Learning Spectral-Like Mesh-Free Discretisations

• URL: http://arxiv.org/abs/2609.02833v1
• Date: Wed, 02 Sep 2026 17:16:16 GMT
• Title: Learning Spectral-Like Mesh-Free Discretisations
• Abstract summary: We introduce Spectral-like Modal Neural Discretisation (SpeND) in which the choice of those degrees of freedom is cast as a learning problem.<n>A hard-constrained projection layer maps the network output onto the affine subspace of consistent weights, so that consistency holds exactly by construction rather than as a penalty.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

Related papers

• Scalable Lindblad Noise Learning via Stochastic Tensor-Network Simulation [48.887627688666974]
  Learning dissipation rates in large-scale open quantum systems is a major obstacle for near-term quantum technologies.<n>Here, we propose a scalable noise-learning framework for Lindblad dissipation rates that combines a simulation method, the Jump Method (TJM)<n>We demonstrate the approach on two noise models in the Ising model, in which independent dissipation rates are learned for each site up to $N_mathrmsite=16$.<n>This combination of scalable numerics and rigorous theoretical guarantees positions TJM-based noise learning as a practical foundation for characterizing dissipation in large quantum devices.
  arXiv  Detail & Related papers  (2026-08-25T15:07:11Z)
• Domain-Decomposed Randomized Neural Networks for Partial Differential Equations in Unbounded Domains [5.423324761643612]
  Partial differential equations on unbounded domains are challenging because the exterior region must be represented without excessive truncation error.<n>We propose a domain-decomposed randomized neural network framework for such problems.
  arXiv  Detail & Related papers  (2026-06-30T08:41:54Z)
• A Convex Quasilinearization Method for Solving Nonlinear PDEs with Physics-Informed Neural Networks [0.0]
  We present a numerical method for the forward solution of nonlinear partial differential equations (PDEs)<n>The method, LiL-Q, is assessed on seven benchmarks, including systems (planestrain elasticity and the incompressible Navier-Stokes equations in two and three dimensions) and steady flow with heterogeneous permeability.
  arXiv  Detail & Related papers  (2026-06-16T17:09:59Z)
• Closed-Form Spectral Regularization for Multi-Task Model Merging [96.82449201305234]
  Model merging combines several independently fine-tuned experts into a single multi-task model without any training data.<n>State-of-the-art merging methods formulate merging as a layer-wise interference problem.<n>We propose SWUDI, a closed-form method that combines a soft exponential filter, which matches the gradient-flow trajectory of iterative descent.
  arXiv  Detail & Related papers  (2026-06-05T14:00:47Z)
• Variational Bayesian Flow Network for Graph Generation [54.94088904387278]
  We propose Variational Bayesian Flow Network (VBFN) for graph generation.<n>VBFN performs variational lifting to a tractable joint Gaussian variational belief family governed by structured precisions.<n>On synthetic and molecular graph datasets, VBFN improves fidelity and diversity, and surpasses baseline methods.
  arXiv  Detail & Related papers  (2026-01-30T03:59:38Z)
• DInf-Grid: A Neural Differential Equation Solver with Differentiable Feature Grids [73.28614344779076]
  We present a differentiable grid-based representation for efficiently solving differential equations (DEs)<n>Our results demonstrate a 5-20x speed-up over coordinate-based methods, solving differential equations in seconds or minutes while maintaining comparable accuracy and compactness.
  arXiv  Detail & Related papers  (2026-01-15T18:59:57Z)
• CompNO: A Novel Foundation Model approach for solving Partial Differential Equations [0.0]
  Partial differential equations govern a wide range of physical phenomena, but their numerical solution remains computationally demanding.<n>Recent Scientific Foundation Models (SFMs) aim to alleviate this cost by learning universal surrogates from large collections of simulated systems.<n>We introduce Compositional Neural Operators (CompNO), a compositional neural operator framework for parametric PDEs.
  arXiv  Detail & Related papers  (2026-01-12T10:04:48Z)
• Manifold limit for the training of shallow graph convolutional neural networks [1.2744523252873352]
  We study the consistency of the training of shallow graph convolutional neural networks (GCNNs) on proximity graphs of sampled point clouds.<n>We prove $$-convergence of regularized empirical risk minimization functionals and corresponding convergence of their global minimizers.
  arXiv  Detail & Related papers  (2026-01-09T18:59:20Z)
• Polynomial Neural Sheaf Diffusion: A Spectral Filtering Approach on Cellular Sheaves [23.06390960959433]
  Sheaf Neural Networks equip graph structures with a cellular sheaf: a geometric structure which assigns local vector spaces (stalks) and a linear learnable restriction/transport maps to nodes and edges.<n>We introduce Polynomial Neural Sheaf Diffusion (PolyNSD), a new sheaf diffusion approach whose propagation operator is a degree-K in a normalised sheaf Laplacian.
  arXiv  Detail & Related papers  (2025-11-28T23:10:54Z)
• Low-Rank Tensor Recovery via Variational Schatten-p Quasi-Norm and Jacobian Regularization [49.85875869048434]
  We propose a CP-based low-rank tensor function parameterized by neural networks for implicit neural representation.<n>To achieve sparser CP decomposition, we introduce a variational Schatten-p quasi-norm to prune redundant rank-1 components.<n>For smoothness, we propose a regularization term based on the spectral norm of the Jacobian and Hutchinson's trace estimator.
  arXiv  Detail & Related papers  (2025-06-27T11:23:10Z)
• Deep Learning Approximation of Diffeomorphisms via Linear-Control Systems [91.3755431537592]
  We consider a control system of the form $dot x = sum_i=1lF_i(x)u_i$, with linear dependence in the controls. We use the corresponding flow to approximate the action of a diffeomorphism on a compact ensemble of points.
  arXiv  Detail & Related papers  (2021-10-24T08:57:46Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.