Can Deep Learning Achieve Cross-Physics Mapping?

I. INTRODUCTION

• Existing neural-operator frameworks predominantly map within the same governing physics, despite practical sensing and imaging often involving distinct physical domains.Different modalities can preserve, suppress, or redistribute information in different ways.Source: §I · pp. 1–2
• Cross-Physics Mapping (CPM) formulates learned operators between function spaces associated with distinct physical phenomena.The framework aims to connect modalities through physically meaningful latent representations rather than replace their underlying physics.Source: §I · p. 2
• A cross-physics operator can exist when fields admit compatible latent representations linked by a bounded latent transformation, without requiring the same governing equations or dynamical equivalence.The stated neural-operator approximation condition additionally requires continuity on a compact subset.Source: §I · pp. 2–3

II. CROSS-PHYSICS SCALING PRINCIPLE

• Matching dominant dimensionless operator scales is proposed to make source and target fields evolve on compatible regions of the operator manifold without implying dynamical equivalence.The scaling principle also guides choices of characteristic length, observation window, material parameters, and spectral resolution.Source: §II · p. 3

III. DATA GENERATION FOR CROSS-PHYSICS OPERATOR LEARNING

• Each pair shares a latent physical scene, including geometry, subsurface heterogeneity, excitation, and characteristic scales, while diffusion and wave fields are generated independently.The learning task does not assume a pointwise or closed-form transformation between the fields.Source: §III · p. 3

A. Computational domain and latent structural model

• The simulations use a two-dimensional rectangular domain with a smooth inclusion whose location, axes, orientation, and material contrast specify its geometry.The material heterogeneity influences both fields through their respective governing coefficients.Source: A. Computational domain and latent… · pp. 3–4

B. Diffusion-domain simulation

• The diffusion field is generated by a heterogeneous diffusion equation with surface flux excitation and zero initial conditions.The equation is discretized using second-order spatial finite differences.Source: B. Diffusion-domain simulation · p. 4

C. Wave-domain simulation

• The wave field is generated independently by a heterogeneous wave equation, retaining oscillatory phase information and finite-speed propagation unlike the diffusion field.Both simulations use the same accessible excitation surface; the mapping is not generally reducible to a local amplitude transformation.Source: C. Wave-domain simulation · p. 5

D. Dimensionless cross-physics matching

• Matching the Fourier number to the squared dimensionless propagation distance aligns characteristic diffusion and wave evolution scales without asserting analytical equivalence.The resulting characteristic diffusion length and wave propagation distance are comparable over the observation interval.Source: D. Dimensionless cross-physics matching · p. 5

E. Preparation for the training datasets

• The learning task uses only front-surface measurements, leaving subsurface structure to be inferred indirectly from surface dynamics.The networks do not receive the full internal fields or defect map.Source: E. Preparation for the training datasets · p. 6
• Both fields are sampled on a common 501×200 observation grid, preventing solver-grid differences from becoming a trivial interpolation task.The underlying PDE solvers retain their own stable internal time steps before observations are resampled.Source: E. Preparation for the training datasets · p. 6
• The dataset contains 1,000 paired realizations split 800/100/100 by realization, with each pair kept in the same subset.Latent scene parameters are randomized across realizations, and Table I describes variation in diffusivity, thickness, and excitation patterns.Source: E. Preparation for the training datasets · pp. 6–7, Table I · p. 7
• DeepONet encodes the input field into 512 branch coefficients and combines them with coordinate-dependent trunk basis functions to evaluate the output field.This factorization separates input-field encoding from coordinate-dependent field evaluation.Source: Figure 8 · p. 7, E. Preparation for the training datasets · p. 7

IV. METHODOLOGY

• The evaluation uses ResUNet and six neural operators: U-NO, WNO, LNO, GNO, DeepONet, and FNO, whose blocks combine truncated Fourier and pointwise convolutions.The FNO configuration uses 16 × 16 retained modes, width 32, and six operator blocks.Source: §IV · p. 7, Figure 9 · p. 8

A. Cross-Physics Mapping Based on ResUNet

• ResUNet directly learns a mapping from paired diffusion fields to corresponding wave fields at 501 × 200 resolution.It serves as a convolutional baseline for the cross-physics transformation.Source: A. Cross-Physics Mapping Based on… · p. 8
• ResUNet uses a four-level encoder–decoder with residual blocks, a 512-channel bottleneck, and skip connections to combine multiscale features and recover fine-scale information.The decoder interpolates features to the encoder dimensions, supporting skip concatenation and reconstruction at the original resolution despite the odd temporal dimension.Source: A. Cross-Physics Mapping Based on… · p. 8

B. Cross-Physics Mapping Based on U-Shaped Neural Operator (U-NO)

• U-NO replaces convolution-only propagation with spectral integral operators, pointwise mappings, and multiresolution skip connections for diffusion-to-wave mapping.Its global Fourier mixing is applied at multiple resolutions, while skip features retain high-resolution information.Source: B. Cross-Physics Mapping Based on… · pp. 9–10
• U-NO combines spectral and local branches in operator blocks, then organizes them in an encoder–bottleneck–decoder hierarchy with progressively reduced Fourier modes.The decoder interpolates features to match encoder resolutions before concatenating skip features.Source: B. Cross-Physics Mapping Based on… · p. 9

C. Cross-Physics Mapping Based on Wavelet Neural Operator (WNO)

• WNO uses wavelet-domain operator learning to represent both slowly varying diffusion components and localized, rapidly varying wave-like structures.Unlike Fourier modes, its wavelet representation is localized in scale and position.Source: C. Cross-Physics Mapping Based on… · p. 10
• A two-level Haar decomposition separates approximation coefficients from three detail subbands, with independent learnable transformations for the subbands.The LL branch represents large-scale structures, while the detail bands encode localized temporal–spatial variations.Source: C. Cross-Physics Mapping Based on… · p. 10
• Each WNO block combines wavelet transformations with pointwise channel mixing and residual convolutional refinement of neighboring temporal–spatial locations.The model reconstructs transformed features using the inverse wavelet transform; replication padding and cropping preserve the original 501 × 200 resolution.Source: C. Cross-Physics Mapping Based on… · pp. 10–11
• The complete WNO stacks four fixed-width blocks, each performing two-level Haar decomposition, subband mixing, inverse reconstruction, and local refinement.Unlike U-NO, it performs multiresolution processing inside each block rather than through a separate encoder–decoder hierarchy.Source: C. Cross-Physics Mapping Based on… · p. 11

D. Cross-Physics Mapping Based on Latent Neural Operator (LNO)

• LNO encodes the diffusion field into a compact latent representation, performs operator learning there, and decodes the result into the wave field.The architecture tests whether the transformation can be modeled after substantial dimensional compression.Source: D. Cross-Physics Mapping Based on… · pp. 11–12
• The encoder maps the coordinate-augmented 501 × 200 diffusion field to a 128-channel latent field of approximately 63 × 25.Residual convolutional stages and average pooling perform the compression while retaining nonlinear local structures.Source: D. Cross-Physics Mapping Based on… · p. 11
• LNO performs global spectral mixing on learned latent features using a truncated Fourier kernel, with a nominal retained mode count of 24 × 12.The spectral operator acts on the compressed latent field rather than directly on the full-resolution input.Source: D. Cross-Physics Mapping Based on… · pp. 11–12
• Four latent operator blocks process the representation before a residual pointwise projection explicitly maps diffusion latent features toward wave latent features.A progressive decoder then reconstructs the transformed representation at the original 501 × 200 resolution without encoder–decoder skip connections.Source: D. Cross-Physics Mapping Based on… · p. 12

E. Cross-Physics Mapping Based on Galerkin Neural Operator (GNO)

• GNO tokenizes the diffusion field into 4 × 4 patches, producing 6,300 tokens with 96-dimensional embeddings after padding the temporal dimension.Normalized temporal and spatial coordinates are embedded into the tokens to retain their physical locations.Source: E. Cross-Physics Mapping Based on… · pp. 12–13
• Galerkin attention contracts key and value features across tokens before evaluating the global representation at each query, avoiding an N × N attention matrix.The contraction forms a d × d operator, relevant here because patching still yields 6,300 tokens.Source: E. Cross-Physics Mapping Based on… · p. 13
• Four Galerkin operator blocks apply repeated global updates to the token sequence, followed by residual convolutional refinement on the reshaped latent patch field.The refinement complements global token interactions by restoring local temporal–spatial consistency.Source: E. Cross-Physics Mapping Based on… · p. 13
• The refined latent field is upsampled and decoded into a wave prediction, with padded temporal samples cropped to restore the 501 × 200 resolution.The architecture models global temporal–spatial interactions on tokenized fields without explicitly constructing dense pairwise attention matrices.Source: E. Cross-Physics Mapping Based on… · pp. 12–13

F. Cross-Physics Mapping Based on DeepONet

• The operator is approximated as a sum of input-dependent coefficients times learned output-coordinate basis functions, with a latent dimension of 128.The branch network encodes the diffusion field into coefficients, while the trunk network evaluates basis functions over the temporal-spatial output domain.Source: F. Cross-Physics Mapping Based on… · p. 14
• The branch network encodes the complete diffusion field into a global coefficient vector using convolutional stages, adaptive pooling, and a projection network.Adaptive pooling produces a fixed 4 × 4 feature map before the final projection to 128 coefficients.Source: F. Cross-Physics Mapping Based on… · pp. 13–14
• The trunk network maps query coordinates to basis representations shared across input fields, covering 100200 temporal-spatial locations.The branch and trunk outputs are combined by a latent inner product to form the predicted wave field.Source: F. Cross-Physics Mapping Based on… · p. 14
• The branch-trunk factorization tests whether the mapping has an approximately low-rank representation.The paper frames DeepONet as a complementary baseline to spectral, wavelet, latent, and Galerkin representations.Source: F. Cross-Physics Mapping Based on… · pp. 14–15
• Diffusion and wave fields are standardized using training-set statistics, and DeepONet uses the shared composite reconstruction objective.Training uses AdamW, a cosine learning-rate schedule, gradient clipping, and the checkpoint with minimum validation relative ℓ2 error.Source: F. Cross-Physics Mapping Based on… · p. 14

G. Cross-Physics Mapping Based on Fourier Neural Operator (FNO)

• The input is augmented with normalized temporal and spatial coordinates before pointwise lifting into a 64-channel latent field.The spectral blocks preserve the field resolution, so the output projection requires no encoder-decoder reconstruction or spatial interpolation.Source: G. Cross-Physics Mapping Based on… · p. 15
• FNO transforms retained Fourier coefficients with trainable complex-valued weights, enabling each spectral layer to propagate information across the full temporal-spatial field.The implementation retains up to 96 temporal and 48 spatial modes.Source: G. Cross-Physics Mapping Based on… · p. 15
• Each FNO block combines global spectral mixing with a learned pointwise transformation, balancing nonlocal dependencies and local channel corrections.The model uses four consecutive blocks with latent width 64.Source: G. Cross-Physics Mapping Based on… · p. 15
• Temporal and spatial gradient discrepancies are added to the reconstruction objective to encourage accurate transitions and wavefront structures.The Fourier-domain discrepancy is implemented but assigned zero weight in the reported experiments.Source: G. Cross-Physics Mapping Based on… · p. 15
• FNO serves as a canonical spectral baseline that retains full field resolution while repeatedly applying global Fourier-domain transformations.The paper contrasts this direct spectral representation with multiresolution, latent, wavelet, Galerkin, and branch-trunk operator designs.Source: G. Cross-Physics Mapping Based on… · pp. 15–16

A. Diffusion-to-Wave

• U-NO reduces MSE by approximately 39%, MAE by 33%, and relative L2 error by 21% compared with FNO.LNO is second overall, while FNO, WNO, and GNO form a similar second performance group.Source: A. Diffusion-to-Wave · p. 17
• Relative L2 errors are 0.548 for DeepONet and 0.528 for ResUNet, both substantially above the neural-operator results.The neural-operator models listed outperform the conventional ResUNet baseline.Source: A. Diffusion-to-Wave · pp. 17–18
• 1.02×10^10 MSE, 2.18×10^4 MAE, 1.01×10^5 RMSE, 0.307 relative L2 error, and 0.905 R2: U-NO leads Table II.It is the only model with R2 above 0.9.Source: A. Diffusion-to-Wave · p. 17, Table II · p. 19
• For the short-pulse defect and defect-free chirp cases, U-NO best matches the target, while FNO and LNO also recover major field structures.The short-pulse case requires recovery of wavefronts and scattering from a strongly smoothed diffusion input.Source: A. Diffusion-to-Wave · pp. 18, 20, Figure 11 · p. 19, Figure 12 · p. 21
• For the 1s asymmetric long-pulse case, U-NO most closely recovers the multiscale wave field, while other models leave more structured residuals.The target combines long-range correlations, local oscillations, and spatially varying amplitudes.Source: A. Diffusion-to-Wave · pp. 20, 22
• Under spatially nonuniform Gaussian-chirp excitation, U-NO best preserves localized response peaks, their amplitudes, and the surrounding field structure.LNO and FNO remain competitive but show more structured residuals.Source: A. Diffusion-to-Wave · p. 20, Figure 13 · p. 21
• In the multiple-defect long-pulse case, U-NO best recovers the propagation and scattering structures; WNO, GNO, ResUNet, and DeepONet miss more detail.The response involves repeated scattering, reflection, and interactions among defect-generated wavefronts.Source: A. Diffusion-to-Wave · p. 22

B. Wave-to-Diffusion

• GNO achieves the lowest MSE (41.903), RMSE (6.473), and relative ℓ2 error (0.154), while LNO achieves the lowest MAE of 1.707.Across architectures, training and validation errors saturate after an initial reduction, with varying generalization gaps.Source: B. Wave-to-Diffusion · p. 22
• For a complex response with multiple localized features, most models recover the heat field’s principal locations and slowly varying background, but leave model-specific residual patterns.FNO and LNO are particularly clean in this case; GNO shows temporally structured artifacts and DeepONet oversmooths localized responses.Source: B. Wave-to-Diffusion · p. 24
• In another case, FNO, LNO, and WNO recover a smooth thermal field from oblique and interference-like wave structures, whereas ResUNet and DeepONet show larger structured deviations.The qualitative comparison includes asymmetric localized excitation and plates with multiple square defects.Source: Figure 14 · p. 23, Figure 15 · p. 23, B. Wave-to-Diffusion · p. 24

VI. CONCLUSION

• Wave-to-diffusion is more stable (GNO relative ℓ2 error 0.154, R2 0.935), while diffusion-to-wave is harder (U-NO relative ℓ2 error 0.307, R2 0.905).Neural operators generally outperform the conventional convolutional baseline in these cross-physics transformations.Source: §VI · pp. 24–25

B. Wave-to-Diffusion

• Wave-to-diffusion requires suppressing oscillatory wave content while preserving the slowly varying thermal component.For a complex wave input, FNO, LNO, and U-NO recover the smooth background, while GNO retains wave-like structure and DeepONet produces an overly smooth, vertically banded reconstruction.Source: B. Wave-to-Diffusion · p. 24, §VI · p. 25

Appendix A: Derivation of the Cross-Physics Scaling Principle

• The scaling aligns characteristic diffusion and wave distances over a common observation interval; it is a similarity criterion, not analytical equivalence between the equations.The respective dynamics remain distinct: diffusion attenuates spatial frequencies, whereas waves evolve in phase.Source: Appendix A · pp. 25–27
• The Fourier number Fo = αt0/L2 measures observation duration relative to diffusion time, while Λ = ct0/L measures the wave’s fraction of domain traversed.The wave transit time is τw = L/c.Source: Appendix A · pp. 25–26
• Matching diffusion penetration and wave travel yields the dimensionless similarity condition Fo = Λ².The condition is used to keep the two fields’ characteristic spatial reaches comparable during the observation interval.Source: Appendix A · pp. 26–27
• A scaling factor χ permits wave travel to differ from diffusion penetration; χ = 1 is the default equal-reach setting and supports controlled range ablations.The factor separates physical parameters from the choice of cross-domain alignment.Source: Appendix A · p. 26
• For heterogeneous fields, reference background parameters set the global scale, while dimensionless coefficient maps encode local material contrasts.The coefficient maps perturb each physical field around the shared latent structure.Source: Appendix A · p. 26
• The effective propagation speed is a cross-domain scaling parameter dependent on observation time, not a physical wave speed of parabolic diffusion.Changing the observation interval changes the speed required for equal characteristic spatial reach.Source: Appendix A · p. 27

• The virtual-wave formulation represents diffusion through an auxiliary field that evolves under a wave equation, with an arbitrary virtual speed and a local-in-space analytic connection.The virtual field is a mathematical construction, not a physically propagating thermal wave.Source: Appendix B · p. 27
• Relative errors are approximately 0.96−1.92 for diffusion-to-wave and 0.65−1.00 for wave-to-diffusion, with the conventional transform imperfect in both directions.Source: Figure 16 · p. 28, Appendix B · p. 29
• Diffusion maps the virtual-wave signal to temperature through a Gaussian-weighted temporal superposition, so recovering the wave requires regularized inversion of a smoothing, ill-conditioned transform.The kernel width grows as ∆t′ ∼√αt/c, while diffusion attenuates high-frequency components and direct inversion amplifies noise.Source: Appendix B · pp. 28–29
• Diffusion-to-wave inversion sharpens blurred interfaces and transient features but amplifies amplitudes; wave-to-diffusion suppresses oscillations into broad thermal fields and remains many-to-one.Source: Appendix B · p. 29
• The classical transform is most natural for impulsive initial heating, whereas finite pulses, chirps, long heating, nonuniform sources, and heterogeneous structures can fall outside its ideal kernel.Regularization also suppresses spectral components needed to recover oscillatory wave structure.Source: Appendix B · p. 29
• Because the analytical correspondence cannot generally be reduced to the ideal virtual-wave transform, the proposed operator-learning approach learns a broader mapping from shared structure, excitation, and dimensionless scales.Source: Appendix B · p. 29