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Learning Spectral-Like Mesh-Free Discretisations
Lucas Gerken Starepravo, Henry Broadley, Steven Lind, Jack R. C. King
TL;DR
Mesh-free discretisations leave stencil degrees of freedom determined by implicit closure choices rather than by accuracy across the resolvable wavenumber band. SpeND learns those weights from local geometry while enforcing polynomial consistency exactly, and modal analysis shows substantially wider-band resolution than LABFM and fourth-order finite differences at equal stencil size.
Problem
Traditional mesh-free closures leave underdetermined stencil degrees of freedom without selecting accuracy at the wavenumbers containing fine-scale content.
Method
SpeND uses a geometry-conditioned neural network and a hard-constrained projection layer to learn consistent explicit weights from modal targets over a prescribed band.
Results
SpeND follows the spectral response over a substantially wider wavenumber range than explicit LABFM and fourth-order finite differences at equal stencil size.
Takeaways & Limitations
The learned operator resolves substantially more of the wavenumber band while retaining the fourth-order convergence rate imposed by consistency.
Abstract
from arXiv · showhide
Meshfree methods such as smoothed particle hydrodynamics (SPH) with kernel corrections, radial basis function-generated finite differences (RBF-FD), and the local anisotropic basis function method (LABFM) construct discrete differential operators by imposing polynomial consistency on a local stencil. For stencils containing more nodes than there are consistency constraints, the resulting linear system is underdetermined, and the remaining degrees of freedom are fixed implicitly by the choice of kernel, basis preconditioning, or a minimum-norm condition. Polynomial consistency constrains the operator only in the low-wavenumber limit, and no part of the construction selects for accuracy at the wavenumbers where fine-scale content resides. We introduce Spectral-like Neural Discretisation (SpeND), in which the choice of those degrees of freedom is cast as a learning problem: stencil weights are parametrised by a neural network conditioned on the local node geometry, trained to approximate the modal response of a spectral operator over the resolvable band. A hard-constrained projection layer maps the network output onto the affine subspace of consistent weights, so that polynomial consistency holds exactly by construction rather than as a penalty. Training is self-supervised and physics-agnostic, requiring no reference solutions; the objective minimises dispersion and dissipation error over a prescribed band-limited function space. Modal analysis on disordered two-dimensional node distributions shows that the learned fourth-order operator follows the exact response over a substantially wider band than either explicit LABFM at equal stencil size or fourth-order finite differences on a structured grid, whilst recovering the expected fourth-order convergence rate under refinement.
I. INTRODUCTION
Traditional mesh-free operators derive resolving power from kernel, assembly, and polynomial-consistency choices, while compact and multi-kernel approaches leave practical or bandwidth limitations. SpeND instead learns explicit, spectrally targeted discretisations while enforcing consistency exactly.
- Traditional mesh-free resolving power is mainly determined by the kernel, assembly algorithm, and imposed polynomial-consistency order.
- Compact meshfree schemes improve resolving power but require a global linear solve, complicating dynamic h-adaptivity and non-trivial boundary conditions.
- Multi-kernel approaches improve traditional explicit operators, but their gains remain confined to a narrow wavenumber band.
- SpeND learns explicit discretisation weights conditioned on local geometry and projects them onto the exactly polynomial-consistent subspace.
- SpeND training is self-supervised and physics-agnostic, requiring no reference solutions while targeting operator error over a prescribed band-limited function space.
A. Numerical Approximation of Differential Operators
Local mesh-free differential operators are weighted sums over neighborhoods, with weights chosen to reproduce a differential operator on polynomial basis functions. When stencils are larger than the constraint system, traditional closure choices determine the remaining weights and their resolving power.
- Each node uses a local neighborhood, and the discrete differential operator is specified by weighted summation over that neighborhood.The neighborhood contains the N nearest nodes.
- Traditional consistent operators obtain stencil weights by requiring exact reproduction of a differential operator on polynomial basis functions.
- For N > M, the consistency system is underdetermined, leaving an affine solution subspace of dimension N − M.
- Traditional methods fix the remaining degrees of freedom through smoothing kernels, basis preconditioning, or minimum-norm conditions.
- Because the closure does not reference varying wavenumbers, the resulting stencil’s resolving power is determined by whichever closure is selected.
B. Resolving Power
Resolving power measures how well a discrete operator reproduces the continuous operator across wavenumbers and can distinguish stencils of identical order through modal analysis. For derivatives, effective-wavenumber deviations quantify dispersion and dissipation errors.
- Resolving power is the wavenumber range over which a discrete operator reproduces the continuous operator.
- Modal analysis distinguishes stencils of identical order by evaluating their response to independent Fourier modes.
- For D = ∂x, the effective wavenumber is defined from the discrete operator’s response to e^{ik·x_ba} and is normalised to recover k_x exactly.
- The real-part deviation of k_eff from k_x measures dispersion error, while the imaginary part measures dissipation error.
III. METHOD
SpeND predicts stencil weights from normalized local geometry using shared network parameters, making the representation independent of physical cloud scale. The physical scale factor is restored during inference.
- A neural network parametrises discretisation weights from the local stencil geometry, with shared parameters trained once across all stencils.
- Each stencil is normalized to the unit ball before being supplied to the network.
- Normalized weights represent weights scaled by local stencil size, while the scale factor is reintroduced at inference.
- The normalization makes the network invariant to the physical scale of the node cloud.
A. Training
SpeND trains stencil weights against the exact modal response across a prescribed resolvable band, using asymmetric penalties for dispersion overshoot and instability.
- A. Training: For each k in the resolvable band, SpeND trains keff to coincide with the exact response cos(k · xba) for D = ∂x.The target wavevectors tile the half-disk ∥k∥≤kny, kx ≥ 0; the omitted half follows from parity.
- A. Training: The objective averages per-stencil dispersion and dissipation losses over stencils from pre-generated disordered point clouds.A positive γ balances resolving power against stability.
- A. Training: Overshoot penalties weight ℜ{keff} > kx more heavily because propagation ahead of the physical wavefront is treated as more harmful than under-resolution.The asymmetry uses ω = λ> 1 above the exact response and unity otherwise.
- A. Training: Positive imaginary effective wavenumber is penalised more heavily to improve the stability of the discretisation weights.The asymmetric weight is ω+ = λ+ > 1 where ℑ{keff} > 0.
B. Hard Constraints
SpeND enforces polynomial consistency exactly through projection, leaving the network to learn only the unconstrained stencil directions.
- B. Hard Constraints: Every operator expressible by SpeND satisfies the consistency equations by construction rather than through a loss penalty.The learned degrees of freedom are precisely the N − M directions left free by consistency.
- B. Hard Constraints: The consistent weights form an affine subspace of dimension N − M, onto which raw network outputs are projected.The projection maps each unconstrained output to the consistent solution set.
- B. Hard Constraints: The projection uses a minimum-norm correction based on the Moore–Penrose pseudoinverse.This correction places the raw output at its closest point in the solution set.
- B. Hard Constraints: When the raw network output is already consistent, the projection reduces to the identity.The projection imposes no architectural requirement on the network producing the raw weights.
IV. RESULTS
The preliminary experiment uses a graph neural network with hard fourth-order consistency and exact modal targets on disordered two-dimensional node distributions.
- IV. RESULTS: The graph neural network uses star-shaped, bidirectional connections between each central particle and its neighbours.Each neighbourhood is represented as a graph whose nodes are the stencil particles.
- IV. RESULTS: Training uses two-dimensional disordered node distributions and enforces fourth-order consistency for the ∂x operator.The consistency vector is b = [1, 0, . . . , 0]⊤ ∈ R14.
- IV. RESULTS: The modal target is the exact response cos(k · xba) over wavenumbers ∥k∥∈(0, kny).The target is applied to the projected weights during training.
A. Modal Analysis
Modal analysis compares SpeND with explicit LABFM and fourth-order finite differences across two wave angles and reports wider spectral agreement without sacrificing fourth-order asymptotic behaviour.
- A. Modal Analysis: SpeND is compared with explicit LABFM at equal stencil size |Na| = 30 and fourth-order finite differences on a structured grid.The effective wavenumber is normalised by kny for both methods and wavevectors.
- A. Modal Analysis: The analysis includes ky = 0 and ky = kx wave angles, shown as solid and dotted lines respectively.The diagonal in the plot represents the exact response.
- A. Modal Analysis: SpeND follows the spectral response over a substantially wider wavenumber range than LABFM or FDM, departing only near the Nyquist limit.The inset reports absolute error over k/kny ∈(0, 0.6), where SpeND outperforms both methods across the full band.
- A. Modal Analysis: The hard-constrained layer preserves fourth-order asymptotic behaviour, shown by rapid absolute-error decay as k → 0.The improved resolving power is therefore not obtained at the cost of formal order.
B. Convergence
Under refinement on unstructured node distributions, SpeND and explicit LABFM recover fourth-order convergence, while SpeND remains consistently more accurate and resolves substantially more of the practical wavenumber band.
- Both SpeND and explicit LABFM recover the expected fourth-order convergence rate once the highest harmonic is resolved.The test field is a four-term truncated square wave with harmonics up to k = 14π.
- SpeND is approximately 2 times more accurate at the coarsest resolution and 3 times more accurate at the finest.
- SpeND retains fourth-order convergence while resolving substantially more of the wavenumber band at practical resolutions.The convergence test uses a superposition of harmonics with non-uniform amplitudes rather than a single training-objective mode.
V. CONCLUSIONS
SpeND learns explicit mesh-free operators with spectral-like resolving power on unstructured node distributions while preserving fourth-order convergence. Its explicit inference avoids global solves and adds no runtime cost relative to comparable mesh-free methods, but validation beyond the reported setting remains in progress.
- SpeND resolves substantially more of the wavenumber band than explicit LABFM and fourth-order finite differences at equal stencil size while retaining fourth-order convergence.
- Both SpeND and explicit LABFM attain fourth-order convergence, with SpeND 2–3 times more accurate throughout refinement.
- SpeND requires no global solve at inference and has the same runtime cost as SPH, LABFM, or RBF-FD when weights are precomputed and reused.
- Out-of-distribution validation and long-time time-dependent solves for eigenvalue stability and sustained resolving power remain future work.