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

Yuhe Wang, Zhu Wang, Min Wang

arXiv:2610.32167v1physics.flu-dyncs.LGcs.NAmath.NA

TL;DR

Repeated Navier–Stokes solves can be demanding in many-query settings. NBM embeds incompressibility in neural bases, projects the equations, and learns parameter-to-velocity-coefficient maps; its certified residual supports operator learning. The paper reports improved accuracy, generalization, and scalability over DeepONets.

  • Problem

    Repeated solutions become demanding in many-query settings with varying physical parameters, boundary conditions, or forcing.

    Source: NBM formulation for incompressible… · p. 5

  • Method

    NBM represents incompressible velocity in divergence-free neural bases, eliminates pressure from the primary solve, and learns parametric maps directly in velocity-coefficient space.

    Source: NBM formulation for incompressible… · pp. 4–5, Discussion · p. 13

  • Results

    NBM reports substantially improved accuracy, generalization, and scalability over DeepONets, with sub-percent velocity errors below baseline neural operators on parametric problems.

    Source: NBM enables residual-certified… · p. 10, Discussion · p. 13

  • Takeaways & Limitations

    The certified residual serves as a physically meaningful coefficient-space training metric while avoiding repeated spatial computation.

    Source: Discussion · p. 13

    Localized nonsmoothness and nonconvex or complex geometries motivate extensions such as domain decomposition, while growing systems will need scalable least-squares strategies.

    Source: Discussion · pp. 13–14

Abstract

from arXiv · show

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.

Results

The Results section presents NBM’s divergence-free, velocity-only formulation and residual-based error certificate, then evaluates single-instance solves, parametric operator learning, and residual certification in training. It also probes how residual-certified operator learning relates to velocity-energy geometry and loss curvature in Kolmogorov and Taylor–Couette flows.

NBM formulation for incompressible Newtonian flow

  • Fixed neural bases represent incompressible velocity directly; projecting out discrete pressure-gradient directions yields a velocity-only least-squares solve and enables parameter–time-to-velocity-coefficient learning.The bases satisfy incompressibility by construction, and pressure can be recovered afterward; the velocity-only system reduces primary unknowns and supports residual-certified or data-driven operator training.Source: NBM formulation for incompressible… · pp. 4–5

NBM convergence analysis and a posteriori error estimation

  • Under stability and full-rank conditions, NBM’s error separates neural-approximation, VCV-discretization, and least-squares components, while projected residual and discrete velocity error are equivalent up to stability constants.When those constants are uniformly bounded over training inputs, minimizing NBO-RC loss is equivalent, up to those constants, to minimizing average discrete velocity-energy error.Source: NBM convergence analysis and a… · pp. 6–7

NBM delivers residual-certified accuracy as a single-instance PDE solver

  • NBM outperforms OpenFOAM and PINN on analytical benchmarks and exceeds PINN against OpenFOAM references for cavity and shear-layer flows.Standard PDE-loss PINN fails to converge for the Re = 1000 cavity; adding sparse velocity anchors yields a usable solution, but its discrepancies remain larger than NBM’s. Refining bases produces spectral-like error decay, while refining VCVs gives algebraic decay; residuals and errors decrease together.Source: NBM delivers residual-certified… · p. 7, Figure 2 · p. 8, Figure 3 · p. 9

Residual certification shapes velocity-aligned operator-network training

  • Residual certification aligns operator-network training with velocity error: controlled metric swaps show alignment follows the residual metric, not the architecture, trained parameters, or Jacobian alone.Residual sensitivity correlates nearly perfectly with velocity-energy error in coefficient space, and residual-metric contours and rank ordering preserve velocity-energy structure unlike MSE.Source: Residual certification shapes… · pp. 10–12

NBM enables residual-certified operator learning for parametric dynamics

Residual certification shapes velocity-aligned operator-network training

  • Figure 6 examines residual-Gram and velocity-energy directional forms, Hessian/Gauss–Newton discrepancy, curvature, local quadratic basins, and rank correspondence for NBO-RC and NBO-DD.The curvature and basin analyses compare residual certification, velocity energy, and MSE metrics across Kolmogorov and Taylor–Couette flows.Source: Figure 6 · p. 13

Discussion

The work presents NBM for solving and learning incompressible Navier–Stokes dynamics, with incompressibility embedded in the discretization and pressure removed from the primary solve. Its certified residual links residuals to velocity error and serves as a coefficient-space training metric, while the discussion identifies directions for extending the method.

Discussion

  • A triple-resolution convergence analysis and a posteriori estimate link the residual to velocity error, supporting residual-certified operator learning.The residual is also used as a coefficient-space training metric, avoiding repeated spatial computation and improving accuracy, generalization, and scalability over DeepONets.Source: Discussion · p. 13
  • Localized non-smoothness and nonconvex or complex geometries motivate localized formulations, while growing systems call for scalable least-squares strategies.Proposed directions include domain decomposition, partition-of-unity, discontinuous-Galerkin-style local–global formulations, randomization, preconditioning, and high-throughput parallel processing.Source: Discussion · pp. 13–14
  • An NBM-powered Transformer operator is proposed as a next step for high-dimensional parameterizations and multi-operator learning across wider PDE families.The proposed architecture aims to capture complex input interactions without severe complexity growth with spatial resolution.Source: Discussion · p. 14

Methods

The Methods section outlines six nondimensional benchmarks, the control-volume integration of temporal and convective schemes, baseline approaches, and computational complexity analysis. The single-instance benchmarks include exact-solution and non-analytical settings, with distinct boundary and flow conditions.

Methods

  • The Methods section defines six nondimensional benchmark problems and describes control-volume integration, baseline neural approaches, and operator-learning complexity analysis.Further implementation and mechanistic-analysis details are provided in the Supplementary Section.Source: Methods · p. 14

Single-instance benchmark problems

  • The Kovasznay benchmark is a steady two-dimensional exact solution at Re = 500, while Taylor–Green is an unsteady decaying-vortex exact solution at Re = 1000.For Taylor–Green, NBM and PINN use exact-solution Dirichlet boundaries, whereas OpenFOAM uses its built-in periodic setup; all methods share the exact initial condition.Source: Single-instance benchmark problems · p. 14
  • The lid-driven cavity is a stationary benchmark without an analytical solution, using Re = 1000 and no-slip conditions on the walls other than the moving lid.Pressure is fixed at one arbitrary point for uniqueness.Source: Single-instance benchmark problems · pp. 14–15
  • The shear-layer benchmark has no exact solution and uses a perturbed inlet shear layer at viscosity 10^-6, with Reynolds number on the order of 10^4.It is associated with Kelvin–Helmholtz instability; the outlet uses open outflow conditions and the top and bottom use slip conditions.Source: Single-instance benchmark problems · p. 15

Parametric benchmark problems

The parametric benchmarks specify forcing- or rotation-driven flows over parameter–time domains. In the control-volume integration, NBM evaluates field values and derivatives from neural bases and uses the resulting residual structure to incorporate finite-volume schemes.

Parametric benchmark problems

  • Kolmogorov flow uses periodic forcing with amplitude χ over χ ∈ [0.5, 1.5] and t ∈ [0, 3], producing strongly nonlinear vortical dynamics.The domain is [-1, 1]^2, viscosity is 10^-3, and the fluid starts from rest with no-slip boundary conditions.Source: Parametric benchmark problems · p. 15
  • Taylor–Couette flow varies inner-cylinder rotation over Ωi ∈ [100, 200] and t ∈ [0, 0.2], spanning circular Couette flow to axisymmetric Taylor-vortex flow.The transition occurs near Ωi ≈ 160; the selected range remains below the wavy-vortex regime.Source: Parametric benchmark problems · p. 15

Control-volume formulation and NBM integration

  • NBM analytically evaluates control-volume cell values, face values, and face gradients from its neural bases, avoiding conventional local interpolation or gradient approximation for those quantities.The retained control-volume structure permits established temporal and convective schemes, including BDF2 for time discretization.Source: Control-volume formulation and NBM… · p. 16
  • The projected control-volume residual and boundary-condition residuals form a linear least-squares problem in velocity coefficients for the next time step.Transient, diffusive, and linearized convective terms are assembled in the residual, and pressure contributions are eliminated by the velocity-only projection.Source: Control-volume formulation and NBM… · p. 16

Physics-informed neural networks

The section describes PINN and DeepONet baselines and compares their computational costs with NBO-RC. Its complexity analysis contrasts repeated spatial-network evaluation in DeepO-PI-DT with NBO-RC’s coefficient-space training after spatial precomputation.

Physics-informed neural networks

  • PINNs serve as single-instance neural baselines, combining PDE, boundary, initial-condition, and pressure-gauge losses; at Re = 1000, the standard cavity PINN did not converge to a physically meaningful steady solution.Adding sparse OpenFOAM velocity anchors made that PINN-based solver no longer data-free.Source: Physics-informed neural networks · p. 16

DeepONet-based neural operators

  • DeepONet maps parameter–time and spatial inputs to flow fields using branch and trunk networks, with data-driven and physics-informed settings differing in their loss weights.The continuous-time physics-informed variant failed to produce physically meaningful solutions, so the section adopts a discrete-time formulation.Source: DeepONet-based neural operators · pp. 16–17

Computational complexity and scalability analysis of operator learning

  • For the representative Kolmogorov-flow configuration, NBO-RC’s online training update uses the coefficient network and a precomputed low-dimensional quadratic residual.The section estimates approximately 6 × 10^6 MAC-equivalent operations per NBO-RC training record.Source: Computational complexity and… · p. 17
  • DeepO-PI-DT’s online cost scales with spatial resolution through repeated trunk evaluation and differentiation, while NBO-RC’s remains independent of resolution for fixed coefficient dimension after precomputation.The NBO-RC spatial operators are precomputed once; DeepO-PI-DT’s dominant cost grows with the number of physics points.Source: Computational complexity and… · pp. 17–18

Computational environment

The computational environment differs by experiment type, while operator-network training and inference timings use the same H100 GPU environment. The source code and reproduction scripts are slated for release upon publication.

Computational environment

  • Parametric benchmarks use an ACES CPU node for offline precomputation and a single NVIDIA H100 GPU for operator-network training and inference; reported NBO and DeepO timings use FP64 on that same GPU.Single-instance experiments use an Apple M3 system with 24 GB unified memory.Source: Computational environment · p. 18

Code availability

  • The authors state that source code and reproduction scripts will be made publicly available on GitHub upon publication.Source: Code availability · p. 19

Supplementary Information:

The supplied passage is a partial title referring to solving and learning Navier–Stokes dynamics with a neural method.

Supplementary Information:

S1 Notation

This section defines notation for NBM discretization, coefficient learning, error analysis, neural approximation, benchmarks, baselines, computational complexity, and mechanistic analysis. It also introduces coefficient-space and velocity-energy curvatures for a local two-dimensional quadratic analysis of basin geometry.

NBM discretization and pressure-gradient elimination

  • The discretization notation covers velocity and pressure-gradient matrices, weighted residuals, pressure-gradient elimination, and projected least-squares systems.It also defines the continuous velocity-only residual and the discrete VCV residual quantities used in consistency analysis.Source: S1 Notation · p. 25

Neural basis operator learning and computational-complexity notation

  • The learning and complexity notation specifies parameter–time inputs, coefficient predictions, training losses, residual Gram matrices, and NBO and baseline computational costs.The section also defines DeepO workload and automatic-differentiation factors.Source: S1 Notation · pp. 25, 28

NBM error estimate and discrete residual–error certificate

  • Error-analysis notation distinguishes exact and computed discrete solutions, algebraic tolerance, basis stability, quadrature accuracy, and residual-to-error certificate quantities.The section also defines projected weak-form and energy-norm quantities, divergence-free spaces, and continuous velocity error.Source: S1 Notation · pp. 25–26

Neural approximation and random-coverage analysis

  • Approximation and coverage notation defines Sobolev and neural approximation classes, architecture quantities, sampled-basis coverage events, and constants in the resulting estimates.It distinguishes the ideal approximant size and network width from the sampled basis count.Source: S1 Notation · pp. 26–27

Benchmark-specific and diagnostic quantities; PINN and DeepONet baseline notation; Mechanistic-analysis notation

  • Benchmark notation includes physical parameters, velocity and residual metrics, PINN and DeepO predictions and losses, training-point counts, and Hessian and Gauss–Newton diagnostics.The benchmark symbols include relative velocity error and relative residual measures; the mechanistic-analysis symbols describe coefficient-space curvature and metric comparisons.Source: S1 Notation · pp. 28–29

S1 Notation

  • κMSE(v) and κE(v) denote directional curvatures under the Euclidean coefficient-space metric and the velocity-energy matrix E, respectively; η∗ is the trained operator-network parameter vector, and s1 and s2 are scalar displacements.v1 and v2 are selected stiff and weak parameter-space directions; KQ is the 2 × 2 local quadratic-form matrix associated with metric Q.Source: S1 Notation · p. 29
  • Qi denotes the samplewise coefficient-space metric, while Ndir is the number of sampled directions used to normalize percentile ranks.The coordinates describe a local two-dimensional plane and are used to visualize local basin geometry.Source: S1 Notation · p. 29

S2 NBM error estimates

The error analysis combines neural approximation, random basis coverage, and VCV discretization consistency to establish an error bound for the computed NBM solution. Analytic regularity gives a stretched-exponential approximation rate, while the coverage and consistency results supply additional conditions in the analysis.

S2.1 Basic settings

  • The discrete analysis partitions the domain into VCVs, forms momentum residuals on each cell, and measures the residual vector with a weighted norm.The weighting matrix represents a discrete measure; W = I gives the standard Euclidean residual norm.Source: S2.1 Basic settings · p. 31

S2.2 Spectral-like neural approximation

  • Analytic regularity yields a stretched-exponential best-M-term rate; finite Sobolev regularity still permits convergence, generally at an algebraic rate.The analysis applies to scalar solutions or, in two dimensions, a scalar potential generating a divergence-free velocity field.Source: S2.2 Spectral-like neural approximation · pp. 32–33

S2.3 Random coverage of an M-term neural approximant

  • With probability at least 1 − δ, random sampling covers the neighborhoods of an M-term approximant, yielding a nearby element in the random trial space.The coverage argument assumes each neighborhood has sampling probability bounded below and its basis is sufficiently close to the target basis.Source: S2.3 Random coverage of an M-term… · pp. 33–34

S2.4 Expressing the coverage estimate in terms of Nb

  • For sufficiently large Nb, the coverage condition holds with M = MNb, expressing the random-coverage estimate in terms of the number of sampled bases.The passage states this relationship for sufficiently large Nb.Source: S2.4 Expressing the coverage estimate… · p. 35

S2.5 Consistency of the VCV residual discretization

  • Under regularity and uniform q-th order quadrature-consistency assumptions, the VCV residual discretization has an error bound with a constant independent of Nc.The second-order midpoint-type VCV discretization has q = 2 in smooth regions.Source: S2.5 Consistency of the VCV residual… · p. 36

S2.6 Proof of Theorem 1

  • The computed-solution proof combines approximation, random coverage, and VCV consistency, then uses local residual stability and least-squares accuracy to control the solve.The stability estimate is invoked for sufficiently large basis and control-volume counts.Source: S2.6 Proof of Theorem 1 · pp. 37–38

S3 NBM residual–error certificate at neural discrete level

The proof of the discrete residual–error certificate uses positivity of the energy matrix and full column rank of the projected operator to obtain a positive singular-value bound.

S3 NBM residual–error certificate at neural discrete level

  • Positive definiteness of the energy matrix and full column rank of B⊥ give a positive smallest singular value in the certificate proof.The supplied proof passages state these ingredients but do not expose the resulting bound.Source: S3.1 Proof of Theorem 2 · p. 40

S4 NBM residual–error certificate at continuous level

At the continuous level, the velocity-only residual is considered on divergence-free test fields, where pressure-gradient components vanish. The estimate applies after boundary lifting to velocity approximations satisfying the prescribed boundary conditions.

S4 NBM residual–error certificate at continuous level

  • Restricting the residual to divergence-free test functions removes pressure-gradient components because those components vanish against such fields.The neural basis space is divergence-free by construction, while velocity boundary conditions are imposed through boundary residuals.Source: S4 NBM residual–error certificate at… · p. 41
  • The continuous residual–error estimate applies to velocity approximations satisfying the prescribed boundary conditions after boundary lifting.The stated framework treats a linearized implicit step with a divergence-free frozen convective velocity and boundary conditions supporting skew-symmetry.Source: S4 NBM residual–error certificate at… · p. 41
  • For general outflow boundaries, the self-convection boundary contribution is nonnegative, so it does not weaken the positive time-diffusion energy term.The passage also states that the upper bound is unchanged and that neutral outflow/stress boundaries contribute zero stress term in the weak formulation.Source: S4 NBM residual–error certificate at… · p. 43

S5 Additional details of velocity-only construction

Pressure is excluded from the primary velocity-only solve and can instead be recovered afterward by fitting a pressure representation to a pressure-gradient target from the momentum equation.

S5.1 A posteriori pressure recovery

  • Pressure recovery fits a scalar neural pressure representation to a momentum-equation pressure-gradient target evaluated at VCV centers.The fitting objective also includes pressure-basis values at boundary faces and a boundary weight.Source: S5.1 A posteriori pressure recovery · p. 44

S6 Benchmark configurations

The supplied benchmark details identify configurations and precision for the single-instance and parametric baselines, and describe generalization evaluation protocols. The mechanistic analysis compares coefficient-space loss metrics and reports distinct parameter-space curvature geometries for NBO-RC and NBO-DD.

S6.1.1 NBM/OpenFOAM configurations

S6.1.2 PINN configurations

S6.2.1 NBO configurations

S6.2.2 DeepO configurations

  • DeepO configurations specify mini-batch sizes, including 32 states for Kolmogorov PI-CT and 64 for PI-DT and DD; all models use Tanh and FP64.Kolmogorov DeepO-PI-CT uses 500,000 iterations with AdamW and a 10^-3 learning rate.Source: S6.2.2 DeepO configurations · pp. 48–49

S6.2.3 Generalization evaluation protocols

S7 Mechanistic analysis of certificate-induced operator-network training geometry

  • The parameter-space Hessian separates into a Gauss–Newton component based on first derivatives and a non-Gauss–Newton component that becomes small as the loss gradient decreases.Source: S7 Mechanistic analysis of… · p. 50
  • NBO-RC and NBO-DD retain different parameter-space curvature geometries because their Jacobians pull back different coefficient-space metrics.Source: S7 Mechanistic analysis of… · p. 50
  • Metric comparisons hold the operator-network Jacobian fixed while changing the coefficient-space metric; local-basin and rank analyses compare residual, velocity-energy, and MSE geometry.Source: S7 Mechanistic analysis of… · pp. 50–51

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