{u}}=\mathbf{A}_{\mathbf{u}}\boldsymbol{\alpha}-\mathbf{b}$ and $\mathbf{r}^{\nabla p}=\mathbf{G}_{p}\boldsymbol{\beta}$ . The velocity-dependent residual is decomposed into components parallel ( $\mathbf{r}_{\parallel}$ ) and orthogonal ( $\mathbf{r}_{\perp}$ ) to the discrete pressure-gradient space $\mathcal{G}_{h}$ , such that $\mathbf{r}^{\mathbf{u}}=\mathbf{r}_{\parallel}+\mathbf{r}_{\perp}$ . With incompressibility satisfied by the divergence-free velocity representation, applying the Leray projection eliminates the component parallel to the pressure-gradient space, leaving $\mathbf{r}_{\perp}=\mathbf{P}_{\perp}^{\mathbf{W}}\mathbf{r}^{\mathbf{u}}$ and yielding the equivalent velocity-only least-squares formulation: $\min_{\boldsymbol{\alpha},\boldsymbol{\beta}}\left\|\mathbf{r}^{\mathrm{full}}\right\|_{\mathbf{W}}^{2}=\min_{\boldsymbol{\alpha}}\left\|\mathbf{P}_{\perp}^{\mathbf{W}}\mathbf{r}^{\mathbf{u}}\right\|_{\mathbf{W}}^{2}$ . 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. $N_{b}$ ( $N_{c}=50\times 50$ ). f, Relative errors vs. $N_{b}$ ( $N_{c}=50\times 50$ ). g, Relative errors vs. $N_{c}$ ( $N_{b}=500$ ). h, Relative error of $E(k)$ vs. $N_{c}$ ( $N_{b}=500$ ). 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 $\varepsilon_{\rm rel}$ vs. $N_{b}$ , ( $N_{c}=50\times 50$ ). n, Relative errors vs. $N_{b}$ ( $N_{c}=50\times 50$ ). o, Relative errors vs. $N_{c}$ ( $N_{b}=500$ ). p, Relative error of $E(k)$ vs. $N_{c}$ ) ( $N_{b}=500$ ). 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. $N_{b}$ ( $N_{c}=100\times 100$ ). f, Relative differences vs. $N_{b}$ ( $N_{c}=100\times 100$ ). g, Relative differences vs. $N_{c}$ ( $N_{b}=1000$ ). h, Relative difference of $E(k)$ vs. $N_{c}$ ( $N_{b}=800$ ). 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. $N_{b}$ ( $N_{c}=100\times 100$ ). n, Relative differences vs. $N_{b}$ ( $N_{c}=100\times 100$ ). o, Relative differences vs. $N_{c}$ ( $N_{b}=1000$ ). p, Relative difference of $E(k)$ vs. $N_{c}$ ( $N_{b}=800$ ). Note: Lid-driven cavity: $Re=1000$ . Shear-layer flow: $Re\sim 10^{4}$ ; 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 , $e_{\rm rel}:$ relative $L^{2}$ velocity error. c, Kinetic-energy spectra $E(k)$ of NBM reference, OpenFOAM, and learned operators. The inset shows relative $L^{2}$ errors of the high-pass-filtered velocity field for $k>60$ , using OpenFOAM as reference. d, Normalized training-loss histories. e, Normalized relative $L^{2}$ 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 $L^{2}$ velocity-error distributions under random train–test splits of $\zeta_{i,j}=(\chi_{i},t_{j})$ , from $80{:}20$ to $0.1{:}99.9$ . h, Relative $L^{2}$ 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; $e_{\rm rel}:$ relative $L^{2}$ velocity error. b, Meridional kinetic-energy spectra $E(k)$ of NBM reference, OpenFOAM, and learned operators. The inset shows relative $L^{2}$ errors of the high-pass-filtered meridional velocity for $k>60$ , using OpenFOAM as reference. c, Normalized training-loss histories. d, Normalized relative $L^{2}$ 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 $L^{2}$ meridional velocity-error distributions under random train–test splits of $\zeta_{i,j}=(\Omega_{i},t_{j})$ , from $80{:}20$ to $0.1{:}99.9$ . g, Relative $L^{2}$ 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 $\mathbf{G}_{{\rm R},{\rm mean}}$ and the velocity-energy metric $\mathbf{E}$ , evaluated along 50 random unit directions $\mathbf{q}$ in the neural basis coefficient space. b, Evolution of the $95$ th-percentile relative discrepancy $\Delta_{\mathrm{GN}}(\mathbf{v})=\|(\mathbf{H}-\mathbf{H}_{\mathrm{GN}})\mathbf{v}\|_{2}/\|\mathbf{H}_{\mathrm{GN}}\mathbf{v}\|_{2}$ between the full Hessian $\mathbf{H}$ and its Gauss–Newton component $\mathbf{H}_{\mathrm{GN}}$ , evaluated over 20 random unit directions $\mathbf{v}$ in the space of trainable network parameters. Insets compare the final directional curvatures $\mathbf{v}^{T}\mathbf{H}\mathbf{v}$ and $\mathbf{v}^{T}\mathbf{H}_{\mathrm{GN}}\mathbf{v}$ . 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
$\Omega,\partial\Omega$	Spatial domain and its boundary	$\Omega\subset\mathbb{R}^{d}$ .
$d$	Spatial dimension	$d=1$ , $2$ , or $3$ .
$\mathbf{x}$	Spatial coordinate	$\mathbf{x}=(x,y)$ in 2D Cartesian.
$x,y$	Cartesian spatial coordinates
$r,\theta,z$	Radial/azimuthal/axial coordinates	For axisymmetric Taylor–Couette flow.

Table S1: Notation used in the main text and Supplementary Information.

Benchmark	$Re$	$N_{b}$	NBM $N_{c}$	OpenFOAM mesh
Kovasznay	$500$	$500$	$50\times 50$	$50\times 50$
Taylor–Green	$1000$	$500$	$50\times 50$	$50\times 50$
Lid-driven cavity	$1000$	$1000$	$100\times 100$	$100\times 100$
Shear-layer	$O(10^{4})$	$1000$	$100\times 100$	$100\times 100$

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	$N_{b}$	NBM $N_{c}$	Latent dimension $N_{z}$
Kolmogorov	$1000$	$50\times 50$	$300$
Taylor–Couette	$1000$	$30\times 180$	$300$

Table S4: NBO configurations for the parametric benchmarks.

Configuration	Input	Parameter–time module
Kolmogorov DeepO-PI-CT	$(\chi,t)$	Two $4\times 128$ MLPs
Kolmogorov DeepO-PI-DT	$(\chi,t)$	One $4\times 128$ MLP
Kolmogorov DeepO-DD	$(\chi,t)$	One $4\times 128$ MLP
Taylor–Couette DeepO-PI-DT	$(\Omega_{i},t)$	One $4\times 128$ MLP
Taylor–Couette DeepO-DD	$(\Omega_{i},t)$	One $4\times 128$ MLP

Table S5: DeepO configurations for the parametric benchmarks.

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Solving and learning advective multiscale Darcian dynamics with the Neural Basis Method

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

Neural PDE Solvers Physics-Informed ML PDE Operator Learning Neural Surrogate Modeling Computational Fluid Dynamics

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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.

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