Sensitivity-Constrained Neural Operators for Data-Efficient Forward and Inverse Modeling of Partial Differential Equation Systems

1 Introduction

• SC-NOs address neural-operator failures on high-dimensional gridded inputs, where accurate sensitivities are essential for inverse stability, robustness, and reliable out-of-distribution behavior.The paper presents empirical, computational evidence rather than a formal cure for the curse of dimensionality, targeting improved accuracy–cost tradeoffs in studied PDE inference problems.Source: §1 · pp. 2–3

2 Sensitivity-Constrained Neural Operators

• The method augments existing neural-operator families, rather than introducing a new architecture, by matching selected solver-derived input–output Jacobian entries alongside solution fields.Reference sensitivities come from differentiable solvers or discrete adjoints, and sampled entries are resampled across minibatches to amortize supervision.Source: §1 · p. 3, §2 · p. 3

2.1 Neural Operator Formulation

• The formulation represents PDE solvers as operators mapping contextual states and supervised input fields to solution functions, while remaining independent of the chosen differentiable operator architecture.The supervised input may include coefficients, forcing, initial conditions, bed topography, or source deformation fields.Source: §2.1 · pp. 3–4

2.2 Sensitivity-Constrained Training Objective

• SC-NO training combines state-prediction and Jacobian-matching losses, with learnable uncertainty-inspired weights adaptively balancing the two supervision signals.Matching sensitivities constrains the surrogate’s local response to input perturbations, which is particularly relevant when inverse procedures or distribution shifts repeatedly evaluate away from training inputs.Source: §2.2 · pp. 4–5

2.3 Scalable Jacobian Sampling and Amortized Supervision

• Because a full Jacobian contains NuNp entries, SC-NO samples output rows per update and resamples them across training, providing an unbiased full-row-loss estimator under uniform sampling.This amortizes gradient supervision over many input–output directions while avoiding full-Jacobian storage or differentiation at every step; experiments typically sample final prediction-time outputs.Source: §2.3 · pp. 5–6

2.4 Sensitivity Computation by Automatic Differentiation and Discrete Adjoints

• Model sensitivities are computed by automatic differentiation, while reference sensitivities use automatic differentiation for simpler solvers and discrete adjoints for larger or more complex solvers.The advection–diffusion benchmark uses solver differentiation; RANS–Spalart–Allmaras and tsunami experiments use discrete adjoints, avoiding finite-difference scaling with gridded input dimension.Source: §2.4 · p. 6

2.5 Computational Complexity

• With m sampled output rows, sampled supervision uses O(mNp) sensitivity entries per update instead of O(NuNp), making m/Nu the principal training-time coverage–overhead control.The stated asymptotic comparison excludes measured runtime; empirical accuracy–cost comparisons include solution data generation, Jacobian computation, and model training.Source: §2.5 · pp. 6–7, Table 1 · p. 6
• Jacobian supervision is not free: it increases model training time and memory and requires solver-derived sensitivities during data preparation.The paper therefore evaluates total wall-clock cost rather than treating the sensitivity loss as costless.Source: §2.5 · pp. 6–7

3.1 Benchmark PDEs and High-Dimensional Inputs

• Controlled benchmarks use advection–diffusion, RANS–Spalart–Allmaras, and Tohoku shallow-water tsunami systems with spatially distributed inputs and solver-derived sensitivity targets.The tsunami benchmark serves as a capstone for source inversion, forecasting, noise robustness, and spatial OOD evaluation.Source: §3.1 · pp. 7–8

3.2 Baselines and Sensitivity-Constrained Variants

• Sensitivity-constrained FNO, WNO, and DeepONet variants retain their baseline architectures and add Jacobian-matching supervision to the corresponding state-prediction objectives.SC-FNO is emphasized in the main text, while WNO and DeepONet provide architectural controls.Source: §3.2 · pp. 8–9

3.3 Task Definitions and Evaluation Settings

• Evaluation covers forward prediction, inverse reconstruction, OOD generalization, and long-horizon autoregressive rollout under controlled and tsunami benchmark settings.Inverse tasks reconstruct distributed fields or tsunami seafloor deformation from sparse early gauge observations, while OOD tsunami cases use disjoint epicenter regions.Source: §3.3 · p. 9, Long-horizon autoregressive rollout · p. 9

Long-horizon autoregressive rollout.

• The RANS rollout model is trained with teacher-forced one-step pairs and recursively applied to test whether sensitivity supervision reduces error growth from repeated self-generated inputs.This setting evaluates temporal stability over longer trajectories.Source: Long-horizon autoregressive rollout · p. 9

3.4 Computational Cost Accounting

• Total computational cost includes solution-data generation, solver-derived Jacobian preparation, and neural-operator training, with Jacobian cost set to zero only for state-only baselines.Accuracy–cost comparisons use total wall-clock cost, and matched-compute claims are made only when explicitly supported by the results.Source: §3.4 · pp. 9–10

4.1 Forward Prediction with High-Dimensional Inputs

• Forward improvements are strongest in low-data regimes, remain consistent across architectures, and narrow as additional solution samples reduce the state-only models’ performance gap.Representative predictions show better preservation of transported concentration and coherent vorticity structures, with less smoothing and dissipative drift than FNO.Source: §4.1 · p. 10, Figure 3 · p. 11, Figure 4 · p. 12

4.1 Forward Prediction with High-Dimensional Inputs; 4.2 Inverse Reconstruction of Distributed Fields

• Sensitivity-constrained models outperform state-only baselines in forward prediction and inverse reconstruction, with inverse gains larger because reconstruction directly depends on surrogate gradients.These improvements hold across the controlled PDE benchmarks and FNO, WNO, and DeepONet, while FNO benefits most among the tested architectures.Source: §4.1 · p. 10, §4.2 · p. 13

4.2 Inverse Reconstruction of Distributed Fields

• Inverse reconstruction benefits consistently across PDE1 and PDE2, where the unknown distributed fields are initial concentration C0 and initial vorticity ω0.Sensitivity supervision improves gradient quality and yields more accurate and stable optimization-based reconstructions, without making the inverse problems well posed.Source: §4.2 · p. 13, Figure 5 · p. 13

4.3 Long-Horizon Autoregressive Rollout

• For FNO, sensitivity supervision reduces final RANS rollout error from 0.276 to 0.086, an approximately 3.2× reduction, while similar reductions occur for WNO and DeepONet.The one-step model is trained on 0–12.5 s and recursively evaluated through 25 s; SC-FNO remains closer to the reference trajectory beyond the training range.Source: §4.3 · pp. 13, 15, Figure 8 · p. 15

4 Controlled PDE Benchmarks; 5 Empirical Scaling with Input Dimension and Compute

• The benchmark and scaling studies compare adding trajectories with extracting solver-derived Jacobian information from each trajectory under costs including data generation, Jacobian preparation, and model training.This evaluates whether sensitivity supervision provides a better accuracy–cost tradeoff as input degrees of freedom, sample size, compute, and Jacobian density vary.Source: §4 · p. 10, §5 · p. 15, §5.1 · p. 16

5.1 Scaling with Input Degrees of Freedom

• In reported RANS scaling experiments, SC-FNO required lower total computational cost than FNO to reach relative L2 error 0.075 across tested input resolutions.This is a finite-range empirical scaling observation, not a universal complexity law; total cost includes data generation, Jacobian preparation where applicable, and model training.Source: §5.1 · p. 16, Figure 9 · p. 16

5.2 Jacobian Sampling and State-Resolution Ablation

• Jacobian supervision remains beneficial as state-resolution density varies, showing that state values and sensitivities provide complementary information rather than interchangeable supervision.State data constrain solution values, whereas Jacobians constrain local responses to distributed-input perturbations.Source: §5.2 · p. 17, Figure 10 · p. 17
• Sparse or intermediate Jacobian sampling sharply reduces error relative to no supervision, while denser coverage yields additional but saturating improvement, so full Jacobian enforcement is unnecessary.The ablation varies final-time Jacobian coverage from 0 × 0 to 64 × 64 at fixed state-data settings, identifying partial sampling as a practical accuracy–overhead compromise.Source: §5.2 · p. 17, Figure 10 · p. 17

6 Near-Real-Time Inverse–Forward Inference for Tsunami Forecasting

• The benchmark coarsens finite-volume shallow-water simulations and adjoint sensitivities from a 308,001-cell, 154,598-node domain to a fixed 100 × 100 grid for neural-operator training and inference.The source is earthquake-induced seafloor deformation generated with the Okada dislocation model, and simulations span 3600 seconds stored in 51 snapshots.Source: §6 · p. 18

6.1 Forward Operator Evaluation Under Known Source Inputs

• Sensitivity supervision lowers forward error at comparable computational cost for in-distribution and spatial-OOD tsunami events, improving FNO, WNO, and DeepONet across tested backbones.At 1,000 training samples, sensitivity-constrained variants reduce error for all three architectures; SC-FNO also reduces missed inundation relative to FNO, including under the tested epicenter shift, while OOD performance is not unrestricted extrapolation.Source: §6.1 · p. 18, Figure 11 · p. 19

6.2 Gradient-Based Source Inference from Sparse Observations

• SC-FNO recovers the location, polarity, and large-scale geometry of the Okada deformation more accurately than FNO from sparse early gauge histories.The three-stage procedure first performs flexible field-space inversion, then maps the estimate to a nine-dimensional Okada parameter vector, and finally refines those parameters through a differentiable decoder and neural-operator forecast.Source: §6.2 · p. 21

6.3 Forecast Reconstruction from Inferred Source Fields

• SC-FNO more accurately reconstructs tsunami gauge histories after the 30-minute inversion window than FNO across reported gauges and wave amplitudes, demonstrating improved coupled inverse–forward forecasting.The workflow requires approximately 4.5 minutes on a single NVIDIA A100 GPU for three-stage inversion and forward reconstruction, excluding simulation generation, adjoint computation, and training; it is a controlled proof of concept rather than a complete operational warning system.Source: §6.3 · pp. 22–23, Figure 14 · p. 23

6.4 Robustness of Source Inversion to Noisy Observations

• SC-FNO maintains lower source-reconstruction error and higher R2 than FNO across tested synthetic observation-noise levels, including low-to-moderate noise.Its reconstructed tsunami events likewise have lower error and higher R2, although performance degrades as noise increases; the result indicates robustness to tested noise perturbations rather than full uncertainty quantification.Source: §6.4 · pp. 23–24, Figure 15 · p. 24

7.1 What Sensitivity Supervision Changes

• Sampled Jacobian supervision complements state supervision by constraining how solutions change under distributed-input perturbations, with benefits strongest in lower-data regimes and under repeated surrogate use.The evidence supports an improved observed data–compute tradeoff, not a universal scaling law or formal resolution of the curse of dimensionality.Source: §7.1 · pp. 24–25

7.2 Why Sensitivity Supervision Matters More for Inversion

• Sensitivity supervision matters more for inversion because optimization depends on accurate surrogate derivatives with respect to unknown fields, not only on predicted state values.In PDE1, PDE2, and the tsunami case, sensitivity-constrained models more accurately reconstruct distributed sources, although the evidence does not establish that these inverse problems become well-posed.Source: §7.2 · p. 25

7.3 Scope of Sensitivity-Constrained Operator Learning

• SC-NO imposes sampled operator-level Jacobians for distributed fields such as C0(x), ω0(x), f(x), and ∆zb(x), resampling subsets of the full sensitivity tensor across training.This distinguishes the approach from derivative constraints on only a small set of scalar parameters and positions it as practical response supervision for high-dimensional gridded inputs.Source: §7.3 · pp. 25–26

7.4 Limitations and Practical Requirements

• The method requires solver-derived sensitivities, obtained here through automatic differentiation for advection–diffusion and discrete adjoints for RANS and shallow water, creating implementation and computational costs.The net benefit depends on solver and adjoint cost, output dimension, sampling density, and downstream task, so results remain empirical over the tested PDE classes, resolutions, and budgets.Source: §7.4 · p. 26

7.5 Implications for Time-Critical PDE Inference

• The study’s broader implication is that derivative-aligned surrogates can support source inversion, data assimilation, uncertainty exploration, design optimization, and long-horizon prediction when high-fidelity solvers are too expensive for repeated online use.Potential applications include flood inundation, storm surge, plume transport, subsurface flow, and volcanic hazard modeling, provided differentiable or adjoint information can be generated offline.Source: §7.5 · pp. 26–27

8 Conclusion

• Across controlled benchmarks, sensitivity supervision improves forward prediction and yields larger gains in gradient-based inverse reconstruction, while improving the observed accuracy–cost tradeoff for high-dimensional gridded inputs.The method is a training strategy rather than a new operator architecture, and the rollout and tsunami workflows indicate particular value for repeated closed-loop or inverse use.Source: §7.1 · p. 24, §8 · p. 27

7.5 Implications for Time-Critical PDE Inference

• The tsunami workflow demonstrates a minute-scale near-real-time proof of concept for reconstructing seafloor deformation from sparse early gauges and forecasting subsequent waves after offline training.It is a pathway toward time-critical inference rather than a complete operational warning system, which would require broader uncertainty treatment, sensor robustness, and additional validation.Source: §7.4 · p. 26, §8 · p. 27, §7.5 · p. 27

Appendices; A. Theoretical justification for sensitivity-constrained training; B. Data generation; C. PDE1: Advection–diffusion equation with spatially distributed velocity field

• The supplementary materials provide theoretical, data-generation, and neural-operator-learning details, including assumptions, cost functionals, stability and inversion analyses, true sensitivities, random-field parameter generation, and the PDE1 setup.These materials cover the appendices, theoretical justification, data generation, and C. PDE1: Advection–diffusion equation with spatially distributed velocity field.Source: Appendices · p. 28, A. Theoretical justification for… · p. 28, B. Data generation · p. 28

D. PDE2: Turbulent Navier–Stokes equations with Spalart–Allmaras closure

• For PDE2, the study defines a neural-operator learning setup for turbulent Navier–Stokes equations with Spalart–Allmaras closure.Source: D. PDE2: Turbulent Navier–Stokes… · p. 28
• For PDE2, adjoint-based sensitivities are included in the turbulent Navier–Stokes modeling workflow.Source: D. PDE2: Turbulent Navier–Stokes… · p. 28

E. PDE3: 2D shallow water equations for the 2011 Tohoku tsunami

• For PDE3, the tsunami source is represented using an Okada deformation model within the 2D shallow-water formulation.Source: E. PDE3: 2D shallow water equations… · p. 28
• The PDE3 workflow computes adjoint-based sensitivities for the 2011 Tohoku tsunami shallow-water problem.Source: E. PDE3: 2D shallow water equations… · p. 28
• The cited passage identifies neural-operator learning.Source: E. PDE3: 2D shallow water equations… · p. 28
• The tsunami implementation includes fine-to-coarse grid mapping between computational representations.Source: E. PDE3: 2D shallow water equations… · p. 28
• The tsunami workflow performs source inversion in three stages.Source: E. PDE3: 2D shallow water equations… · p. 28

A Supporting Analysis: Mechanisms of Sensitivity-Constrained Training · Neural Operators

• Neural operators learn resolution-invariant mappings between infinite-dimensional input-function and solution spaces, using nonlocal and local latent updates.The framework represents PDE inputs such as initial conditions, boundary conditions, and parameters as functions and maps them to solution functions.Source: A Supporting Analysis: Mechanisms of… · p. 29, Neural Operators · p. 29
• The operator formulation parameterizes Gθ: U → V with learnable weights and separates spatial, temporal, and parameter dependence in its latent representation.This formulation underlies the sensitivity analysis by treating the model as a differentiable operator with respect to system parameters.Source: A Supporting Analysis: Mechanisms of… · p. 29, Neural Operators · p. 29

Neural Operators

• Different neural-operator parameterizations trade off representation mechanisms, including Fourier spectral transforms, wavelet decompositions, and DeepONet trunk-branch factorization.The passages identify these as distinct architecture strategies for approximating operator mappings.Source: Neural Operators · pp. 29–30

Sensitivity-Constrained Neural Operators (SC-NO) · Cost Functionals

• SC-NO training augments solution loss with solver-derived sensitivity loss, with λ controlling the strength of sensitivity enforcement.Sensitivities can be computed on-the-fly using a differentiable solver or adjoint model.Source: Sensitivity-Constrained Neural… · p. 30, Cost Functionals · p. 31

Theoretical Justification · Assumptions and Notation

• Under bounded-domain and uniform-convergence assumptions, the theoretical argument links small empirical losses to uniform control of both solution and sensitivity errors.The analysis assumes bounded true solutions and sensitivities and an overparameterized, well-trained model.Source: Assumptions and Notation · pp. 30–31

Improved Generalization

• Sensitivity regularization constrains the learned parameter-to-solution Lipschitz constant, yielding a lower-complexity hypothesis class and tighter generalization bounds on unseen parameters.Driving εs toward zero makes the learned Lipschitz constant approach the bounded physical sensitivity Mp.Source: Improved Generalization · p. 31

Long-Term Prediction Stability

• For long-term prediction, matching sensitivities with respect to initial conditions reproduces physical perturbation growth rates up to O(εs) and avoids additional spurious instabilities.The analysis contrasts SC-NO with state-only training that may learn an incorrect sensitivity operator with an excessively large growth exponent.Source: Long-Term Prediction Stability · p. 32

Inversion Accuracy

• In parameter inversion, the surrogate gradient error is bounded by solution and sensitivity errors, so reducing both drives optimization toward an O(εu + εs) neighborhood of the true optimum.With local μ-strong convexity, the analysis predicts more accurate and stable gradient-based convergence than solution-only training.Source: Inversion Accuracy · p. 32

B Data Generation

• Each training sample contains an input function, its high-fidelity PDE solution, and the true Jacobian with respect to the spatially varying parameter field.The input domain is X = R2 with discrete time steps over [0, T].Source: B Data Generation · p. 33

Structure of the Input Function

• The input combines tu initial solution time steps with the spatially varying parameter field p(x), and the operator predicts the additional tG time steps.Thus, a(x, t, p) = (ainit(x, t), p(x)).Source: Structure of the Input Function · p. 33, Dataset Structure with True… · p. 33

Dataset Structure with True Sensitivities

• The final-time Jacobian forms a matrix mapping parameter changes at every spatial location to solution responses, capturing local and nonlocal PDE-induced dependencies.Evaluating sensitivity at t = T represents temporal propagation from t = 0 to T while focusing the sensitivity loss on one time step.Source: Dataset Structure with True… · p. 34

Scaled Gaussian Random Field for Parameter Generation

• Parameters p(x) are generated as bounded, scaled Gaussian random fields whose Laplacian-based construction and tanh normalization provide spatial smoothness, local correlation, and predefined-range outputs.This yields structured stochastic parameter fields for neural-operator training.Source: Scaled Gaussian Random Field for… · p. 34

Problem setup

• The advection–diffusion benchmark models concentration transport with spatially varying Gaussian-random velocity fields, constant diffusion D = 0.005, and periodic boundaries.The domain is X = [0, 1.0] × [0, 1.0] over t ∈ [0, 1.0].Source: Problem setup · p. 35

Numerical Solver for the Advection-Diffusion Equation

• The differentiable advection–diffusion solver uses finite differences, central diffusion, upwind advection, periodic ghost cells, RK4 integration, and gradient tracking for ∂U/∂p.It uses a 150 × 150 grid downsampled to 50 × 50, step size 10^-4, and 100 recorded time steps.Source: Numerical Solver for the… · p. 35

Neural Operator Learning for the Advection-Diffusion Equation

• For PDE1, SC-NO maps an initial solution context and spatial velocity field to subsequent time steps while matching the solution Jacobian ∂u/∂p.The operator learns a solution mapping spanning [tu, T] across PDE instances.Source: Neural Operator Learning for the… · p. 35

Problem setup

• The turbulent Navier–Stokes benchmark couples vorticity–stream-function dynamics with the Spalart–Allmaras model, parametric body forcing, wall conditions, and Reynolds numbers 500–1000.The domain is the unit square over T = 25, with molecular viscosity ν ∈ [0.001, 0.002].Source: Problem setup · p. 37

Numerical Solver for the Turbulent Navier–Stokes Equations

• Its differentiable solver uses second-order finite differences, a fast Poisson solve, RK4 integration, and adjoint-based gradients for initial vorticity and forcing.It stores 250 snapshots at Δt = 0.1 over [0, 25.0].Source: Numerical Solver for the Turbulent… · p. 38

Neural Operator Learning for the Turbulent Navier–Stokes Equations

• The turbulent-flow neural-operator experiments include full-horizon sequential forecasting, multiresolution initial-vorticity tests, and one-step autoregressive rollouts evaluated over 250 steps.Rollout training uses teacher forcing, controlled input noise, and short-range transitions before testing long-horizon stability and error accumulation.Source: Neural Operator Learning for the… · p. 38

Adjoint-Based Sensitivity Computation for the Turbulent Navier–Stokes Equations

• Continuous adjoints compute exact gradients of final vorticity with respect to initial vorticity and forcing using one primal forward solve and one backward solve.The adjoint equations are integrated backward from terminal conditions derived from the objective functional.Source: Adjoint-Based Sensitivity Computation… · p. 39, Adjoint Terminal and Boundary Conditions · p. 40, Sensitivity with Respect to Forcing… · p. 40

Overview

• The Tohoku tsunami workflow uses SWE neural operators to infer Okada-parameterized seafloor deformation from noisy first-30-minute buoy observations and forecast 50 subsequent time steps.FNO and SC-FNO are pretrained on high-resolution, observationally validated finite-volume simulations, with final-stage bed-topography sensitivities computed adjointly.Source: Overview · p. 42, Modeling Seafloor Deformation Using… · p. 43

Adjoint System Formulation

• The adjoint formulation incorporates forward-dynamics constraints and differentiates shallow-water bed dependence, where explicit deformation effects enter the momentum equations through bed-slope terms.The resulting local Jacobian has continuity rows without explicit bed dependence and localized x- and y-momentum sensitivities.Source: Adjoint System Formulation · p. 45, Backward Recursive Computation of… · pp. 47–48

Backward Recursive Computation of Adjoint Variables

• The adjoint system is initialized at the final time and recursively propagated backward, producing final water-depth sensitivities to bed topography for tsunami inversion and forecasting.Local finite-volume Jacobians capture flux, source, neighboring-cell, and localized bed-slope effects consistently with the adjoint framework.Source: Backward Recursive Computation of… · pp. 46–47

Neural Operator Learning for the Shallow Water Equations

• The tsunami neural operator maps five initial water-stage steps and seafloor deformation to 46 future steps, using adjoint final-stage bed-topography sensitivities for SC-NO training.Training data come from dTSUNAMI simulations on a 308,001-cell mesh with 51 snapshots, subsequently coarsened to a 100×100 grid; evaluation includes in-training and OOD earthquake epicenters.Source: Neural Operator Learning for the… · p. 48

Mapping Between Fine Solver Mesh and Coarse Neural Operator Grid

• Bathymetry-aware hydrostatic upscaling reconstructs nonnegative fine-scale depths while enforcing coarse mean-depth and volume constraints, outperforming interpolation near shorelines.Interpolation performs comparably only in moderate-to-deep regions; the proposed reconstruction avoids spurious shallow-water inundation and supplies solver-level fields without modifying neural-operator predictions.Source: Mapping Between Fine Solver Mesh and… · pp. 49–51

Three-Stage Inversion Framework for Tsunami Source Reconstruction

• The three-stage inversion first optimizes deformation fields from early gauge observations, encodes them into nine Okada fault parameters, and refines those parameters through differentiable decoding.Stage 3 backpropagates gauge loss through the decoder and neural operator, constraining refinement to a physically meaningful parameterized subspace.Source: Three-Stage Inversion Framework for… · pp. 51, 53, Figure E.5 · p. 52
• SC-FNO converges toward physically consistent tsunami source reconstructions, whereas baseline FNO fails meaningful parameter refinement and leaves noisier deformation fields.In field-space inversion, SC-FNO converges faster with smoother gradients and better rupture localization; in parameter-space refinement, it continues reducing gauge-signal mismatch toward the true Okada deformation.Source: Three-Stage Inversion Framework for… · pp. 51, 53, Figure E.6 · p. 53, Figure E.7 · p. 54
• SC-FNO forecasts complete tsunami evolution with closer observed amplitude and phase agreement, maintaining stability and reducing residual errors relative to FNO.Both models reproduce overall waveform trends after recovered-source propagation, but SC-FNO more closely matches observed gauge signals throughout the simulation.Source: Three-Stage Inversion Framework for… · p. 53

Additional results for PDE3

• Additional PDE3 results compare error metrics across models trained with different sample sizes in-training.Source: Table E.1 · p. 55
• Additional PDE3 results also compare error metrics across models trained with different sample sizes out-of-training.Source: Table E.2 · p. 55

• Relative L2 error and mean absolute error (MAE) are the primary continuous field-error metrics over evaluated spatial–temporal solution fields.MAE is normalized by the number of evaluated spatial–temporal degrees of freedom, yielding an average rather than total absolute error.Source: F Performance Metrics · p. 56
• R2 is additionally reported for inverse-source and tsunami reconstruction experiments, covering reconstructed inputs, predicted solution trajectories, or tsunami sources depending on the experiment.Source: F Performance Metrics · p. 56
• Tsunami-impact evaluation classifies spatial cells as dry below 0.01 m water depth and wet at or above 0.01 m, then reports false-negative ratio.The false-negative ratio distinguishes reference-wet cells predicted as dry from those predicted as wet.Source: F Performance Metrics · p. 56
• Inundation accuracy additionally counts true-positive, true-negative, false-positive, and false-negative wet–dry classifications.Source: F Performance Metrics · p. 56

• Models were trained with Adam, batch size 16, initial learning rate 0.001, and a scheduler decaying by γ = 0.95 every 100 epochs.Source: G Hyperparameters and Settings · p. 57
• The spatiotemporal domain used an nx × ny spatial grid, with nt = tu + tG combining input and predicted time steps.Source: G Hyperparameters and Settings · p. 57
• Model-specific hyperparameters were summarized in separate tables for FNO, WNO, and DeepONet across the PDEs.Source: Table G.1 · p. 57, Table G.2 · p. 57, Table G.3 · p. 57

Cost measurement

• Computational cost is measured using per-epoch training time, 500-epoch total training time, and preprocessing overhead including sample generation and Jacobian computation where applicable.Peak GPU memory was not a primary comparison metric.Source: H Computational Cost Analysis · p. 58

PDE1

• Table H.1 reports the total computational cost breakdown over 500 epochs for PDE1.The supplied passage identifies the table scope but does not provide its numerical values.Source: Table H.1 · p. 58

PDE2–PDE3

• Tables H.2 and H.3 report total computational cost breakdowns over 500 epochs for PDE2 and PDE3, respectively.The supplied passages identify the table scopes but do not provide numerical values.Source: Table H.2 · p. 58, Table H.3 · p. 59

• Abdolmehdi Behroozi handled conceptualization, methodology, software, formal analysis, investigation, data curation, visualization, and original-draft writing.Source: CRediT authorship contribution statement · p. 60
• Chaopeng Shen contributed conceptualization, methodology, supervision, project administration, funding acquisition, and writing—review and editing.Source: CRediT authorship contribution statement · p. 60
• Daniel Kifer contributed methodology, supervision, and writing—review and editing.Source: CRediT authorship contribution statement · p. 60
• Kathryn Lawson contributed methodology, investigation, and writing—review and editing.Source: CRediT authorship contribution statement · p. 60