AMOR: Adaptive Modal Operator Refinement for Operator Learning on Arbitrary Meshes and Irregular 3D Geometries

1. Introduction

• Repeated spatial solves make many-query tasks expensive, especially for time-dependent PDEs where costs recur over time steps.Neural operators learn mappings between function-space representations to support evaluation across parameter values and discretization resolutions after training.Source: §1 · p. 2
• FNO’s FFT implementation assumes an equispaced Cartesian grid, and its tensor-product spectral weights scale as O(m^d), increasing memory and compute costs in three dimensions.Resampling irregular meshes can also staircase boundaries and require interpolation back to the original nodes.Source: §1 · p. 2
• AMOR replaces FNO’s Fourier spectral convolution with reduced-basis blocks evaluated on mesh nodes, supporting arbitrary node sets and fixed or learned bases through a common encode/decode interface.Separate input and output bases are coupled by a dense learned transformation across retained modes.Source: §1 · pp. 2–3
• For continuous point-separating bases that contain the constant function, AMOR is a universal approximator of continuous operators on compact sets of bounded inputs.Cross-mode coupling decodes modes from a constant channel, supplying positional encoding absent from componentwise lifting and projection.Source: §1 · p. 3
• AMOR is competitive with FNO overall and scales favorably across two- and three-dimensional benchmarks, including highly irregular meshes.Its reduced-coefficient parameterization avoids FNO’s explicit tensor-product mode scaling.Source: §1 · p. 3
• Existing reduced-basis operator work such as PODNO uses a shared POD basis and structured grids, whereas AMOR extends to separate input/output bases and non-tensor-product meshes.Other mesh-flexible approaches include graph operators, mesh-based simulators, and coordinate-based models, with trade-offs in cost or training requirements.Source: §1 · pp. 3–4

2.1. Operator Learning Framework

• AMOR learns a map from L2 input fields to L2 output fields, using a stack of lifted hidden operator layers and a shared shallow projection.The framework follows Lanthaler et al.; its activation assumptions include global Lipschitz continuity and nonpolynomial behavior.Source: §2 · p. 4, §2.1 · pp. 4–5

2.2. AMOR as a Neural Operator

• AMOR uses finite-rank reduced-basis integral paths, while hidden states may leave reduced spaces and dense modal couplings do not impose translation equivariance.When the target operator has translation symmetry, AMOR must learn it from data; the bases may be fixed or data-adaptive.Source: §2.2 · p. 5, §2.3 · p. 6

2.3. Universal Approximation

• For compact input sets with bounded componentwise L∞ norms, AMOR with fixed continuous, point-separating bases containing constant modes approximates any continuous operator to arbitrary accuracy.The theorem permits any depth L ≥ 2 and finite widths; input and output bases may be chosen independently.Source: §2.3 · p. 6
• The universality proof uses off-diagonal modal couplings to supply position internally, turning a constant channel into separating mode values without coordinate inputs.The theorem requires continuous bases that separate points and contain normalized constant modes; scalar spans containing 1, x1, …, xd separate points.Source: §2.3 · pp. 6–7

2.4. AMOR’s Architecture and Implementation

• Factorized coefficient weights reduce parameter count to a fraction 1/kdec + 1/dv of the dense count when that fraction is below one.The discrete implementation uses a Gram-matrix inner product to represent field geometry, with consistent or lumped mass matrices depending on the mesh.Source: §2.4 · pp. 7–8

2.5. Moving Between Meshes

• On a target mesh, the mass-weighted dual encoder preserves transported modal coordinates, and its encode-decode map is the orthogonal projector onto the prolonged basis span.This preserves coordinates on transported modal spaces, not full network evaluations; nonlinearities and transfer can prevent agreement for general fields.Source: §2.5 · pp. 10–11

2.6. Reduced Basis Constructions

• Reduced bases are orthonormalized in the mesh-weighted inner product; constructions include snapshot-based POD/PCA and scalar Laplacian modes lifted across field components.Theorem 4 additionally requires continuum continuity and point separation, which discrete orthonormality alone does not establish.Source: §2.6 · pp. 11–12
• Harmonic bases derive from a finite-element stiffness matrix or symmetric graph Laplacian, with boundary conditions and graph construction forming part of the basis specification.The construction depends on the geometry and chosen operator, not training snapshots; repeated eigenvalues require aligning independently fitted mesh bases before reusing mode-indexed weights.Source: §2.6 · p. 12
• Learned bases are orthonormalized during each forward pass, while optional Cayley rotations preserve orthonormality but do not enlarge the represented reduced space.Full column rank is required, and near rank deficiency affects numerical conditioning; the Cayley parametrization does not cover orthogonal matrices with eigenvalue −1.Source: §2.6 · p. 12
• Polynomial bases evaluate total-degree tensor-product Jacobi polynomials on affinely rescaled coordinates; degree at least one yields spans that separate spatial points.Classical orthogonality on the reference box need not hold on general domains or in the discrete mass inner product, so sampled polynomials are orthonormalized in the chosen numerical inner product.Source: §2.6 · p. 12
• Derivative-informed subspaces (DIS) use a Gaussian input prior and Jacobian actions to construct input and output modes; discarded input eigenvalues bound best retained-coordinate mean-squared error.Output modes are an Mout-weighted POD of Jacobian actions on prior samples. The construction uses Jacobian–vector and adjoint products without forming the Jacobian, and applies to unstructured meshes with a finite-element prior covariance.Source: §2.6 · pp. 12–13

3. Methods

• Eight benchmark problems span elliptic, hyperbolic, parabolic, and solid-mechanics PDEs on structured and unstructured meshes.Source: §3 · p. 13

3.1. Data

• Table 2 summarizes the eight datasets; solver, element type, boundary conditions, and parameter ranges are detailed in Appendix B.1.Source: §3.1 · p. 13, Table 2 · p. 13
• The benchmark datasets include elliptic problems with homogeneous Dirichlet conditions and Gaussian-process draws for forcing and material fields at three correlation lengths.Source: §3.1 · p. 13
• The Helmholtz dataset solves a fixed-frequency 600 Hz system with PML absorption and three variants of a bi-Laplacian material field.Source: §3.1 · p. 13
• Kolmogorov-flow datasets solve 2D Navier–Stokes at Re ∈ {40, 500, 5000}, varying spatial resolution and snapshot interval relative to flow decorrelation time.Source: §3.1 · p. 13
• The 3D linear-elastostatics dataset uses a clamped, uniformly loaded unit cube and a log-normal Young’s-modulus field at fixed Poisson ratio, mapping E to displacement u.Source: §3.1 · pp. 13–14

• The public 3D lid-driven-cavity Navier–Stokes dataset varies internal geometry across ellipsoids, toroids, boxes, and cylinders.Source: FlowBench LDC · p. 14
• The Airfoil and Elasticity Geo-FNO benchmarks predict Mach-number and internal-stress fields on geometry-varying meshes, using a curvilinear structured mesh and sample-dependent point cloud, respectively.Source: FlowBench LDC · p. 14

Aircraft rib.

• All three aircraft-rib datasets solve the same static linear-elasticity problem on a single tetrahedral mesh with 72,615 nodes, a clamped root, and a log-normal random field E(x).Source: Aircraft rib · p. 14
• The multipatch-bracket and distributed-aero versions also vary Poisson ratio, and the loading configurations are motivated by—but do not reproduce—reported aerospace structural-test scenarios.Source: Aircraft rib · p. 14
• The three loading regimes are fixed-tip axial traction, uniform traction away from the clamp, and 1–3 randomly placed patches with independent directions; the combined dataset has 40,000 samples.The separate cantilever-tip and distributed-aero datasets each have 10,000 samples, while multipatch bracket has 20,000; total load is rescaled to keep deflection within 0.01–0.3% of rib length.Source: Aircraft rib · p. 14, Figure 2 · p. 14

3.2. Model configurations and training

• AMOR and FNO use the same channel width and number of nonlocal layers but different nonlocal-operator parameterizations; on these benchmarks, AMOR retains 99.9% of training variance except where ranks are validation-selected.Source: §3.2 · pp. 14–15
• Models use AdamW, at most 300 epochs, early stopping on validation relative L1 error, and 70/20/10 data splits; aircraft-rib results use a single seed.Source: §3.2 · p. 15

4.1.1. AMOR is competitive with FNO on structured grids

• AMOR beats FNO on nonlinear-diffusion and Helmholtz variants and outperforms PCANet on every structured-grid problem, while FNO leads on Kolmogorov flow.The Kolmogorov-flow gap persists across all three variants; learned and Laplacian bases reduce but do not close it.Source: §4.1.1 · p. 15, Table 3 · p. 16

4.1.2. Choice of Basis

• Learned bases give the lowest errors, reducing PCA error by 19–42% but increasing training time per epoch 3.5–6.4×; the best precomputed basis depends on the problem.Polynomial bases work best for nonlinear diffusion, DIS for Helmholtz, and Laplacian for Kolmogorov flow.Source: §4.1.2 · pp. 15, 17, Table 4 · p. 17

4.1.3. Scaling to three dimensional problems

• AMOR is competitive with FNO on both 3D benchmarks; it is more accurate and faster on Elastostatics, while FNO is more accurate and faster on FlowBench.The reported comparisons are on the 3D FlowBench and Elastostatics benchmarks.Source: Table 5 · p. 17, §4.1.3 · p. 17
• AMOR’s inference time grows approximately linearly with basis count and 5.3× across tested widths, versus FNO’s cubic mode scaling and 11× width increase.For widths 8 to 64, times rise from 5.0 to 26.1 ms/sample for AMOR and 5.1 to 55.9 ms/sample for FNO; Figure 3 shows the tested scaling comparisons.Source: Figure 3 · p. 18, §4.1.3 · pp. 18–19

4.2.1. Benchmark problems from Geo-FNO

• AMOR reduces Geo-FNO’s average relative error by 31% on Airfoil and 42% on Elasticity, and has the lowest reported L2 error on both irregular-mesh datasets.The Airfoil and Elasticity benchmarks use irregular meshes; their reported error norms follow the corresponding source methods.Source: Table 6 · p. 19, §4.2.1 · p. 19

4.2.2. Elasticity of an aircraft rib

• Aircraft-rib test error rises with load complexity: below 1% for tip loading, 1.17% for uniform aerodynamic loading, and 10.4% for multipatch loading.A pooled model generalizes across all three regimes at a moderate accuracy cost relative to per-scenario models; multipatch loading remains the dominant error source.Source: §4.2.2 · p. 20, Table 7 · p. 20

4.3. Transfer to unseen meshes

• Without retraining, AMOR transfers to unseen meshes: nonlinear-diffusion errors remain near the training-grid result, and the rib model beats coarse finite-element prolongation on three finer meshes.Diffusion relative L1 errors are 2.00%, 1.87%, and 1.88% on 50², 150², and 200² grids, versus 1.91% on the training grid; rib errors are 9.45–15.02% versus 9.79–17.16%.Source: §4.3 · p. 20

5. Limitations

• A fixed global basis can require so many modes for transport-dominated or multiscale solution manifolds that its efficiency advantage is lost.Kolmogorov flow illustrates the limitation; dynamics-aware constructions may be needed beyond a static spatial basis.Source: §5 · pp. 20–21
• Separately fitting input and output bases can miss structure that is simple for the operator but not for individual snapshots.This may occur in bifurcating or composed families; joint or coupled basis fitting is proposed as an untested extension.Source: §5 · p. 21
• Mesh transfer requires one-time basis prolongation and construction of a target-mesh Gram-weighted dual encoder, unlike FNO’s resolution-independent Fourier basis.The authors report that this step is cheap and transfer accuracy is strong in the settings studied.Source: §5 · p. 21
• Without factorization, AMOR typically has more parameters than FNO on the studied 2D benchmarks, though this ordering can narrow or reverse in 3D.Inference memory on constrained hardware can still favor FNO.Source: §5 · p. 21

6. Conclusions

• AMOR replaces FNO’s Fourier spectral convolution with arbitrary-mesh reduced-basis blocks and supports separate input and output bases through a common encode/decode interface.For bases containing the constant function and separating points, it is a universal approximator of continuous operators.Source: §6 · p. 21
• For time-dependent, transport-dominated dynamics, the authors identify dynamics-adaptive constructions—not a better static spatial basis alone—as the next step.Suggested directions include trajectory-window PCA, jointly trained nonlinear encoders, coupled basis/operator learning, and multi-step losses.Source: §6 · p. 21
• AMOR operates directly on a highly irregular 3D aircraft-rib finite-element mesh with tens of thousands of nodes, without remeshing or changing learned weights across resolutions.The reported experiments span three physically distinct, experimentally motivated loading regimes, as described in the paper context.Source: §6 · p. 21
• The authors identify AMOR as a candidate for 3D industrial surrogate modeling when simulation data use tetrahedral, hexahedral, or unstructured meshes requiring additional geometric handling for FFT-based FNO.Examples named include crashworthiness, patient-specific hemodynamics, and additive-manufacturing process optimization.Source: §6 · p. 22

Code and Data availability

• The paper reports public code and dataset availability, including author-generated nonlinear-diffusion, 3D-elastostatics, and aircraft-rib datasets.It also identifies sources for Helmholtz, FlowBench, Kolmogorov-flow, airfoil, and elasticity datasets.Source: Code and Data availability · p. 22

Appendix A.1. FNO as a Neural Operator

• FNO transforms inputs with the FFT, applies a matrix to each retained Fourier mode, then transforms the result back.The parameterization uses the convolution theorem for efficient global receptive fields but standard FFT implementation assumes an equispaced grid.Source: Appendix A.1 · pp. 29–30
• Geo-FNO relaxes the equispaced-grid assumption by learning a deformation from the physical domain to a regular latent grid on which FNO operates.Source: Appendix A.1 · p. 30

Appendix A.2. Comparison with universality results for FNO, PODNO and PCA-Net

• FNO universality results concern periodic domains or averaging layers, while PODNO’s generalized spectral operator requires non-polynomial globally Lipschitz activation and suitable Sobolev basis functions.Theorem 4 instead requires continuous bases that separate points and contain the constant function.Source: Appendix A.2 · p. 30
• PCA-Net’s universality result is restricted to a single encoding-decoding step, whereas AMOR combines data-driven bases with deep compositions of nonlocal and nonlinear layers.The required PCA-Net network size for general Lipschitz operators can grow exponentially with truncation dimension; whether deeper AMOR compositions avoid that bound remains open.Source: Appendix A.2 · p. 30

Appendix A.3. FiLM Conditioning

• Aircraft-rib experiments use FiLM for scalar or low-dimensional load metadata not already represented as nodal fields; structured-grid benchmarks do not use it.The metadata include scalar cantilever tip load, or Poisson ratio and a case or patch indicator; traction enters as a nodal field.Source: Appendix A.3 · p. 30

Appendix A.4. Proof and Technical Details for Universal Approximation

• The AMOR universality proof combines finite-rank approximation, averaged modal features, and realization of the encoder and decoder by AMOR layers.It adapts the averaging argument to vector-valued inputs and uses modes as positional encoding when the input basis separates points.Source: Appendix A.4 · pp. 31–33

Appendix A.4.5. Proof of the Main Theorem

• The construction realizes averaged input features in the transition output, then approximates the resulting continuous map using output functions.The proof applies Lemmas 11, 13, and 14 in sequence.Source: Appendix A.4.5 · p. 36
• The theorem also holds for factorized AMOR because the construction’s coefficient tensors have rank at most one.The construction uses only encoding mode m = 1.Source: Appendix A.4.5 · p. 36
• Under the theorem’s assumptions, fixed input and output bases allow arbitrarily accurate approximation, but the result bounds neither required width nor conditioning or training error.The proof does not quantify how basis size or quality affects approximation efficiency.Source: Appendix A.4.5 · p. 37
• The proof’s hypotheses include constant modes and point-separating modes; failing these sufficient conditions does not rule out accurate approximation of a particular target.Periodic modes may identify opposite cell faces, while bases vanishing on the boundary lack the constant mode; both cases require separate arguments.Source: Appendix A.4.5 · pp. 36–37
• For the explicit fixed-linear-encoder/decoder form, inputs with equal encoded coefficients are indistinguishable, whereas AMOR retains nonlinear processing before averaging and after decoding.Removing all pointwise and residual routes around the initial linear encoding creates an input-information bottleneck.Source: Appendix A.4.5 · p. 38

Appendix B. Computational Details

• Nonlinear diffusion uses a cell-centred finite-volume scheme on a 100 × 100 grid, while Helmholtz uses P1 elements on a triangulated PML-extended rectangle.The nonlinear datasets vary input-field smoothness; the Helmholtz datasets vary prior parameters while retaining the same fixed point source.Source: Appendix B · p. 38

Appendix B.1.3. Kolmogorov flow. All three datasets are supplied by [68, 69]. All models

• Models predict a one-step Euler residual, with the input basis remaining snapshot-based and the output basis fit to increments Δu = u(t + Δt) − u(t).Source: Appendix B.1.3 · p. 38
• The datasets include Geo-FNO elasticity and airfoil fields, 3D linear elastostatics, FlowBench cavity simulations, and aircraft-rib loading scenarios.The setups range from a 972-node elasticity mesh and a 221 × 51 airfoil mesh to a tetrahedral rib mesh with 217,845 displacement degrees of freedom.Source: Appendix B.1.3 · pp. 38–39

Appendix B.2. Architecture and Hyperparameters

• FNO and AMOR share nonlocal-layer count and channel width, but FNO learns per-mode matrices while AMOR couples retained modes with dense or factorized coefficient tensors.AMOR uses factorization in most experiments; basis sizes are selected by explained variance or validation sweeps, depending on dataset.Source: Appendix B.2 · pp. 39–40

Appendix B.3. Training Hardware

• Training uses dataset-specific batch sizes and a common GPU resource within each dataset across models.The hardware is V100-16GB for nonlinear diffusion and Geo-FNO datasets, and V100S-PCIE-32GB for the other listed datasets.Source: Appendix B.2 · p. 40, Appendix B.3 · p. 40

Appendix C.1. Mesh generalization on the nonlinear diffusion dataset

• Relative L1 error remains between 1.87% and 2.00% across 50 × 50 to 200 × 200 grids when the 100 × 100-trained AMOR model transfers without retraining.The decoder modes are interpolated and paired with a dual encoder; the evaluation uses a separate 1000-sample test set at each resolution.Source: Appendix C.1 · p. 40, Figure C.4 · p. 42

Appendix C.2. Visualization of Representative Model Predictions

• Representative visualizations compare predictions and spatial errors for FNO, AMOR, and PCA-Net on Helmholtz and Kolmogorov-flow variants.Airfoil and elasticity visualizations show AMOR predictions, ground truth, and pointwise relative absolute errors for selected samples.Source: Helmholtz · p. 40, Figure C.5 · p. 42, Figure C.6 · p. 43

Appendix C.3. Computational Scaling in Three Dimensions

• Inference-time sweeps compare FNO mode counts with AMOR basis sizes selected to match training-set reconstruction errors, and also vary channel width.Other hyperparameters and compute resources are held fixed within each sweep.Source: Appendix C.3 · p. 44, Figure C.11 · p. 47
• Training-time sweeps vary mode count and channel width for both models, with other hyperparameters and compute resources fixed.FNO’s sweep counts modes per axis; AMOR’s counts basis functions in its input and output bases.Source: Appendix C.3 · p. 44, Figure C.12 · p. 47

Appendix C.4. Aircraft Rib

• Figure C.13 shows AMOR’s predicted displacement and pointwise error against ground truth for all three loading cases.The error increases with load-scenario complexity.Source: Appendix C.4 · p. 48, Figure C.13 · p. 48
• A single model on the pooled 40,000-sample dataset reaches 6.57%/7.34% overall relative L1/L2 error, with scenario errors increasing under joint training.The increases are 0.06% → 0.57% for tip, 1.17% → 2.27% for aero, and 10.40% → 11.93% for multipatch; tip values are not directly comparable because the pooled mapping includes load as an input.Source: Appendix C.4 · pp. 48–49, Table C.10 · p. 49
• On the three finer tip-loading meshes, AMOR improves on coarse-FEM-solution prolongation by 0.3–4.3 percentage points in relative L1 error.Both errors increase from h11 to h5; the baseline compares paired coarse and native fine-mesh solutions rather than using a single finest-mesh reference.Source: Appendix C.4 · p. 49, Table C.11 · p. 50