Convolutional Neural Operators for robust and accurate learning of PDEs

• PDE solution operators map function-space inputs, including initial and boundary conditions, coefficients, and source terms, to solutions in suitable function spaces.This operator perspective motivates learning solution operators directly from data.Source: §1 · p. 1
• Conventional finite-difference, finite-element, finite-volume, and spectral methods can be prohibitively expensive for high-dimensional or many-query problems, motivating fast machine-learning surrogates.The cited applications include uncertainty quantification, inverse problems, PDE-constrained control, and optimization.Source: §1 · p. 1
• CNNs offer locality, computational and data efficiency, multiscale processing, and a connection to finite-difference discretizations, motivating their return to operator learning.Existing operator-learning methods nevertheless face issues including limited expressivity, aliasing, and possible inconsistency in function spaces.Source: §1 · p. 2
• The paper proposes CNO, a U-Net-based operator architecture whose CNN modifications enforce structure-preserving continuous-discrete equivalence for genuine, alias-free operator learning.The motivation is that naive CNN operator learning can produce grid-resolution-dependent results and aliasing errors.Source: §1 · p. 2
• CNOs are proved to be representation-equivalent neural operators and universal approximators for operators corresponding to a large class of PDEs.The universality result establishes approximation to desired accuracy.Source: §1 · p. 2
• CNO is on-par with or outperforms tested baselines across Representative PDE Benchmarks, including both in-distribution and out-of-distribution tests.The benchmarks span linear elliptic and hyperbolic, nonlinear parabolic and hyperbolic PDEs, with possibly multiscale solutions.Source: §1 · p. 2

Bandlimited Approximation

• CNOs restrict inputs and outputs to bandlimited function spaces because Sobolev spaces do not generally support continuous-discrete equivalence required for robust operator learning.Any target operator can be approximated arbitrarily accurately by an operator between bandlimited spaces, whose discrete representation can be defined using the sinc basis.Source: §2 · pp. 3–4

Definition of CNO

• The CNO architecture lifts an input function, applies layers combining upsampling or downsampling, convolution, and activation, then projects the result to the output space.Its blocks map bandlimited functions to bandlimited functions throughout the composition.Source: §2 · pp. 4, 6

Convolution Operator

• CNO convolutions are parametrized by discrete kernels in physical space, making them local rather than Fourier-space parametrizations as in FNO.The kernels are defined on grids fine enough to satisfy the Whittaker-Shannon-Kotelnikov sampling theorem [58].Source: §2 · p. 5

Activation Layer

• To prevent aliasing, CNO activation layers upsample before applying nonlinearities and downsample afterward to return outputs to the desired bandlimit.Pointwise nonlinearities can generate arbitrarily high frequencies and otherwise violate the underlying function-space constraints [14] [2].Source: §2 · p. 5

Instantiation through an Operator U-Net architecture

• The modified Operator U-Net uses downsampling and upsampling blocks, ResNet and invariant blocks, channel-wise patching, and skip connections to form a genuinely multiscale operator-learning architecture.Skip connections transfer high-frequency content across encoder-decoder levels while deeper processing filters it through sinc-based operations.Source: Figure 1 · p. 3, §2 · p. 6

Continuous-Discrete Equivalence for CNO

• CNOs are representation-equivalent neural operators, so they support evaluation at multiple grid resolutions without aliasing errors.This property follows from the construction of their discretized convolution, sampling, activation, and residual operations.Source: §2 · pp. 6–7

PDE setting

• The result applies to PDE solution operators G†: X∗ ⊂ H^r(D) → H^r(D) for abstract PDEs on D = T^2 whose differential operator depends on x through a coefficient function a ∈ H^r(D).The solution operator maps the coefficient function a to the PDE solution u.Source: §3 · p. 7

Assumptions

• The approximation theorem requires G† to be continuous and to satisfy a modulus-of-continuity condition with p ∈ {2, ∞} and 0 ≤ σ ≤ r − 1.The modulus ω is monotonically increasing and tends to zero as its argument tends to zero; the condition holds automatically when X∗ is compact and G† is continuous.Source: §3 · p. 7

Universality theorem

• Theorem 3.1 states that for any ε > 0 and operator G† satisfying the stated assumptions, there exists a CNO G achieving the desired approximation for every a ∈ X∗.The theorem assumes σ ∈ N0, p ∈ {2, ∞}, r > max{σ, 2/p}, and B > 0.Source: §3 · p. 7

Extensions

• A more general version in Supplementary Material B also incorporates additional source terms in the PDE.The main theorem is presented for the abstract PDE (3.1), while the supplementary result extends its scope.Source: §3 · p. 7

Training Details and Baselines

• Models are compared using ensemble hyperparameter training, selecting each configuration with the smallest validation error before reporting test errors against DeepONet (DON), FNO, Galerkin Transformer, FFNN [18], ResNet [18], and U-Net.The protocol randomly samples hyperparameter configurations for each model to identify the best-performing version on every benchmark.Source: §4 · pp. 7–8

Representative PDE Benchmarks

• The Representative PDE Benchmarks suite targets diverse, reproducible, computationally demanding two-dimensional Cartesian PDEs with sufficiently many spatial and temporal scales.The suite excludes one-dimensional problems because traditional solvers are fast and three-dimensional problems because data are expensive to obtain and store; multiscale problems require fine grids for accurate classical solutions.Source: §4 · pp. 8–9

Results

• CNO is the best-performing architecture on every RPB task except Allen–Cahn, outperforming FNO, U-Net, and ResNet across the benchmark suite.FNO outperforms FFNN, DeepONet [39], and Galerkin Transformer [8] on nearly all in-distribution tasks, while U-Net and ResNet are competitive; CNO nearly matches or exceeds them overall and improves on FNO by almost 20× for Poisson.Source: Table 1 · p. 8, Transport Equation · p. 11

Out-of-Distribution Testing

• CNO generalizes to unseen out-of-distribution data with test errors increasing by at most approximately 2×, while outperforming baselines except on Allen–Cahn.FNO generalizes poorly on transport because it lacks translation equivariance, whereas U-Net, ResNet, and CNO preserve it; DeepONet, FFNN, and Galerkin Transformer also perform poorly on transport.Source: Transport Equation · pp. 11–12

Resolution Invariance

• CNO provides resolution-invariant Navier–Stokes errors, unlike FNO and U-Net, whose errors increase by up to 25% and a factor of 3, respectively, across resolutions.CNO’s spectrum has more accurate high-frequency decay; FNO exhibits amplified frequency content consistent with aliasing, while U-Net amplifies high-frequency modes.Source: Transport Equation · p. 12, Figure 2 · p. 12

Efficiency

• CNO models achieve smaller validation errors than FNO at equal model size and training time, with a best-performing CNO model training almost twice as fast.The comparison uses parameter count and per-epoch training time on the Navier–Stokes benchmark.Source: Transport Equation · p. 13, Figure 3 · p. 13

Scaling laws

• CNO scales with training data at r = 0.37, faster than FNO at r = 0.28 and GT at r = 0.27, requiring approximately 14.3K samples for 1% error versus 60.1K and 133.2K.A smaller 0.82 M-parameter CNO scales at r = 0.3, worse than the best-performing 7.8M-parameter CNO, indicating model size can bottleneck data scaling.Source: Transport Equation · p. 13, Figure 3 · p. 13

5 Discussion.

• CNO significantly outperforms FNO and other baselines on most representative PDE benchmarks, including out-of-distribution tests, while targeting performance, efficiency, resolution invariance, and data scaling.The architecture is designed as a representation equivalent neural operator and is supported by a universality theorem for a broad class of PDE operators.Source: §5 · p. 14
• CNO preserves continuous-discrete equivalence for bandlimited functions, unlike CNN-based PDE solvers lacking this structure and FNO implementations that can fail to enforce it.CNO uses local spatial convolution and adapts anti-aliasing ideas from to PDE operator learning, whereas FNO convolves in Fourier space.Source: §5 · p. 14

A.1 Approximation of Operators mapping between Sobolev spaces by operators mapping between spaces of bandlimited functions.

• Any continuous operator between the stated Sobolev spaces can be approximated arbitrarily accurately by an operator between bandlimited spaces after discarding sufficiently high-frequency input and output components.Fourier projection and stability of the original operator provide the approximation mechanism through an appropriate frequency cutoff.Source: §A.1 · pp. 16–17

A.2 Continuous-Discrete Equivalence for Operator G∗from Section 2.1

• For bandlimited signals sampled at or above the Nyquist rate, continuous operators admit a unique discrete representation whose diagram commutes and whose aliasing error is identically zero.Sampling below the Nyquist rate produces nonzero aliasing error, while changes of sampling grids can be handled through corresponding basis changes.Source: §A.2 · pp. 18–19

A.3 Multi-channel versions of elementary operators for CNO (2.3)

• CNO’s multichannel convolution, upsampling, downsampling, and activation operators apply their single-channel counterparts independently across input or output channels.Convolution kernels are indexed by input and output channels, while sampling and activation mappings preserve the multichannel signal structure.Source: §A.3 · pp. 19–20

A.4 Discrete operators for CNO

• Discrete CNO convolution uses padded kernels to preserve the s × s spatial dimensions while summing contributions across input channels into output channels.The implementation supports circular or zero padding and kernels with shape k × k × d_in × d_out.Source: §A.4 · p. 20
• Discrete upsampling inserts zeros and applies an interpolation filter, whereas downsampling filters high frequencies before retaining every N-th sample; multichannel operations are channel-wise.Perfect filters can cause Gibbs ringing near discontinuities, so implementations generally use windowed-sinc filters instead.Source: §A.4 · p. 21
• The discrete activation applies a pointwise scalar nonlinearity, independently per channel, and experiments found that the fixed refinement factor N = 2 is sufficient for accurate predictions.The practical implementation must balance ideal filtering against ringing artifacts caused by infinite-impulse-response perfect filters.Source: §A.4 · pp. 21–22

Layer decomposition

• CNO layers decompose into convolution, nonlinear activation, and projection operators between spaces of bandlimited functions.This decomposition provides the elementary mappings used to establish representation equivalence.Source: §A.5 · p. 22

Activation correspondence

• The activation layer is constructed from projection and embedding operators whose discretizations implement downsampling and upsampling while preserving continuous–discrete equivalence.The pointwise activation is assumed to map into a higher-bandwidth bandlimited space, up to negligible frequencies.Source: §A.5 · p. 23

Proposition conclusion

• The exact correspondence of continuous and discrete constituent operators establishes CNOs as Representation equivalent Neural Operators (ReNOs) in the sense of [2], proving Proposition 2.1.The argument also extends to bandlimited periodic functions with finitely many nonzero Fourier coefficients.Source: §A.5 · p. 23

B Proof of Theorem 3.1 of Main Text

• Setting B.1 extends the theorem beyond coefficient-only coordinate dependence by allowing functions a, f1, …, fℓ and requiring a continuous PDE solution operator with stability control.The assumptions include a continuous operator eG, a restriction G† for fixed forcing inputs, and a continuity condition that holds automatically when X* is compact and eG is continuous.Source: B Proof of Theorem 3.1 of Main Text · pp. 23–24

The CNO is implemented through an encoder

• The proof first constructs a continuous operator by combining grid samples, Fourier features, interpolation, and neural-network approximation of the PDE operator.Bandlimited functions are recovered from fine-grid values, making the discrete representation equivalent to trigonometric-polynomial interpolation before applying universal approximation.Source: B Proof of Theorem 3.1 of Main Text · p. 24, The CNO is implemented through an… · p. 25

B Proof of Theorem 3.1 of Main Text

• For every ε > 0, a CNO approximates any operator satisfying Setting B.1 on bounded subsets of Hr(T2) to arbitrarily small error.The theorem assumes r > max{σ, 2/p}, p ∈ {2, ∞}, and establishes the existence of K and N.Source: B Proof of Theorem 3.1 of Main Text · p. 24, The CNO is implemented through an… · p. 26

The CNO is implemented through an encoder

• The encoder maps a to its grid evaluations together with Fourier features, and an invariant block maps this representation to the discrete CNO output.The encoder is EK(a) = (a(xN), (cos(k · xN), sin(k · xN))1≤∥k∥∞≤K); when ℓ = 0, K = 0 and no Fourier features are needed.Source: The CNO is implemented through an… · p. 25
• Translation equivariance lets a shallow CNN realize the continuous approximation, while lifted filters with stride N/M reproduce all translated subgrid evaluations on the fine grid.The resulting output approximates G†(a) across (N/M)^2 translated M × M subgrids, and interpolation filtering yields the CNO form exactly.Source: The CNO is implemented through an… · pp. 26–27
• The construction requires choosing M independently of N; setting N = M can prevent convergence, whereas the proof yields sparse convolution kernels with stride N/M.Alternative size-controlled approximation results could avoid this construction but would require a more complicated proof and stronger regularity assumptions on G†.Source: The CNO is implemented through an… · p. 27
• The implementation applies activations to an upsampled discrete signal, breaking exact equivariance in practice while retaining a universality route for polynomial activations.Polynomial activations recover exact equivariance at sufficiently high upsampling rates; rational activations are also discussed as possible alternatives.Source: The CNO is implemented through an… · p. 27
• A continuous CNO output enables automatic differentiation for physics-informed losses, but convergence-rate guarantees require stronger stability and regularity assumptions than Theorem B.2 provides.The needed ingredients include a polynomially decaying continuity modulus and stronger regularity of G and G†.Source: The CNO is implemented through an… · p. 27

B.1 Auxiliary results

• The auxiliary results state a universal approximation theorem: finite sums are dense in C(R^d) exactly when the activation σ is not an algebraic polynomial.The theorem assumes σ is locally essentially bounded and has a discontinuity set whose closure has zero Lebesgue measure.Source: §B.1 · p. 28
• Interpolation with the periodic sinc filter produces exactly the same result as trigonometric polynomial interpolation, first proved for odd one-dimensional N and extended to even N and multiple dimensions.The multidimensional extension follows by tensorisation.Source: §B.1 · pp. 28–29

C.1 Training and Implementation Details

• The supplementary technical section introduces the baselines used in the main text before detailing their architectures and experiment-specific settings.The supplied passages identify the baseline descriptions as part of the training and implementation details.Source: §C.1 · p. 29

C.1.1 Feed Forward Dense Neural Networks

• The dense neural-network baseline uses composed affine layers and scalar nonlinearities, with uniform hidden width, residual blocks, and shortcut connections [18].Inputs and outputs are encoded on Cartesian grids; Leaky ReLU is used, residual-block depth is fixed at k = 2, and width and layer count are selected by cross-validation.Source: §C.1.1 · pp. 29–30

C.1.2 ResNet

• The ResNet baseline combines convolutional downsampling, residual blocks, and transposed-convolution upsampling with batch normalization and ReLU activations [18].Its initial layer uses a 7 × 7 kernel, later downsampling layers use 3 × 3 stride-2 kernels, and channel counts increase during downsampling.Source: §C.1.2 · p. 30

C.1.3 UNet

• The UNet baseline follows the architecture proposed in while selecting the first-layer channel count through cross-validation and preserving the original progressive channel pattern.Subsequent layers are adjusted to remain aligned with the selected first-layer channel count.Source: §C.1.3 · p. 30

C.1.4 Convolutional Neural Operator

• CNOs use windowed-sinc filters to obtain finite impulse responses and reduce Gibbs ringing, with cutoff, transition-band width, and tap count explicitly controlled.The filters are constructed by multiplying ideal sinc filters by finite windows; two-dimensional filters are implemented through successive row and column convolutions.Source: §C.1.4 · pp. 30–31
• CNO experiments fix Nσ = 2, ch = 0.8, and Ntap = 12 because larger Nσ increases computation, ch ≥0.8 changes test error little, and more taps offer limited benefit.Increasing Ntap from 12 to 20 improves test error by approximately 1.5%, but raises per-epoch training time from 4.37s to 5.26s, exceeding a 20% increase.Source: §C.1.4 · pp. 31–32
• CNO equivariance is only approximate under practical activation-layer approximations, but sufficiently high upsampling appears adequate in practice; exact equivariance can use polynomial or rational activations.The exact-activation alternative may have limited practical utility.Source: §C.1.4 · p. 32

C.1.5 Galerkin Transformer

• The Galerkin Transformer is an encoder-decoder operator learner with convolutional downsampling, coarse positional encoding, and a softmax-free Galerkin-type attention bottleneck.Its decoder upsamples transformer features using convolutional and spectral convolutional layers.Source: §C.1.5 · p. 34

C.1.6 DeepONet

• DeepONet encodes sampled input functions with a branch network and combines its coefficients with trunk-network basis functions to represent the output function.The numerical implementation uses a convolutional branch and feed-forward trunk, with layer counts, residual blocks, and basis count selected by cross-validation.Source: §C.1.6 · p. 35

C.1.7 Fourier Neural Operator

• FNOs compose a lifting operator, repeated Fourier layers, and a nonlinear projection to map input functions to outputs.The Fourier layers use truncated DFT coefficients, trainable Fourier-mode matrices, inverse DFT reconstruction, residual weights, and GeLU activation; lifting dimension, layer count, and kmax are cross-validated.Source: §C.1.7 · pp. 35–36
• The FNO projection is implemented as a shallow neural network with one hidden layer, 128 neurons, and GeLU activation before producing the output field.The model’s tunable lifting dimension dv, number of Fourier layers L, and Fourier cutoff kmax are selected through cross-validation.Source: §C.1.7 · p. 36

C.2 Training Details

• Except GT, models use ADAM for 1000 epochs with L1 loss, step-wise learning-rate decay, weight decay, and cross-validated hyperparameters.FFNN, U-Net, and DeepONet use batch size 10, whereas FNO and CNO use batch size 32.Source: §C.2 · p. 36
• GT uses ADAM for 1000 epochs with a weighted L2 loss and a OneCycleLR schedule whose maximum learning rate is 5 · 10−4.The schedule uses div_fac = 104 and pct_start = 0.3, with max_lr selected empirically.Source: §C.2 · p. 36
• Models retain the parameter state with the lowest validation relative L1 error, while early stopping interrupts training after 50 epochs without improvement.Validation error is evaluated at every epoch.Source: §C.2 · p. 36
• Random-search cross-validation tests 30 configurations for FFNN, U-Net, and DeepONet, 48–72 for GT, 24–48 for CNO, and 36–72 for FNO.The selected configuration minimizes validation relative L1 error, and the searched model-size ranges are reported in Table 5.Source: §C.2 · p. 36
• The best-performing hyperparameter configurations for each model and benchmark are reported in Tables 6, 10, 7, 11, and 12.These configurations are the outcomes of the random-search procedure.Source: §C.2 · p. 37
• CNO is robust to random initialization, exhibiting a very low standard-deviation-to-mean ratio across all RPB benchmarks over 10 training runs.The reported statistics are means and standard deviations of relative median L1 test errors for CNO, FNO, and U-Net.Source: §C.2 · p. 37, Table 4 · p. 37

C.3 Details about the description and numerical results in each benchmark

• The appendix documents all experiments comprising the RPB benchmarks presented in the main text.This subsection provides benchmark-specific experimental descriptions and numerical results.Source: §C.3 · p. 37

C.3.1 Poisson Equation

• The Poisson experiment samples uniformly distributed coefficients aij from [−1, 1], with K = 16 and r = 0.5, and uses exact solutions as targets.Training and testing data are normalized to [0, 1] using the training normalization constants for testing.Source: §C.3.1 · pp. 37, 39
• For Poisson testing, training uses 1024 samples, each in- and out-of-distribution test set uses 256 samples, validation uses 128 samples, and solutions are sampled at 64 × 64 resolution.The operator maps source terms f to exact solutions u; out-of-distribution inputs increase the number of modes from K = 16 during setup to K = 20.Source: §C.3.1 · pp. 37–39
• CNO is clearly best on the Poisson equation, while UNet is a distant second and FNO has test errors more than an order of magnitude higher than CNO.CNO also approximates multiple solution frequencies accurately, whereas FNO performs poorly, especially on the out-of-distribution example.Source: §C.3.1 · p. 40, Figure 5 · p. 42
• CNO captures the Poisson solution’s complete frequency spectrum with high accuracy, whereas FNO and UNet incur substantial errors, particularly in high-frequency components.The ground-truth solution contains multiple scales, and FNO’s errors may be attributable to aliasing.Source: §C.3.1 · p. 41, Figure 7 · p. 42

C.3.2 Wave Equation

• For the wave equation, the learned operator maps initial conditions to u(·, T = 5), with c = 0.1, K = 24, and r = 1 during training.Out-of-distribution testing decreases r to 0.85, changing the relative amplitudes of modes and the solution dynamics.Source: §C.3.2 · p. 43
• The wave-equation dataset contains 512 training samples, 256 samples for each test regime, and 128 validation samples, all sampled at 64 × 64 resolution.Testing uses the same [0, 1] normalization constants as training.Source: §C.3.2 · p. 43
• CNO is the best-performing model for the wave equation in both in-distribution and out-of-distribution testing, reinforcing the main-text benchmark conclusion.The comparison uses empirical test-error distributions and representative predictions for CNO, FNO, and UNet.Source: §C.3.2 · p. 43, Figure 8 · p. 43

C.3.3 Transport Equation

• The transport experiments approximate the operator mapping initial data to the solution at T = 1 using 512 training, 256 validation, and 256 in-distribution and out-of-distribution samples.Both smooth Gaussian and discontinuous disk inputs are normalized, with shifted centers defining out-of-distribution tests.Source: §C.3.3 · pp. 43, 44, 46
• For smooth transport, CNO is slightly superior in-distribution and has a significant out-of-distribution advantage over UNet, while FNO, DeepONet, and FFNN generalize poorly.Smooth transport uses radially symmetric Gaussian inputs whose centers shift between training and out-of-distribution testing.Source: §C.3.3 · p. 44
• For discontinuous transport, UNet performs as well as CNO out-of-distribution, whereas FNO, DeepONet, FFNN, and GT generalize poorly because they fail to be translation equivariant.Discontinuous samples are generated at 128 × 128, Gaussian-filtered, and downsampled to 64 × 64 to reduce aliasing and Gibbs effects.Source: §C.3.3 · pp. 46–47

C.3.4 Allen-Cahn Equation

• For Allen–Cahn, FNO is marginally superior to CNO and UNet in-distribution, while UNet is best out-of-distribution.The data are generated by finite differences at 64 × 64 resolution because exact solutions are unavailable.Source: §C.3.4 · p. 47

C.3.5 Navier-Stokes

• CNO is clearly the best-performing model for Navier–Stokes on both in-distribution and out-of-distribution tests, significantly outperforming UNet and FNO.The benchmark learns initial velocity-to-velocity evolution from spectrally simulated, multiscale data.Source: §C.3.5 · p. 51

C.3.6 Darcy Flow

• CNO is best for Darcy flow in both in-distribution and out-of-distribution testing.The operator maps Gaussian-process diffusion coefficients to steady-state solutions, with different length scales defining the two test regimes.Source: §C.3.6 · p. 52

C.3.7 Flow past airfoils

• For flow past airfoils, CNO and UNet outperform FNO on out-of-distribution testing.The learned operator maps perturbed airfoil shape functions to steady-state density, with evaluation errors computed outside the airfoil.Source: §C.3.7 · p. 54

C.3.8 On the Choice of the RPB benchmarks.

• The RPB suite spans linear elliptic, linear hyperbolic, nonlinear parabolic, nonlinear hyperbolic, and non-local advection-diffusion PDEs.The benchmark rationale includes Poisson, wave, transport, Allen-Cahn, Compressible Euler, and Incompressible Navier-Stokes problems.Source: §C.3.8 · pp. 54, 57
• The benchmarks prioritize readily available, reproducible data and analytical solutions while avoiding three-dimensional and one-dimensional problems with inconvenient access or limited computational payoff.Two-dimensional problems are selected carefully because traditional methods can be too fast when the spatial problem is only one-dimensional or insufficiently complex.Source: §C.3.8 · p. 57
• This benchmark is relatively easy to approximate: at T = 30, at most 20 Fourier modes in each direction achieve an error of approximately 1%.A 20 × 20 grid therefore suffices, while traditional spectral or finite-difference methods require 10−3 seconds or lower.Source: §C.3.8 · pp. 57–58
• On the standard Navier-Stokes dataset, FNO and CNO obtain test errors of 1.15% and 0.96%, respectively, but are at most only one order of magnitude faster than traditional methods.Both models have inference times of approximately 10^-4 seconds, so training and data-generation overheads leave little payoff for this benchmark.Source: §C.3.8 · p. 58
• Three orders of magnitude or more of speedup makes the thin shear layer benchmark worthwhile for evaluating CNO as a PDE surrogate.The thin shear layer requires nearly 100 Fourier modes for 2% L1-error, whereas CNO achieves approximately 3% error in 10^-4 seconds versus 10^-1 seconds for a spectral-viscosity solver.Source: §C.3.8 · p. 58

C.4 Testing at Different Resolutions.

• CNO requires compatible grid representations, with sampling rate s ≥ 2w; signals exceeding the grid’s Nyquist limit must be filtered before downsampling.Discrete inputs are upsampled when s′ ≤ s and filtered and downsampled when s′ > s, while incompatible continuous inputs must first be transformed to the computational grid.Source: §C.4 · p. 59
• Resolution testing uses Navier-Stokes data generated at 128 × 128, downsampled to lower resolutions, with UNet, FNO, and CNO models trained at 64 × 64.CNO is applied after the prescribed representation transformations, whereas FNO and UNet are applied directly to unresized inputs following.Source: §C.4 · p. 59
• CNO demonstrates the highest stability across Navier-Stokes resolutions and is approximately invariant to resolution, unlike UNet and FNO.UNet strongly depends on training resolution, while FNO shows a less pronounced dependence; CNO respects continuous-discrete equivalence.Source: §C.4 · p. 59

C.5 Ablation Studies.

• Replacing CNO’s filtered upsampling, downsampling, and activation layers with nearest-neighbor upsampling, average pooling, and pointwise activations produces the CNO w/o Filters ablation.This ablation resembles a regular UNet with additional Encoder–Decoder ResNets and does not preserve continuous-discrete equivalence.Source: §C.5 · p. 60
• The ablations use the best-performing CNO hyperparameters and report both in-distribution and out-of-distribution test errors across every benchmark experiment.The second ablation removes the middle Encoder–Decoder ResNets while retaining corresponding-level UNet concatenations and the deepest ResNet.Source: §C.5 · p. 60
• The original CNO outperforms ablations in 12 of 16 tests, while matching the best models closely in the remaining four.The first ablation is generally worse, indicating that standard CNN operations introduce aliasing and resolution dependence by violating continuous-discrete equivalence.Source: §C.5 · p. 60
• Removing the middle Encoder–Decoder ResNets can make CNO slightly better in one out-of-distribution test, but otherwise worsens errors by less than 10% to almost 300%.The model retains continuous-discrete equivalence, yet the results show that ResNets are important for performance and generalization.Source: §C.5 · pp. 60–61
• The first ablation trains approximately 1.75 times faster for Poisson, 1.35 times faster for Darcy Flow, 1.25 times faster for Navier–Stokes, and about 1.1 times faster for several other equations.Training time is approximately equal for Discontinuous Transport.Source: §C.5 · p. 61
• The second ablation trains approximately 1.75 times faster for Poisson, 1.25 times faster for Navier–Stokes and Darcy Flow, and 1.1 to 1.15 times faster for several other benchmarks.Other benchmarks require similar training time.Source: §C.5 · p. 61

C.6 Error vs. number of training samples.

• For Poisson, CNO outperforms FNO across all training-set sizes; for Wave, FNO is better with few samples, whereas CNO becomes better with many samples.CNO’s error decays at approximately rate 0.5 with respect to the number of training samples.Source: §C.6 · p. 61

• Input and output samples are illustrated for the Poisson and Wave equations.Source: Figure 28 · p. 63, Figure 29 · p. 63
• Input and output samples are also shown for Continuous Transport, discontinuous transport, and the Allen-Cahn equation.Source: Figure 30 · p. 64, Figure 31 · p. 64, Figure 32 · p. 65
• The dataset depictions further include the Navier-Stokes equations, Darcy flow, and compressible Euler equations.Source: Figure 33 · p. 65, Figure 34 · p. 66, Figure 35 · p. 66