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Learnable Splines for Unstructured Neural Operators

Vignesh Gopakumar, Zhuoyuan Wang, Ander Gray, Valentin Duruisseaux, Josh Williams, William Michael Edward Ellis, Nuno Nobre, Matthew Field, James Panayis, Alexander I. Blair, Andrew Davis, Stanislas Pamela, Robert Akers, Marc Peter Deisenroth, Anima Anandkumar

arXiv:2610.22446v1cs.AIcs.LGphysics.comp-ph

TL;DR

Engineering simulations require repeated, costly solves on irregular meshes, while Fourier neural operators assume uniform rectangular grids. L-SplineFNO couples conditional learning with a learnable spline transfer to structured Fourier processing. Across six engineering benchmarks it achieves strong accuracy, with the strongest results on four cases and close performance on Aircraft, while remaining less accurate than mesh attention on the bracket case.

  • Problem

    Irregular engineering meshes and the cost of repeated high-fidelity solves motivate surrogates that retain information on unstructured domains while addressing the limits of grid-based operators.

    Source: §1 · p. 2

  • Method

    L-SplineFNO conditions an FNO backbone on global design and operating parameters and uses a learnable structure-preserving spline transfer between unstructured meshes and a latent grid.

    Source: §1 · p. 3

  • Results

    L-SplineFNO has the strongest predictive accuracy on heat sink, building fire, heated duct, and HIVE, remains close to leading attention models on Aircraft, and trails mesh attention on bracket.

    Source: §4.1 · p. 9

  • Takeaways & Limitations

    The spline reconstruction provides inexpensive analytic gradients and rigorous bounds over whole cells and uncertain parameter boxes.

    Source: §5 · p. 12

    The spline transfer can smooth sharp features, and the fixed linear decoder limits outputs to the span of the spline basis, especially affecting sharp interfaces and surfaces embedded in volumetric grids.

    Source: §5 · p. 12

Abstract

from arXiv · show

Neural operators approximate solution operators of partial differential equations at a fraction of the cost of numerical solvers. Efficient grid-based architectures such as the Fourier neural operator (FNO) assume uniform rectangular grids, whereas engineering simulations use large, irregular, and locally refined meshes. We introduce L-SplineFNO, a memory-efficient neural operator that couples an FNO backbone with a sparse, learnable tensor-product B-spline transfer between unstructured meshes and a latent Cartesian grid. A local normalised scatter avoids the geometry-dependent solve of analytic spline transformation while preserving compact support, partition of unity, and spline smoothness by construction. The neural operator, conditioned on global design and operating parameters, learns the mapping of geometry to the field solution. Across six three-dimensional engineering benchmarks, L-SplineFNO achieves strong predictive accuracy while requiring substantially less memory and compute than baseline operators, scaling to 26.6 million nodes in training and 46 million in inference on a single A100 GPU. It also supports zero-shot super-resolution and efficient closed-form spatial derivatives. Uniquely, the analytic spline decoder yields rigorous prediction bounds over continuous space and uncertain parameters, via Bernstein convex hulls and Taylor models, demonstrated on a 29.2-million-node MAST-U tokamak model.

1 Introduction

Engineering simulations repeatedly solve parameterized field problems on irregular meshes, while grid-based FNOs require uniform rectangular grids. L-SplineFNO combines conditional learning with a learnable structure-preserving spline transfer to retain structured Fourier processing and provide continuous-space and parameter bounds.

1 Introduction

  • Repeated simulation is costly because design sweeps and uncertainty studies require many evaluations, while a single high-fidelity solve can take hours on a compute cluster.Safety-critical settings also require guaranteed prediction ranges under uncertain operating conditions.Source: §1 · p. 2
  • Compact support, partition of unity, and spline smoothness give the transfer local stencils, constant preservation, and closed-form evaluation and differentiation at off-grid queries.The stencil size is set by spline degree rather than a search radius.Source: §1 · p. 2
  • Bernstein convex hulls and Taylor models provide rigorous prediction bounds over continuous space and uncertain parameter boxes.The decoder’s nonnegative Bernstein basis keeps predictions within coefficient convex hulls.Source: §1 · p. 3
  • The operator incorporates global design and operating parameters; benchmark settings range from parameters that restate geometry to a fixed mesh where parameters are the only sample-specific signal.Source: §1 · p. 3
  • L-SplineFNO replaces analytic SplineFNO’s geometry-dependent linear solve with a learnable structure-preserving B-spline scatter to a latent FNO grid and analytic reconstruction at arbitrary queries.Source: §1 · p. 3
  • Across six 3D thermal, structural, and aerodynamic benchmarks, L-SplineFNO shows strong predictive accuracy, linear mesh-size scaling, inexpensive analytic derivatives, and deployment on a 29.2-million-node MAST-U mesh.The evaluation compares representative mesh-native, latent-attention, and gridded operators under a common conditioning pipeline and matched parameter counts.Source: §1 · p. 3

2 Background and Related Work

Conditional operator learning maps geometry and parameters to steady-state solution fields. Existing approaches extend FNOs to unstructured geometries through mesh-native processing or latent-grid transfers, while prior B-spline and conditioning methods serve distinct roles.

Conditional operator learning

  • For steady problems without an input field, the solution operator maps geometry and parameters to the solution, with all sample-specific information entering through the domain and parameter vector.Source: §2 · p. 3

Neural operators

  • FNOs use Fourier-space convolution with O(N log N) FFT complexity, but their operation is restricted to uniform rectangular grids.Source: §2 · p. 4

Operator learning on unstructured geometries

  • Unstructured operator approaches include mesh-native models, latent-grid transfers, and precomputed geometry representations, with trade-offs in memory, fidelity, and end-to-end training.No representation is uniformly best across accuracy, cost, memory, and deployment.Source: §2 · p. 4

B-splines in scientific machine learning

  • Prior scientific B-spline methods use splines as output representations, unlike L-SplineFNO’s learnable transfer between an unstructured mesh and a structured latent grid.Source: §2 · p. 4

Conditioning on physical parameters

  • Physical parameters may repeat geometric information or encode hidden loads and material properties; FiLM and related approaches inject parameters through feature modulation, attention, or generated weights.Source: §2 · p. 4

3 Spline-Gridded Fourier Neural Operators

The framework lifts geometry- and parameter-conditioned mesh features onto a uniform latent grid for an FNO, then reconstructs outputs at query points. L-SplineFNO replaces the analytic transfer solve with a local learnable scatter.

3 Spline-Gridded Fourier Neural Operators

  • The method carries mesh features to a uniform latent grid for a standard FNO and reconstructs outputs at arbitrary query points.Region-specific lifts, scatter kernels, and output projections can share one latent grid and FNO; an analytic density feature augments scattered features.Source: §3 · p. 4, §3.3 · p. 7

3.1 Conditional Input Lifting

  • The model conditions nodal features on mesh geometry and a global vector of design and operating parameters, without a supplied input field.Coordinates bypass FiLM modulation and determine the compact spline stencils.Source: §3.1 · p. 5

3.2 SplineFNO

  • Analytic SplineFNO transfers unstructured features to a uniform latent grid, processes them with an FNO, and reconstructs features at output coordinates.Its sparse, analytic spline transfers have no learned parameters; learning occurs in the FNO.Source: §3.2 · pp. 5–6

3.3 Learnable SplineFNO (L-SplineFNO)

  • L-SplineFNO replaces the geometry-dependent spline solve with a local learnable scatter that approximates the analytic transformation.The transfer retains compact support, partition of unity to numerical precision, and C^(p−1) regularity; analytic reconstruction produces outputs at query points.Source: §3.3 · pp. 6–7
  • L-SplineFNO outperforms baselines on most experiments while remaining significantly compute-efficient.Table 1 reports relative L2 error and training time for each architecture under FiLM conditioning.Source: Table 1 · p. 7

4 Experiments

Across the benchmark suite, L-SplineFNO combines strong predictive accuracy with a favorable accuracy–cost trade-off. The section also evaluates analytic gradients, large-mesh scalability, and rigorous bounds on a full-device MAST-U model.

4.1 Benchmarks

  • L-SplineFNO is strongest on heat sink, building fire, heated duct, and HIVE, stays close to leading attention models on Aircraft, and trails direct mesh attention on the bracket.Compared with analytic SplineFNO, learning the spline transfer improves accuracy across all six cases with only a modest model-size change.Source: §4.1 · p. 9

4.2 Analytic Gradients

  • Closed-form spline differentiation gives pointwise field gradients without a reverse-mode coordinate tape, with query cost O(N(p + 1)^d).Full-model autograd measures coordinate sensitivity, which can differ from the field gradient for globally coupled operators.Source: §4.2 · pp. 9–10

4.3 Scalability

  • L-SplineFNO trains on up to 26.6M nodes and runs inference on 46.0M, beyond the capacity of the compared methods.On an A100 80GB GPU in FP32 with batch size one, runtime grows near-linearly with node count for each model over its feasible range, making memory the capacity constraint.Source: Figure 4 · p. 9, §4.3 · p. 10

4.4 Virtual Tokamak of MAST-U

  • On the full 29.2M-node MAST-U structural mesh, L-SplineFNO reaches 1.8% relative vector L2 error on a single H100 GPU.The model predicts displacement under thermal expansion using the thermal expansion coefficient and wall temperature as inputs.Source: §4.4 · pp. 10–11
  • Bernstein coefficient-ratio extrema enclose the spline prediction throughout each cell, including between mesh nodes; spatial subdivision tightens the bound.Second-order Taylor models propagate parameter uncertainty through the trained network, and parameter subdivision shrinks nonlinear remainders.Source: §4.4 · pp. 11–12
  • The reported intervals bound the surrogate prediction, not its error against a reference solution; only validated FP64 results carry an end-to-end rounding certificate.Ordinary FP32 or FP64 can evaluate the same enclosure formulas more cheaply, but without that certificate.Source: §4.4 · p. 12

5 Discussion

The discussion weighs L-SplineFNO’s efficiency and continuous guarantees against smoothing and basis limitations, while the spline construction preserves locality, partition of unity, and smoothness. It also outlines extensions for interfaces, physics-informed training, active learning, and transient fields.

5 Discussion

  • L-SplineFNO scales linearly in mesh size, performs strongly on most benchmarks, and provides analytic gradients and rigorous whole-cell and parameter-box bounds.Its principal accuracy exception is bracket stress concentrations, where mesh-based attention remains more accurate.Source: §5 · p. 12
  • The spline transfer can smooth interface kinks and small-feature stress concentrations, while its fixed linear decoder imposes an accuracy floor that higher spline degrees do not remove.Higher degrees also raise three-dimensional stencil costs and can exhaust memory on large meshes.Source: §5 · p. 12
  • Repeated knots could represent interface discontinuities, while derivatives, parameter enclosures, and FiLM conditioning suggest extensions to physics-informed training, active learning, and transient fields.The current enclosures bound the surrogate rather than the numerical simulation; calibrated surrogate-to-solver error bounds are proposed for future work.Source: §5 · p. 13

A Notation

  • Table 2 groups symbols by topic and points to their definitions; Table 3 collects symbols with multiple meanings, resolved by context.Bold lowercase denotes a vector, calligraphic letters denote collections, subscript G marks latent-grid quantities, and hats mark predictions.Source: A Notation · p. 20, Table 2 · p. 20, Table 3 · p. 23

B Learnable-Spline Construction

  • The construction lifts one-dimensional cardinal B-spline kernels to multiple dimensions by tensor products, with the default degree set to cubic.The learned kernel adds constrained compact spline corrections to the fixed analytic spline.Source: B Learnable-Spline Construction · p. 24, §B.2 · p. 25

B.1 From 1D to d-D: Tensor-Product Construction

  • Tensor-product splines retain compact support, partition of unity, and C^(p−1) smoothness, enabling constant-field reproduction and bounded local stencils.For degree p, each mesh point interacts with at most (p + 1)^d grid points.Source: §B.1 · pp. 24–25

B.2 From Analytic to Learned Kernels

  • Learned corrections preserve compact support, partition of unity, and C^(p−1) smoothness, but may have negative lobes and lose the analytic spline’s convex-combination property.The correction is constrained through a null-space parameterisation; its partition-of-unity condition holds to numerical precision.Source: §B.2 · pp. 25–26

B.3 Stable Normalised Scatter and Density Feature

  • Signed learned weights can cancel during accumulation, so the normalised scatter safeguards its denominator and adds an analytic density feature to retain mesh-coverage information.The density feature is initialized with zero weight, leaving the initial model output unchanged.Source: §B.3 · pp. 26–27

B.4 Continuous spatial bounds and parameter uncertainty

The bounds combine Bernstein enclosures of the spline decoder with second-order Taylor models of parameter dependence. Spatial subdivision tightens the enclosures, while validated binary64 includes arithmetic uncertainty for the stored-weight surrogate.

B.4 Continuous spatial bounds and parameter uncertainty

  • Positive denominator Bernstein coefficients make each decoder prediction a convex combination of coefficient ratios, yielding continuous spatial enclosures rather than bounds based on sampled points.The method applies to the rational tensor-product cubic spline on each decoder cell.Source: §B.4 · p. 27
  • The median enclosure width halves between subdivision depths zero and three, while each further level roughly doubles cost.Each subdivision splits a box at the midpoint of its longest side and covers every original decoder cell.Source: §B.4 · p. 27, Table 4 · p. 28
  • Second-order Taylor polynomials with bounded remainders propagate parameter dependence through the network, and shared parameter symbols preserve dependencies across points, channels, and spline coefficients.The method transforms signed polynomial coefficient fields into spatial Bernstein form before computing parameter extrema.Source: §B.4 · p. 28
  • At depth three, joint displacement-magnitude ranges are 41.06–42.80 mm for steel, 39.47–42.22 mm for copper, and 35.26–39.21 mm for graphite.The ranges use an 8 × 8 parameter partition and cover the original uncertainty box.Source: §B.4 · p. 29
  • Validated binary64 increases median and maximum widths by approximately 1.53% and 1.72%, respectively, while enclosing the real surrogate defined by stored weights.Ordinary FP32/FP64 propagation does not provide an end-to-end rounding certificate; sampled containment checks do not establish that guarantee.Source: §B.4 · pp. 30–31, Table 5 · p. 31
  • The enclosures bound surrogate predictions, not prediction error against a reference solution or PDE, solver-convergence, or physical-modelling error.The guarantee concerns the surrogate prediction itself.Source: §B.4 · p. 32

C L-SplineFNO Implementation

L-SplineFNO transfers point features to a Cartesian grid with sparse tensor-product spline stencils, processes them with an FNO, and gathers grid features at query points. The implementation uses guarded normalization, optional conditioning and multiscale or region-specific paths, and chunked computation to manage cost and memory.

C L-SplineFNO Implementation

  • The FNO processes the scattered grid features, which are gathered at query points using fixed analytic spline weights; source and query clouds may differ in location and size.The output projection maps gathered features to physical output variables.Source: C L-SplineFNO Implementation · pp. 33–34
  • Each point uses four neighboring grid nodes per axis, with axis-wise normalized weights multiplied to form a compact tensor-product stencil.Weights are normalized along each axis when the safeguard floor is inactive; learned and analytic kernels use different floors.Source: C L-SplineFNO Implementation · p. 33
  • Guarded scatter normalizes weighted feature sums using signed and absolute weight totals, bounding outputs under signed-weight cancellation and leaving uncovered nodes zero.Constant features are reproduced where neither normalization floor changes the denominator.Source: C L-SplineFNO Implementation · p. 34
  • FiLM conditioning modulates the 96 sinusoidal coordinate features, while normalized coordinates bypass modulation; a geometry-derived density channel is appended after transfer normalization.The density channel is initialized to have no effect on features.Source: C L-SplineFNO Implementation · p. 35
  • Optional model paths support multiple scales and region-specific branches, with region routing based on supplied node labels and a shared latent backbone.Separate region kernels accumulate their scatter sums before normalization.Source: C L-SplineFNO Implementation · p. 35
  • At fixed width and stencil size, scatter and gather scale linearly with point counts, while factored stencils, chunked recomputation, shared-geometry reuse, and matrix-free decoding reduce transfer overhead.The encoder avoids a grid Gram matrix and factorization; chunking controls temporary buffers rather than total model or training memory.Source: C L-SplineFNO Implementation · p. 35

D Physics and Data

The six benchmarks span shared- and per-sample-mesh geometries and differ in how much conditioning information is already visible in the point cloud. The heated duct adds a multi-region setting with distinct interface behavior.

D Physics and Data

  • The six benchmarks distinguish shared-mesh cases (Aircraft, HIVE, Heated duct) from per-sample-mesh cases (Heat sink, Bracket, Building fire).The distinction concerns whether a single node ordering applies across cases.Source: D Physics and Data · p. 36
  • Bracket and Building fire each combine geometry-inferable and hidden parameters, while Heat sink parameters describe shapes already visible to the model.Bracket pressure and fire ventilation parameters are hidden from the point cloud; fire coordinates and bracket geometry are inferable.Source: D Physics and Data · p. 36
  • Aircraft, HIVE, and Heated duct have non-redundant parameters: their shared geometry does not distinguish operating conditions across runs.Aircraft cases share coordinates across flight conditions; HIVE and Heated duct likewise use one mesh across runs.Source: D Physics and Data · p. 36
  • The heated duct has three separate meshes coupled at two conjugate interfaces, where temperature is continuous but its gradient is not.Heated duct and Building fire use three encoding and decoding branches with a shared latent backbone.Source: D Physics and Data · p. 36

D.1 Conventions common to all datasets

The benchmarks use steady-state conditioned operator learning, with geometry as the only per-node input and global parameters supplied separately. Common conventions govern mesh handling, coordinate encoding, and reporting of test error.

D.1 Conventions common to all datasets

  • Each sample provides inputs, targets, global parameters, and a steady-state flag; variable-size Heat sink and Bracket meshes are sampled or duplicate-padded to a fixed node budget.Repeated real nodes preserve geometry for gridded backbones, while padding reduces the effective number of distinct nodes.Source: §D.1 · pp. 36–37
  • All six benchmarks are geometry-only: coordinates are processed into 99 channels, comprising three normalized coordinates and 96 sinusoidal features.Coordinates come from case-specific inputs when positions vary, or mesh metadata for fixed meshes; conditioning modulates only sinusoidal features under the listed methods.Source: §D.1 · p. 37
  • Field and parameter values are normalized per variable, and splits use seeded random permutations except for the heated duct’s designed split.Normalizers and the coordinate processor are fitted on the pooled corpus before splitting; Building fire uses cell-size-aware positional encoding.Source: §D.1 · p. 37
  • Reported test errors are percentage relative L2 errors in physical units, averaged per sample; this equals 100×NRMSE under the paper’s definition.Because the denominator is target magnitude rather than variation, the measure is generous to fields with large constant offsets, including the thermal cases.Source: §D.1 · p. 38

D.2 Heat sink

The heat-sink benchmark tests whether a conditional operator can infer temperature from varying three-dimensional geometry. Its thin fins and variable unstructured meshes create a demanding setting for grid-based methods.

D.2 Heat sink

  • The benchmark models steady conduction through a finned aluminium CPU heat sink, with forced convection laterally and free convection on top.Material conductivity, heat load, film coefficient, and ambient temperature are fixed; geometry alone varies and determines the solution.Source: §D.2 · p. 38
  • Fins 1.5–2.5 mm thick at 3.25–6.5 mm pitch alias on uniform latent grids, and at least 16 cells along x are needed to separate eight fins.The pitch follows from a fixed 26 mm baseplate and the number of fins.Source: §D.2 · p. 38
  • The 625 heat-sink cases have distinct tetrahedral meshes ranging from 4,671 to 32,645 nodes, with median mesh size 12,673.At the used budget of 16,384 nodes, 193 larger cases are sampled and the remaining 432 are duplicate-padded.Source: §D.2 · p. 38
  • The learning setup uses four design parameters and a 70/30 split, with three coordinate input channels and temperature as the target.The four-factorial design includes all 625 combinations exactly once; the temperature values span 296.46–343.76 K.Source: §D.2 · p. 38, Table 9 · p. 38

D.3 Bracket

The bracket benchmark models steady elasto-plastic deformation and predicts nodal stress. Its parameters separate geometry already visible in the mesh from applied pressure that is not, while the dataset uses globally normalised designs and meshes with no node correspondence.

D.3 Bracket

  • The benchmark predicts nodal stress for steady elasto-plastic deformation of a hollow lug fixed at one end and pressure-loaded over a marked area.It uses quadratic hexahedral elements, isotropic hardening, and the small-deformation assumption.Source: §D.3 · p. 39
  • The three geometry parameters restate information visible in the node coordinates, whereas applied pressure is mesh-invisible and can vary between designs with identical geometry.The loader exposes the redundant and non-redundant parameter halves for independent ablation.Source: §D.3 · p. 39
  • The 3,000 designs span four continuous parameters—length, thickness, hole radius, and applied pressure—and the released coordinates, parameters, and stress are globally min–max normalised to [0, 1].Because coordinate and parameter scaling factors are unavailable, physical units cannot be recovered; global normalisation preserves mean per-design spans and relative scale between designs.Source: §D.3 · p. 39
  • Each design has a separate hexahedral mesh with 5,054–15,210 nodes (median 9,405) and no node correspondence between designs.At a node budget of 8,192, 2,122 larger meshes are subsampled and 878 are duplicate-padded; average distinct-node coverage is 97%.Source: §D.3 · p. 39
  • Figure 11 presents stress error maps from various architectures modelling the bracket.The supplied passage identifies the visual’s subject but does not state a comparative outcome.Source: Figure 11 · p. 40
  • Normalised stress has mean 0.197 and standard deviation 0.121, with its 99th percentile at 0.534 and case means spanning [0.086, 0.383].The upper half of the normalised range is populated only by stress concentrations at the hole and fixed end.Source: §D.3 · p. 40

D.4 Aircraft

This section examines how operators use geometry and conditioning across aircraft and other cases, including flight-condition interpolation on largely seen shapes, smoke transport, conjugate heat transfer, and parameter-only conditioning. It also covers limits associated with split design, prediction-domain choices, and multi-material interfaces, and how the operator supplies spatial structure when six parameters capture the bulk thermal level.

D.4 Aircraft

  • Aircraft test scores measure flight-condition interpolation on mostly seen airframes, since each airframe’s conditions share coordinates and only µ distinguishes them.The geometry still distinguishes airframes; pressure is the most heavy-tailed target (Kurt = 108) and drives the joint score.Source: §D.4 · pp. 40–41

D.5 Building fire: smoke and heat transport in a building

  • Two fire-position parameters are recoverable from the mesh, but five ventilation parameters are invisible in node coordinates, so geometry alone reveals fire location but not ventilation.The fire-centred refinement box makes the fire position recoverable; the retained mesh still moves with the burner after the box is excluded.Source: §D.5 · pp. 43–44
  • The fire prediction domain excludes the burner’s refinement-box gas nodes, leaving 92,837 nodes to predict heat and combustion-product transport through occupied storeys.Although those nodes are 16.5% of the gas-cell cloud, they carry roughly two thirds of error energy because they contain the flame; excluding them shifts the task away from the prescribed source.Source: §D.5 · pp. 43–44

D.6 Heated duct: conjugate heat transfer

  • The heated duct combines three disjoint material meshes in one cloud, making material membership inferable from coordinates while testing whether a single C2 basis can represent interface kinks.Temperature is continuous across conjugate interfaces but its gradient jumps; on the [48, 48, 32] grid, 24.0% of covered knots receive mass from multiple regions.Source: §D.6 · pp. 45–46
  • In the heated duct, qh and Tin rescale or shift temperature exactly, while only Uin, kh and km change the field shape Θ.The decoupled momentum equation yields T(x; qh, Uin, Tin, kh, km) = Tin + qh Θ(x; Uin, kh, km).Source: §D.6 · p. 46

D.7 HIVE monoblock

  • HIVE is a conditioning-pathway test: its shared mesh and absent field input leave all run-to-run variation, including coil displacement, to be conveyed through µ.The model fits a 6 → R194691 map from roughly 148 training points; a companion configuration adds the initial temperature field to measure its value against parameter conditioning.Source: §D.7 · pp. 48–49
  • A linear fit of μ to case mean explains R2 = 0.89 and scores 6.9%, versus 9.3% for a global constant and 6.5% for a per-case constant.The passage notes that the ∼300 K pedestal flatters Equation (39), while the six parameters set the bulk thermal level. Tfinal spans [300.00, 557.23] K with mean 341.43 K and standard deviation 36.67 K.Source: §D.7 · p. 49
  • The rise to convergence spans [0.03, 162.58] K with mean 39.34 K, while Tinitial spans [294.06, 405.23] K.Tfinal has Skew = 1.30 and Kurt = 4.78; case-mean temperatures span [310.98, 414.67] K.Source: §D.7 · p. 49

E MAST-U: full-device structural reconstruction

The section presents a full-device MAST-U structural surrogate evaluation and describes six baseline architectures for unstructured neural operators. The MAST-U study uses a 70-solve parameter scan on a shared 29.2-million-node mesh, with test cases interpolating within the training parameter ranges.

E MAST-U: full-device structural reconstruction

  • A whole-device surrogate targets repeated structural queries because a static solve takes about 500 CPU-hours and component boundary conditions are not known in advance.The stated first step toward a structural digital twin conditions the surrogate on two parameters of a thermal-load study.Source: E MAST-U: full-device structural… · p. 49
  • The 70-solve corpus shares a 29,183,754-node mesh, and its ten test cases measure interpolation within the training parameter ranges.The study predicts nodal displacements for a thermal-load parameter scan; element-wise stresses are not targets.Source: E MAST-U: full-device structural… · pp. 49–50
  • 3.40% relative vector L2 error on one held-out native-mesh solve, versus 1.60% on its sampled nodes; the evaluation also changes the source set.All 29,183,754 native nodes are used as sources and queries for the full-mesh result.Source: E MAST-U: full-device structural… · p. 50

F Models

  • The six baselines comprise mesh-native point-set or latent-token models and gridded operators; the latter transfer mesh data to a uniform latent grid for FNO processing.The spline encode/decode pair replaces the nearest-cell scatter used by NUNO and the radius-graph kernel integral used by GINO.Source: F Models · p. 51

F.1 Transolver

  • Transolver groups nodes into learned feature-based slices, applies attention among slice tokens, then broadcasts results back to the mesh.Its Physics-Attention groups physically similar states regardless of spatial proximity.Source: §F.1 · pp. 51–52

F.2 General Neural Operator Transformer (GNOT)

  • GNOT uses separate trunk and branch streams with linear attention and coordinate-gated expert blocks, and its pointwise trunk supports queries at unseen coordinates.In this setup, the single branch receives the same node tensor as the trunk.Source: §F.2 · pp. 52–53

F.3 Universal Physics Transformer (UPT)

  • UPT pools the point cloud into a fixed number of latent tokens, processes those tokens, and decodes at arbitrary query positions.Its processor cost is independent of the mesh node count.Source: §F.3 · p. 53

F.4 Multi-Scale Patch Transformer (MSPT)

  • MSPT preserves local detail with full attention inside patches and communicates globally through pooled supernodes that persist across blocks.Spatially coherent ball-tree patches correspond to compact regions of the unstructured domain.Source: §F.4 · p. 53

G Conditioning Methods

The section compares conditioning methods applied to encoded coordinate features before the model’s input lift. Across the reported ablations, L-SplineFNO averages the best performance and compute efficiency, while FiLM performs well relative to concatenation and cross-attention.

G Conditioning Methods

  • Conditioning acts on encoded coordinate features before input lifting, while the normalized coordinates themselves remain unmodulated.The reported methods do not condition width-c lifted features or intermediate backbone states.Source: G Conditioning Methods · p. 56

G.1 FiLM

  • FiLM applies parameter-dependent channel-wise scale and shift uniformly across spatial locations, preserving feature width and spatial structure.Zero-initialized heads recover the unconditioned backbone behavior at initialization.Source: §G.1 · p. 56

G.2 Concatenation

  • Concatenation broadcasts each scalar parameter to every node or grid point, increasing the input feature width from cf to cf+k.The backbone is left to interpret the appended parameter channels.Source: §G.2 · p. 56

G.3 Cross-Attention

  • Across the reported ablations, L-SplineFNO averages the best performance and compute efficiency, and FiLM performs remarkably well against concatenation and cross-attention.The provided table captions identify six benchmark cases but do not supply their cell values.Source: §G.3 · p. 57, Table 18 · p. 58, Table 19 · p. 59

H Scalability

The section reports approximately linear runtime and memory growth for L-SplineFNO over the measured point-cloud sizes, alongside zero-shot evaluation on finer held-out meshes. The reported super-resolution tests retain high predictive agreement at resolutions above training.

H Scalability

  • L-SplineFNO’s runtime and memory grow approximately linearly over measured sizes, and its lower memory use permits larger point clouds; NUNO is faster in successful steady-state trials.The measurements use fixed latent resolution and do not establish accuracy scaling or a unique asymptotic advantage.Source: H Scalability · p. 64, Figure 16 · p. 64
  • L-SplineFNO maintains broadly stable runtime and peak-memory cost per million nodes as point-cloud size increases.Figure 17 presents these per-node costs on logarithmic axes.Source: Figure 17 · p. 65

I.1 Zero-shot super-resolution

  • Without finetuning, the model retains high predictive agreement on held-out native meshes at 1.91×, 1.79–1.86×, and 2.53× training resolution for Heat Sink, Bracket, and Aircraft, respectively.The tests use meshes with at least 95% of each dataset’s maximum available nodes.Source: §I.1 · p. 66, Table 24 · p. 66
  • Coordinate-dependent spline scatter and decoding operate independently of mesh node count and ordering, enabling deployment on finer meshes without retraining.The reported super-resolution results concern tasks selected for mesh variability across their datasets.Source: §I.1 · p. 66

I.2 Multiscale depth and spline degree

The ablations vary scatter scale and spline degree while holding the main backbone and training setup fixed. Neither change provides systematic accuracy gains, while both increase computational cost, supporting the single-scale cubic configuration as the reported accuracy–efficiency compromise.

Multiscale depth

  • Multiscale scatter adds coarser branches over the same domain, interpolates them back to the fine grid, and fuses them before the FNO blocks.The added branches increase parameter count by at most 0.13%.Source: §I.2 · p. 66
  • Multiscale depth gives no systematic accuracy gain, while training time rises 21–44% for 1–2 scales and 40–89% for 1–2–4 scales.Aircraft improves under 1–2 in every seed, but the heat sink is consistently harmed; other reported changes are small or inconsistent.Source: §I.2 · p. 66, Figure 18 · p. 67

Spline degree

  • Increasing spline degree raises continuity from C2 for cubic to C4, C6, and C8, while expanding support from four to six, eight, and ten knots per axis.The degree ablation retains one scatter scale and does not change the FNO backbone.Source: §I.2 · p. 66
  • Higher spline degrees do not improve accuracy overall; degree five helps Aircraft but raises errors on Bracket, Heated Duct, HIVE, and Heat Sink.Degrees seven and nine were each run at a single seed, and their errors were never meaningfully below cubic.Source: §I.2 · p. 67
  • In three dimensions, spline stencil entries grow from 64 at degree three to 1000 at degree nine, driving a systematic training-time penalty.The streamed, memory-bounded chunks allow every tested degree to complete, so the cost appears in training time rather than memory; cubic is the reported accuracy–efficiency compromise.Source: §I.2 · pp. 67–68

I.3 Analytic Gradient

The analytic spline decoder evaluates continuous-field gradients by differentiating its normalized basis. Across six cases, it is faster and uses less additional peak memory than decoder autograd.

I.3 Analytic Gradient

  • The decoder defines a continuous field over query coordinates, and the analytic gradient differentiates the normalized basis, including boundary normalization.The readout bias contributes no derivative.Source: §I.3 · pp. 68, 70
  • Each query touches at most 4^d grid points; coefficient reuse and batched basis evaluation give O(N4^d) query work for fixed output channels, in addition to grid projection.Material-specific derivatives are evaluated within each material, with coordinate and output normalization scales converting results to physical units.Source: §I.3 · p. 70
  • 1.84–3.67× faster and 52–68% less additional peak memory than decoder autograd; maximum physical-gradient L2 disagreement was 3.04 × 10−6.Figure 19 shows per-case costs; Table 27 measures derivative evaluation and unit conversion, excluding encoding and prediction.Source: Figure 19 · p. 68, §I.3 · p. 70, Table 27 · p. 70
  • With the encoded latent grid fixed, decoder autograd and the analytic spline compute the same pointwise field gradient; full-model VJP measures a different coordinate sensitivity.Moving an input node can change the encoded grid and therefore predictions elsewhere.Source: Table 27 · p. 70, §I.3 · p. 70

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