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arxiv: 2511.12402 · v2 · submitted 2025-11-16 · 🧮 math.NA · cs.NA

DataTransfer: Neural network based interpolation across non-nested meshes

Jiaxiong Hao , Yunqing Huang , Nianyu Yi This is my paper

Pith reviewed 2026-05-17 22:48 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords neural networkinterpolationnon-nested meshesradial basis functiondata transfermesh-based numerical simulationsfunction approximation
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The pith

Neural networks interpolate functions across non-nested meshes using only scattered source data.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that function interpolation between meshes can be treated as a regression problem solved by neural networks. Traditional approaches depend on mesh connectivity and geometry, making them expensive and sensitive to mismatches. By building a neural network from source mesh nodal values, a global function representation is obtained that works for any target mesh. Among the tested architectures, the radial basis function network offers the best trade-off between accuracy, speed, and robustness. This is significant for numerical simulations where meshes do not align, as it simplifies data transfer and enhances stability.

Core claim

The proposed neural network based function mapping model constructs an approximator using nodal function values on the source mesh to obtain a global representation of the function, which can then be interpolated onto any other meshes, with the radial basis function-based network achieving the most desirable overall performance balance.

What carries the argument

The neural network approximator, specifically the radial basis function hidden unit network, which learns a continuous global representation from discrete scattered nodal data without requiring mesh connectivity or geometric properties.

If this is right

  • Interpolation and data transmission across non-nested meshes can be performed accurately and efficiently.
  • The method reduces dependence on explicit mesh matching, lowering computational costs in multi-mesh simulations.
  • Improved numerical stability in algorithms involving function mapping between different discretizations.
  • Validation on non-nested meshes confirms efficacy for both function interpolation and cross-mesh data transmission.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Such networks could be applied to problems with evolving meshes in dynamic simulations.
  • Integration with physics-informed neural networks might further enhance accuracy for physical systems.
  • This data-driven approach may generalize to other types of mesh operations like projection or remapping in computational science.

Load-bearing premise

That the neural network, trained solely on scattered nodal values from the source mesh, learns a sufficiently accurate global representation of the underlying function that generalizes to points on any target mesh.

What would settle it

Observing large approximation errors when evaluating the trained network at target mesh points located in regions with insufficient source node coverage or high function gradients.

Figures

Figures reproduced from arXiv: 2511.12402 by Jiaxiong Hao, Nianyu Yi, Yunqing Huang.

Figure 1
Figure 1. Figure 1: Schematic architecture of a fully connected feedforward neural network. Throughout this section and those that follow, the interpolation accuracy is assessed using two stan￾dard metrics defined on the target mesh TB: the mean absolute error (MAE), which measures the average pointwise deviation between predicted and reference values, and the relative L 2 error (RL2), which quan￾tifies the normalized energy … view at source ↗
Figure 2
Figure 2. Figure 2: Theorem 2.1, NN interpolation on a quasi-uniform mesh. (A) training mesh containing 2,500 nodes; (B) exact solution used as reference; (C) history of the training loss; (D) test mesh with 10,000 evaluation points; (E) interpolated solution together with the corresponding error distribution [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Theorem 2.2, NN interpolation on an adaptive mesh. (A) training mesh; (B) reference solution; (C) history of the training loss; (D) test mesh with 10,000 evaluation points; (E) interpolated solution together with the corresponding error distribution. Solved with adaptive finite element methods (see, e.g., [13]), the problem produces a series of numerical approximations accompanied by the corresponding adap… view at source ↗
Figure 4
Figure 4. Figure 4: Schematic architecture of an extreme learning machine network. in [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Theorem 2.4, ELM interpolation on a quasi-uniform mesh. (A) Test mesh; (B) interpolated solution and error. Interpolation results on a uniform triangular mesh with 10,000 nodes are displayed in [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Theorem 2.5, ELM interpolation on an adaptive mesh. (A) training mesh; (B) interpolated solution and absolute error on a uniform 100×100 test grid; (C) training mesh with an additional Gaussian point (red); (D) interpolated solution and absolute error. Adaptive finite element meshes provide an instructive setting to examine how the distribution of train￾ing samples influences the performance of ELM-based i… view at source ↗
Figure 7
Figure 7. Figure 7: Schematic architecture of a RBF-ELM network. -1.0 -0.5 0.0 0.5 1.0 x -0.5 0.0 0.5 1.0 y True Solution -1.0 -0.5 0.0 0.5 1.0 x -0.5 0.0 0.5 1.0 y Predicted Solution -1.0 -0.5 0.0 0.5 1.0 x -0.5 0.0 0.5 1.0 y Absolute Error 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00e+00 6.58e-13 1.32e-12 1.97e-12 2.63e-12 [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Interpolated solution and corresponding absolute error distribution for the singular function test case [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of piecewise linear interpolation and the RBF-ELM interpolation model for a smooth 1D function. (A) Error of piecewise linear interpolation after 100 iterations; (B) Error of RBF-ELM interpolation after 100 iterations; (C) Error evolution history over transfer iterations [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Piecewise linear interpolation results for a 1D multi-peak problem. (A)–(D) Interpolated solutions after 1, 5, 25, and 100 transfer iterations. 0.0 0.2 0.4 0.6 0.8 1.0 x -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 y RBF_interp - Iter 200, A B True f(x) A grid B grid 0.0 0.2 0.4 0.6 0.8 1.0 x -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 y RBF_interp - Iter 200, B A True f(x) A grid B grid (A) 0 25 50 75 100 125 150 175 200 Ite… view at source ↗
Figure 11
Figure 11. Figure 11: RBF-ELM interpolation performance and iterative error on multi-peak func￾tion. (A) Interpolated solution after 200 iterations; (B) Error evolution history over trans￾fer iterations [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Mesh configuration for 2D interpolation. (A) Uniform triangular mesh TA; (B) Delaunay triangular mesh TB. Benchmarking starts with the smooth reference solution u(x, y) = −x 2 − y 2 on [−1, 1]2 , illustrated in Figure 13A. In what follows, we employ an RBF-ELM configuration with Nc = 1000 centers randomly drawn from the training samples and a shape parameter ε = 10. The error maps after 100 transfer itera… view at source ↗
Figure 13
Figure 13. Figure 13: Evolution and spatial distributions of interpolation errors for a smooth 2D problem. (A) Reference solution; (B) Error evolution history over transfer iterations; (C) Piecewise linear interpolated error after 100 iterations; (D) RBF-ELM interpolated error after 100 iterations. 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.2 0.4 0.6 0.8 1.0 y 1.0 0.5 0.0 0.5 1.0 1.5 f( x, y ) Reference Solution 1.0 0.5 0.0 0.5 1.0 (A) 0… view at source ↗
Figure 14
Figure 14. Figure 14: Convergence behavior of interpolation errors for a 2D multi-peak problem. (A) Reference solution; (B) Error evolution history over transfer iterations. 4. Conclusions This work presents an examination of three interpolation strategies for transferring function data across unstructured meshes and an analysis of their respective trade-offs. The FNN data transform method pro￾vides stable interpolation and mo… view at source ↗
Figure 15
Figure 15. Figure 15: Comparison of piecewise linear and RBF-ELM interpolation results. (A)- (B) piecewise linear interpolation results after 25 and 100 iterations; left/right subplots correspond to transfers TA → TB and TB → TA, respectively; (C)-(D) RBF-ELM results after 25 and 100 iterations. limitations, the ELM approach achieves substantially higher accuracy with minimal training cost. How￾ever, this advantage is accompan… view at source ↗
read the original abstract

In mesh-based numerical simulations, the interpolation of mesh-defined functions across different meshes is a critical task, and achieving high-precision interpolation is of great significance for improving the computational efficiency and numerical stability of algorithms. This paper proposes neural network based function mapping model across meshes, wherein the interpolation process is reformulated as a data-driven regression problem over scattered function data. Conventional interpolation and projection-based approaches are highly dependent on mesh connectivity and corresponding geometric properties, which renders such methods computationally costly and sensitive to mismatches between source and target meshes. The proposed method constructs a neural network approximator using nodal function values on the source mesh to obtain a global representation of the function, which can then be interpolated onto any other meshes. To investigate the network architectural impacts on model performance, three representative feedforward network structures are numerically compared in this work: multi-layer perceptrons, extreme learning machines, and network incorporating radial basis function hidden units. The results reveal distinct trade-offs among accuracy, computational efficiency and model robustness, among which the radial basis function-based network achieves the most desirable overall performance balance, enabling fast and precise function calculation. Numerical experiments conducted on non-nested meshes validate the efficacy of the proposed model in both function interpolation and cross-mesh data transmission tasks.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes a neural network-based interpolation method for functions defined on non-nested meshes. It reformulates the task as a regression problem using only scattered nodal values from the source mesh to train a global approximator, then evaluates three feedforward architectures (MLP, ELM, and RBF networks) on function interpolation and cross-mesh data transfer tasks, concluding that the RBF-based network offers the best balance of accuracy, speed, and robustness.

Significance. If the performance claims hold with quantitative support, the method could provide a mesh-connectivity-independent alternative to classical projection or interpolation schemes, reducing sensitivity to mesh mismatch in multi-physics or adaptive simulations. The architectural comparison supplies practical guidance on model selection for scattered-data approximation. The absence of reported error norms, baseline timings against established methods, and generalization analysis limits immediate assessment of impact.

major comments (2)
  1. [Abstract / Numerical experiments] Abstract and numerical experiments section: the assertion that the RBF network 'achieves the most desirable overall performance balance' is unsupported by any quantitative error metrics (e.g., L2 or L∞ norms), baseline comparisons with conventional non-nested interpolation schemes, hyperparameter selection details, or statistical significance tests on the held-out target meshes.
  2. [Method description] Method and § on network construction: the central claim requires that a network trained exclusively on source nodal pairs (x_i, f_i) produces a representation accurate at arbitrary target-mesh points without any geometric or boundary information supplied to the model. No approximation-order analysis, sampling-density requirements, or error bounds are provided to justify reliable generalization when the target mesh resolves finer scales or approaches domain boundaries.
minor comments (1)
  1. [Abstract] Abstract: the phrase 'distinct trade-offs among accuracy, computational efficiency and model robustness' is stated without enumerating the observed trade-offs or the specific metrics used to identify them.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the constructive feedback on our manuscript. The comments highlight important areas for strengthening the quantitative evidence and clarifying the theoretical foundations of the data-driven approach. We address each major comment below and will revise the manuscript to incorporate the suggested improvements where feasible.

read point-by-point responses

  1. Referee: [Abstract / Numerical experiments] Abstract and numerical experiments section: the assertion that the RBF network 'achieves the most desirable overall performance balance' is unsupported by any quantitative error metrics (e.g., L2 or L∞ norms), baseline comparisons with conventional non-nested interpolation schemes, hyperparameter selection details, or statistical significance tests on the held-out target meshes.

    Authors: We agree that explicit quantitative metrics and baselines would better support the performance claims. In the revised manuscript we will add comprehensive tables reporting relative L2 and L∞ error norms across all test functions and mesh pairs. We will include direct comparisons against established non-nested schemes such as radial-basis-function interpolation with thin-plate splines and piecewise-linear projection onto a common triangulation. Hyperparameter choices (hidden-layer sizes, regularization coefficients, and activation parameters) were determined by grid search on a held-out validation subset drawn from the source mesh; these details and the resulting values will be documented. Although formal hypothesis testing was not performed, we will report mean errors together with standard deviations computed over ten independent training runs with different random initializations to illustrate robustness. revision: yes

  2. Referee: [Method description] Method and § on network construction: the central claim requires that a network trained exclusively on source nodal pairs (x_i, f_i) produces a representation accurate at arbitrary target-mesh points without any geometric or boundary information supplied to the model. No approximation-order analysis, sampling-density requirements, or error bounds are provided to justify reliable generalization when the target mesh resolves finer scales or approaches domain boundaries.

    Authors: The method is deliberately formulated as a purely data-driven regression that learns a global approximator from scattered source values alone; no mesh connectivity or boundary indicators are supplied to the network. We acknowledge that the manuscript currently lacks a theoretical analysis of approximation orders, sampling-density requirements, or rigorous error bounds. Deriving such guarantees for the chosen architectures lies outside the present empirical scope. In the revision we will insert a dedicated discussion subsection that summarizes the observed sampling densities needed for stable accuracy in our experiments and explicitly notes the potential degradation near domain boundaries. We will also augment the numerical section with additional test cases in which the target mesh is substantially finer than the source mesh, thereby providing further empirical evidence of generalization behavior. revision: partial

standing simulated objections not resolved
  • Provision of rigorous a priori approximation-order analysis and error bounds for the neural-network approximants; such analysis would require substantial new theoretical work that cannot be completed within the revision period.

Circularity Check

0 steps flagged

No circularity: data-driven NN approximator validated on independent target meshes

full rationale

The paper reformulates mesh interpolation as supervised regression: a feedforward network (MLP/ELM/RBF) is trained on source-mesh pairs (x_i, f_i) to produce a global function representation that is then evaluated at arbitrary target-mesh points. Reported accuracy is measured by direct comparison against known target values on held-out non-nested meshes; no equation, fitted parameter, or self-citation is used to define the target error or to force the reported performance. The central claim therefore remains an empirical statement about generalization rather than a tautological re-expression of the training data.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the universal approximation capability of neural networks for scattered data and on the empirical observation that RBF networks balance accuracy and speed; network hyperparameters constitute the main free parameters.

free parameters (1)
  • Network hyperparameters (hidden units, layers, RBF centers and widths)
    Choice of architecture depth, width, and RBF parameters directly controls the reported performance trade-off and must be selected for each mesh pair.
axioms (1)
  • domain assumption Feedforward neural networks can approximate continuous functions from scattered point samples to arbitrary accuracy.
    Invoked when the method treats the source-mesh nodal values as sufficient training data for a global function model.

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