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Out-of-distributional risk bounds for neural operators with applications to the Helmholtz equation

J. Antonio Lara Benitez, Takashi Furuya, Florian Faucher, Anastasis Kratsios, Xavier Tricoche, Maarten V. de Hoop

arXiv:2301.11509v3cs.LGcs.NAmath.NAstat.ML

TL;DR

Neural operators can generalize poorly on nonlinear PDE operators, and their generalization under distribution shifts remains an important focus. The paper introduces a transformer-inspired neural-operator family with stochastic depth and analyzes its complexity and out-of-distribution risk. It reports improved high-frequency Helmholtz performance and proposes a hypernetwork surrogate for the forward operator.

  • Problem

    Deep FNOs can perform poorly on some nonlinear PDE operators despite theoretical universality, while the paper focuses on generalization from finitely many noisy inputs.

    Source: §1 · pp. 1–2

  • Method

    The paper modifies neural operators into a sequential transformer-inspired architecture with stochastic-depth Bernoulli switches and analyzes its complexity and Gaussian-measure out-of-distribution risk.

    Source: §1 · p. 2, §2 · p. 3

  • Results

    The sNO + εI family improves performance on high-frequency Helmholtz problems without increasing parameter count in the no-stochastic-depth case or compromising traditional neural operators' approximation capabilities.

    Source: §9 · p. 29

  • Takeaways & Limitations

    The authors report that their results suggest the Helmholtz forward operator can be captured effectively by a hypernetwork surrogate, with potential use in Bayesian inversion.

    Source: §9 · p. 29

    The derived bounds are not tight, and obtaining lower bounds remains challenging.

    Source: §9 · p. 29

Abstract

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Despite their remarkable success in approximating a wide range of operators defined by PDEs, existing neural operators (NOs) do not necessarily perform well for all physics problems. We focus here on high-frequency waves to highlight possible shortcomings. To resolve these, we propose a subfamily of NOs enabling an enhanced empirical approximation of the nonlinear operator mapping wave speed to solution, or boundary values for the Helmholtz equation on a bounded domain. The latter operator is commonly referred to as the ''forward'' operator in the study of inverse problems. Our methodology draws inspiration from transformers and techniques such as stochastic depth. Our experiments reveal certain surprises in the generalization and the relevance of introducing stochastic depth. Our NOs show superior performance as compared with standard NOs, not only for testing within the training distribution but also for out-of-distribution scenarios. To delve into this observation, we offer an in-depth analysis of the Rademacher complexity associated with our modified models and prove an upper bound tied to their stochastic depth that existing NOs do not satisfy. Furthermore, we obtain a novel out-of-distribution risk bound tailored to Gaussian measures on Banach spaces, again relating stochastic depth with the bound. We conclude by proposing a hypernetwork version of the subfamily of NOs as a surrogate model for the mentioned forward operator.

1. Introduction

The introduction motivates stronger generalization guarantees for neural operators, especially under input-distribution shifts, and presents a transformer-inspired architecture with stochastic depth. It previews empirical results on Helmholtz problems and theoretical analyses of complexity and out-of-distribution risk.

1. Introduction

  • Neural operators are motivated by costly PDE solvers, but deep FNOs can perform poorly on some nonlinear PDE operators despite theoretical universality.The introduction also notes that out-of-sample generalization from finitely many noisy inputs is a central focus.Source: §1 · pp. 1–2
  • The paper extends neural-operator generalization theory beyond finite-dimensional parameters and Fourier kernels to alternatives such as wavelets and spherical harmonics..Source: §1 · p. 2

Our Contributions

  • The authors derive an out-of-distribution risk bound for centered Gaussian inputs on Banach spaces, with stochastic-depth random variables controlling the bound as network depth increases.The bounds rely on non-tight Lipschitz-constant estimates, so further analysis is needed to explain observed out-of-distribution behavior fully.Source: §1 · p. 2, Our Contributions · p. 3
  • The proposed sNO + εI keeps the non-local integral-operator layer and local MLP layer in a sequential, transformer-like layout, with Bernoulli switches controlling information propagation.The switches have depth-decreasing probabilities and add randomness during training.Source: §1 · p. 2, Our Contributions · p. 3
  • The proposed architecture generalizes to out-of-distribution inputs where earlier networks do not, including anisotropic covariance, changed input range, and roughness despite smooth isotropic training fields.The benchmark also reports smaller in-distribution test loss after modifying FNOs toward sFNO + εI.Source: Our Contributions · p. 3
  • The paper proposes a hypernetwork version of the architecture as a surrogate for the Helmholtz forward operator, mapping source and wave speed to wavefield values at receiver locations.The authors suggest applications to Bayesian inversion and algorithms needing repeated forward-operator evaluations.Source: Our Contributions · p. 3
  • Stochastic depth controls the expected Rademacher-complexity bound regardless of layer count when pℓ = O(ℓ^-(1+ε)); the sNO + εI bound stays controlled with depth while other neural-operator bounds diverge.The analysis applies independently of discretization and integral-operator basis.Source: Our Contributions · p. 3

2. Proposed networks: “Metaforming the neural operator”

The proposed operators rearrange neural-operator layers sequentially and add transformer-inspired normalization, skip connections, and stochastic depth. The authors report improved prediction across settings and a relative L2-norm improvement over traditional FNOs for convolutional kernels.

Sequential neural operators (sNO)

  • sNO places the integral operator before a local MLP, following a sequential non-local then local arrangement inspired by transformer layers.The integral operator acts as the token mixer, while the MLP mixes features.Source: §2 · p. 4, Figure 1 · p. 4
  • For convolutional kernels, sNO significantly improves relative L2-norm over traditional FNOs at similar parameter count.The comparison is reported in Section 4.2 and Figure 7.Source: §2 · p. 4

sNO + εI

  • The sNO + εI variants add an identity skip connection, with stochastic depth included in one version to access deeper models.The paper labels variants without and with stochastic depth as versions 1 and 2, respectively.Source: §2 · p. 4
  • The proposed sFNO + εI improves loss and wavefield prediction over NOs and sNOs across the reported settings.The comparison covers Tables 5 to 10.Source: §2 · p. 5

sNO + εI with stochastic depth

  • Stochastic depth randomly drops layers using Bernoulli variables, and the authors conjecture this may enable greater optimization exploration.In the experiments, survival probabilities decay linearly from 1 at the first layer to 0.7 at the last.Source: §2 · p. 5

3. Parametric time-harmonic wave equations, forward operator and data generation

The section sets out the Helmholtz wave setting, defines the wave-speed-to-solution and receiver-based forward maps, and describes how wave speeds are generated. The experiments use independently sampled Gaussian random fields with controllable smoothness and correlation properties.

3.1. Time-harmonic wave equations

  • The paper studies two-dimensional time-harmonic acoustic waves, with a free-surface Dirichlet condition and absorbing conditions on the rest of the domain boundary.The configuration follows a geophysical setting.Source: §3.1 · pp. 5–6, §3 · p. 5

3.2. Wave speed to solution map, G : c 7→p

  • The map G sends a wave-speed model c to the Helmholtz pressure field p for fixed source and frequency.The experiment uses a fixed point source at 15 Hz and computes solutions with the HDG method and hawen.Source: §3.2 · pp. 6–7
  • The forward operator maps wave speed and source to pressure values at discrete receiver locations.The receiver set Σ defines the restriction of the pressure field to recorded locations.Source: §3.2 · p. 7

3.4. Wave speeds as Gaussian random fields (GRF). Whittle–Matérn field

  • Wave speeds are generated by transforming independent Whittle–Matérn Gaussian random field realizations, whose smoothness and correlation length can be controlled.The chosen field construction provides nonnegative Sobolev regularity and supports tests that vary input-distribution properties.Source: §3.4 · pp. 7–8

4. Training and testing in-distribution for G

The section evaluates neural operators on the 15 Hz Helmholtz coefficient-to-solution map, reporting wavefield reconstructions, training-loss landscapes, and activation ablations. Its experiments compare architectures under controlled parameter and training settings.

4. Training and testing in-distribution for G

  • The 15 Hz in-distribution experiment uses a test set as large as the training set to estimate generalization more reliably.The section notes that experiments at other frequencies and domain configurations appear in Appendix D.4.Source: §4 · p. 8

4.1. Neural network ”prediction” of the wavefield

  • Each architecture yields a superior wavefield reconstruction; Figure 6 compares reconstructed fields and relative errors against reference solutions.The displayed domain is 1.27×1.27 km2, and the figure shows the real part of the wavefield.Source: §4.1 · p. 9, Figure 6 · p. 9

4.2. Hyperparameters of the neural networks

  • The stochastic-depth version has 22 layers with 12 Fourier modes per layer, while the other networks have essentially the same parameter counts.Stochastic depth uses linearly interpolated survival probabilities from 1 for the first layer to 0.7 for the last.Source: §4.2 · pp. 9–10

4.3. Training of the experiment

  • Training uses 25,000 of 50,000 generated samples, with 5,000 for validation and 20,000 for testing; the documented runs use 100 epochs.Although 300 epochs gave the best results for sFNO + εI with stochastic depth, the reported comparison keeps the same parameters across networks.Source: §4.3 · p. 10

4.4. Multiple random initialization

  • Across six random initializations, the reported test-loss trend is consistent, with relative L2 test losses tabulated for each realization.Figure 7 visualizes the test-loss distributions.Source: §4.4 · p. 10, Table 2 · p. 11, Figure 7 · p. 11

4.5. Visualization of the loss landscape

  • The loss landscapes differ: sFNO and sFNO + εI share deeper, well-delineated basins, while (sFNO + εI)v2 has the deepest central anisotropic basin.FNO has a shallow irregular crease; the stochastic-depth v2 landscape has a simpler topology than that crease.Source: §4.5 · p. 11, Figure 8 · p. 12

4.6. Ablation study

  • With identity Fourier activation, sNO has lower relative L2 loss than FNO, while stochastic-depth (sFNO + εI)v2 shows no noticeable gain over its v1 counterpart.The no-activation comparison is reported for the 15 Hz experiment.Source: §4.6 · pp. 12–13, Figure 9 · p. 12, Table 3 · p. 13
  • With nonlinear activation, (sFNO + εI)v2 consistently outperforms other architectures without overfitting; residual FNO also significantly improves over FNO.For ReLU and Leaky-ReLU, the benefit of the skip connection is difficult to distinguish from sNO.Source: §4.6 · pp. 12–13, Figure 10 · p. 13, Figure 11 · p. 13

5. Testing out-of-distribution analysis (OOD) for G

The experiments test trained neural operators on wave-speed distributions shifted from training in correlation, affine range, and smoothness. Stochastic-depth models show resilience to these shifts, while the paper also proposes a hypernetwork surrogate for the Helmholtz forward operator.

5.1. OOD experiments with different correlation and affine transformation

  • Across the OOD families, sNO + εI with stochastic depth is notably robust; in Table 10, sFNO + εIv2 has test losses half those of any other neural operator.The families vary correlation and affine range; the anisotropic cases use unequal correlation parameters.Source: §5.1 · pp. 14–15, Table 10 · p. 14
  • In family 6, only sFNO + εIv2 accurately predicts admissible wave propagation among the compared models.The FNO prediction visibly deviates from the ground truth in the highlighted region.Source: §5.1 · p. 15, Figure 12 · p. 15

5.2. OOD experiments changing the smoothness of the field

  • The smoothness experiments vary ν to test wave speeds from different Sobolev classes, first changing smoothness alone and then combining shifts in other parameters.The final two families vary all parameters, and the rough and smooth anisotropic fields are challenging for the neural operators.Source: §5.2 · pp. 15–16
  • The paper proposes a hypernetwork surrogate for the Helmholtz forward operator, with point-source inputs and fixed-size pressure measurements at receiver locations.Equation (19) offers a potential relaxation of the point-source assumption for more general sources.Source: §6 · pp. 16–17
  • The forward-operator setup maps wave speed and source information to pressure values at receiver locations, with the output discretization fixed by those receivers.The paper identifies distinct discretizations for wave speed and point sources as an architectural challenge.Source: §6 · pp. 17, 19
  • The forward-operator experiment reports a relative L2-error of 3 × 10^-2 for its hypernetwork implementation.The metanetwork is a two-layer residual network with leaky ReLU activation; the authors note they did not use a more robust feature extractor.Source: §6 · p. 19

7. Out-of-distribution under Gaussian sampling

This section develops out-of-distribution risk bounds for neural operators under centered Gaussian sampling on Banach spaces, using Wasserstein coupling, Gaussian concentration, and Lipschitz estimates related to Cameron–Martin geometry. It presents an out-of-sample generalization bound and compares the bounds for the two stated model classes.

7.1. Preliminaries

  • The analysis uses centered Gaussian measures on Banach spaces and their Cameron–Martin spaces, whose compact unit balls connect small-ball probabilities with metric entropy.The section also notes that Whittle–Matérn fields belong almost surely to appropriate Sobolev spaces and induce centered Gaussian measures there.Source: §7 · p. 20, §7.1 · pp. 20–21

7.2. Out-of-distributional generalization

  • Wasserstein coupling can compare data-generating and out-of-distribution laws even when they are mutually singular, unlike classical divergences.The risk comparison is controlled by the Wasserstein distance between measures and the Lipschitz constants of the hypotheses and loss.Source: §7.2 · p. 21, §7.3 · p. 23
  • Theorem 7.3 bounds out-of-distribution risk for the NO and sNO+εI_v2 hypothesis classes with high probability, under Gaussian small-ball regularity and a Wasserstein coupling condition.The bound depends on the coupling distance, the Gaussian small-ball function, and architecture-specific Lipschitz bounds.Source: §7.2 · p. 21
  • For standard NOs, the Lipschitz upper bound can diverge with depth when (C_w + C_k)C_σ > 1, whereas the sNO+εI_v2 bound remains bounded with Bernoulli stochastic depth and Z_ℓ = 1.The theorem also gives rates for Gaussian sampling measures when the Cameron–Martin unit-ball metric entropy has specified asymptotic growth.Source: §7.2 · p. 22

7.3. Derivation of Theorem 7.3

  • The proof extends a transport-based change-of-measure approach to the infinite-dimensional setting, combining Wasserstein risk comparisons with concentration of the empirical Gaussian sampling measure.The concentration result yields a high-probability risk bound involving the coupling distance and the Gaussian small-ball function.Source: §7.3 · pp. 23, 25
  • Theorem 7.3’s out-of-sample generalization bound follows by applying Lemma 7.6 to the standard NO and sNO+εI hypothesis classes.Under Assumptions 8.4 and 8.6, the derivation uses Lipschitz estimates for these classes.Source: §7.3 · p. 25
  • The risk bounds for (sNO+εI)v2 are much tighter than those for sNO+εI because the former has a tighter constant.Source: §7.3 · p. 25
  • The authors leave comparable lower bounds to future work, noting that tight constants can take years to establish, as illustrated by and.Source: §7.3 · p. 25

8. Analysis of the generalization error bound of the networks

The section compares generalization-error bounds for standard neural operators and stochastic-depth variants using Rademacher complexity. Under stated conditions on the stochastic-depth variables, the proposed (sNO + εI)v2 bound remains controlled as depth grows and is smaller than the standard architecture’s bound.

8. Analysis of the generalization error bound of the networks

  • The analysis focuses on network complexity and generalization bounds because standard and proposed neural operators share universality.The section states that universality does not distinguish the architectures; its focus is their complexity and corresponding generalization-error bounds.Source: §8 · p. 25

8.1. Preliminaries

  • The generalization error is split into empirical approximation and complexity errors, with the latter bounded using Rademacher complexity.The framework assumes a Lipschitz loss and bounded network outputs.Source: §8.1 · p. 26

8.2. Related Works of generalization error bound (GEB)

  • Unlike earlier operator bounds relying on finite-dimensional trainable parameters, this analysis treats broader kernel functions and reports sharper Rademacher-complexity bounds than Kim and Kang [62].The passage contrasts an order O(1/n^(1/(d̂+1))) bound with O(1) in Kim and Kang [62].Source: §8.2 · p. 26

8.3. Rademacher complexity of neural operators

  • For standard neural operators, the Rademacher-complexity bound is derived under bounded-weight, bounded-kernel, and Lipschitz-kernel assumptions.The proof estimates covering numbers for weight matrices and Lipschitz-continuous kernel functions, then applies Dudley’s theorem.Source: §8.3 · p. 27

8.4. Rademacher of sNO and intermediate architectures

  • For the proposed network family, the Rademacher-complexity bound incorporates stochastic-depth variables and the local MLP layers.The family is defined using Bernoulli switches and identity terms in its layer composition.Source: §8.4 · pp. 27–28

8.5. GEB and comparison among architectures

  • Under the stated stochastic-depth conditions, (sNO + εI)v2 has a smaller generalization-error bound than the standard architecture, and its bound need not blow up with depth.The result assumes summable decay of the Bernoulli activation probabilities; the bound remains controlled as depth increases regardless of Cw, Ck, and Cσ.Source: §8.5 · p. 29

9. Summary and discussion

The discussion reports improved high-frequency Helmholtz performance for the sNO+εI family and relates stochastic depth to stability and complexity control. It also identifies non-tight bounds as a limitation and presents forward-operator surrogacy as a prospective application to inverse problems.

9. Summary and discussion

  • The sNO+εI family improves high-frequency Helmholtz performance without increasing parameter count in the no-stochastic-depth case or compromising traditional neural operators’ approximation capabilities.The analysis also keeps the training dataset size unchanged and tests on datasets comparable in size to the training data.Source: §9 · p. 29
  • The (sFNO + εI)v2 variant is resilient to different wave-speed realizations, while the analysis links stochastic-depth variables to out-of-distribution bounds and hypothesis-class complexity control.The bounds are for Gaussian measures in abstract Banach spaces; Rademacher-complexity estimates address in-distribution generalization.Source: §9 · p. 29
  • The derived bounds are not tight, and the authors identify deriving lower bounds as challenging.They express optimism about further progress on theoretical guarantees beyond approximation properties.Source: §9 · p. 29
  • The experiments suggest that a surrogate can capture the Helmholtz forward operator, with Bayesian inversion and derivative-free ensemble Kalman methods proposed as possible applications.The cited algorithms are described as requiring multiple evaluations of the costly forward operator.Source: §9 · p. 29

A.4. Neural Operator.

The appendix defines neural operators through lifting, integral-operator layers, and projection, then reviews the measure-theoretic and statistical-learning tools used in the generalization analysis. Its proof sketches estimate covering numbers to bound Rademacher complexity.

A.4. Neural Operator.

  • A neural operator maps an input function through a lifting map, composed nonlinear integral-operator layers, and a projection map.Each layer applies an activation to the sum of a linear map and an integral operator.Source: §A.4 · p. 38

A.4.1. Fourier Neural Operators (FNOs).

  • When the integral kernel is convolutional and suitable function-space conditions hold, FFT implementation yields the Fourier neural operator described by Li et al. [82].The appendix also states that adding local operations does not change the universality property of standard and sequential neural operators [66, Theorem 11].Source: §A.4.1 · p. 39, §A.4.2 · p. 39

A.5. Bochner integral

  • The appendix introduces Bochner integration as the Banach-space integral used to define expected error, and reviews Gaussian measures as a typical PDE measure.It describes Gaussian measures through their mean and covariance operator.Source: §A.5 · p. 39, §A.6 · p. 39

A.7. Cameron-Martin space

  • For Gaussian measures, the Cameron–Martin space identifies translations yielding absolutely continuous translated measures; in infinite dimensions, that space can have measure zero.The appendix also characterizes it as a reproducing-kernel Hilbert space.Source: §A.7 · p. 40

A.8. Gaussian random field

  • The appendix constructs a Gaussian random field from a Gaussian measure with covariance operator (−∆)^−α on the unit square, for α > 1.The resulting draws are almost surely continuous and thus pointwise defined.Source: §A.8 · p. 40

A.9. Statistical learning

  • The statistical-learning framework uses expected and empirical risk and decomposes generalization error into empirical risk and a complexity term.Rademacher complexity is estimated through covering numbers using Dudley’s theorem; the kernel analysis invokes the doubling dimension of D × D.Source: §A.9 · pp. 41–42

B.1. Proof of Lemma 8.3 Proof.

  • Under Assumption 8.2(i) and (ii), employing Wainwright [112, Theorem 4.10] or Gin´e and Nickl [41, Theorem 3.4.5] yields an inequality with probability greater than 1 − 2e^−δ.Source: §B.1 · p. 43

B.2. Proof of Theorem 8.5 Proof.

  • The standard-neural-operator proof bounds Rademacher complexity by estimating covering numbers for weights and Lipschitz kernels, then applying Dudley’s theorem.The analysis uses covering-number estimates for the weight and kernel classes.Source: §B.2 · pp. 43, 45

B.3. Proof of Theorem 8.7

The proof follows the standard neural-operator argument for the stochastic-depth family, estimating its loss-class covering number through successive layer differences and then applying a Rademacher-complexity bound.

B.3. Proof of Theorem 8.7

  • The proof uses the same general argument as the standard neural-operator bound, with the stochastic-depth architecture defining the hypothesis class.The displayed family composes identity-plus-switch terms with local MLP and kernel operations.Source: §B.3 · p. 46
  • The proof then uses Dudley’s theorem and estimates for the weight and kernel classes to obtain the complexity bound.The estimates are combined to conclude the theorem’s bound.Source: §B.3 · pp. 46, 48, 49
  • The proof estimates differences between loss functions induced by two networks in the stochastic-depth class to bound its covering number.It applies the Lipschitz-loss assumption in evaluating these differences.Source: §B.3 · pp. 46–47

B.4. Proof of Corollary 8.8 Proof.

The proof expresses G(a) as a composition of operator and kernel layers, then uses Theorem 8.5 to conclude inequality (44).

B.4. Proof of Corollary 8.8 Proof.

  • G(a) is written as a composition of the operator and kernel layers, with the resulting function measured in L2(D;R^du).The displayed expression applies the layers successively to input a.Source: §B.4 · p. 49
  • The displayed composition includes both Wℓ and Kℓ at each layer.Source: §B.4 · p. 49
  • The passage says that, after the σCa term, combining with Theorem 8.5 yields inequality (44), without describing a role for Lemma 8.3 or specifying the inequality’s bound.Source: §B.4 · p. 49

B.5. Proof of Corollary 8.9 Proof.

The proof applies Assumption 8.6 to a composition with stochastic-depth factors, then combines the estimate with Theorem 8.7 to conclude inequality (45).

B.5. Proof of Corollary 8.9 Proof.

  • The proof applies Assumption 8.6 to estimate G for inputs a in supp(µa).Source: §B.5 · p. 49
  • The displayed composition inserts stochastic-depth factors ZℓId + Xℓfℓ and ZℓId + Xℓσ ◦ Kℓ across the layers.Source: §B.5 · p. 49
  • Combining the estimate with Theorem 8.7 yields inequality (45).The supplied passages do not give the inequality's explicit bound.Source: §B.5 · pp. 49–50

C. Remark for Sections 8.3, 8.4, 8.5.

The appendix notes a finite-basis kernel implementation and boundedness properties of the generalization bounds, then documents loss-landscape and out-of-distribution visualizations across several test settings.

C. Remark for Sections 8.3, 8.4, 8.5.

D.1. Loss Landscape Visualization

  • The loss-landscape appendix provides close-up and color-based planar views near the found minimizer for comparing landscape topology.Source: §D.1 · p. 52, Figure D.20 · p. 52

D.2. Out-of-distribution

  • The out-of-distribution visualizations use networks trained at 15 Hz on isotropic Whittle–Matérn fields with λ = (1, 1) and wavespeeds [1500, 5000].The reconstructions are compared with reference solutions across multiple wave-speed realizations.Source: §D.2 · p. 52, Figure D.21 · p. 53

OOD 1

  • For OOD 1, the isotropic test setting uses λOOD1 = (0.20, 0.20) and wavespeeds [1500, 5000], where FNO still struggles to capture wave propagation.Source: OOD 1 · p. 52, Figure D.21 · p. 53

OOD 2; OOD 3

OOD 4; OOD 5; OOD 6

D.3. OOD of the velocity BP 2004.

  • The 2004-BP velocity benchmark tests trained networks on a scaled velocity model to assess behavior on inputs that differ substantially from the training distribution.The source and 15 Hz frequency follow prior experiments, while the model size is adjusted for GPU capacity.Source: §D.3 · p. 55

D.4. Experiments at 7, 12 and 15 Hz.

This appendix describes additional experiments at 7, 12, and 15 Hz, including a larger-domain configuration and visualized wave-speed and pressure-field examples.

D.4. Experiments at 7, 12 and 15 Hz.

  • The additional datasets include a 12 Hz lower-frequency case and a 7 Hz case with the source beneath the surface and a larger domain.Source: §D.4 · p. 56
  • The appendix avoids compensating for the network by increasing training epochs or dataset size.Source: §D.4 · p. 56
  • Figures D.25 and D.26 visualize 15 Hz pressure-field reconstructions for out-of-distribution settings with changed covariance parameters and wavespeed ranges.Source: Figure D.25 · p. 57, Figure D.26 · p. 58
  • For the 7 Hz experiment, the data are pressure fields solving Equation (8), represented with wave speeds on a 128×128 grid at 30 m spacing.Source: Experiments of 7 Hz (different… · p. 57

D.5. Experiments at 12 Hz.

The 12 Hz experiment uses data solving Equation (8), while the reconstruction appendix presents dataset details and architecture comparisons across several frequencies. Its figures show test-loss comparisons and pressure-field reconstructions, but the supplied captions do not report which architecture performs best.

D.5. Experiments at 12 Hz.

D.6. Wavefield reconstruction at 15Hz

  • Figure D.27 presents BP 2004 [12] using the networks trained in row 1 of Table 2.Source: Figure D.27 · p. 59
  • The Experiment 1 full-wave dataset contains 40,000 wave-speed and acoustic-wave pairs on a 128×128 grid over a 3.81×3.81 km2 domain.The grid step is 30 m, and the source is buried in the domain.Source: Figure D.28 · p. 59
  • Figures D.29 and D.31 compare test-set relative L2 loss across architectures at 7 Hz and 12 Hz, with each architecture trained nine times.Source: Figure D.29 · p. 59, Figure D.31 · p. 61
  • Figures D.30, D.32, and D.33 show pressure-field reconstructions and relative-error maps for different architectures and wave-speed realizations at 7, 12, and 15 Hz.Source: Figure D.30 · p. 60, Figure D.32 · p. 61, Figure D.33 · p. 62

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