ESPC-Transolver: An Equivalent-Surface Physics-Guided Coupling-Aware Transolver for 3D Multiple Scattering

1. Introduction

• The central question is whether shared local scattering responses improve prediction on unseen geometry families compared with geometric conditioning.The challenge is inferring unseen objects’ scattering properties while modeling phase- and amplitude-sensitive interactions among objects.Source: §1 · p. 2
• Canonical responses to common spherical-wave excitations provide physically aligned input channels across different geometry families.A local DtN map and modal scattering conversion express the responses on enclosing spheres.Source: §1 · p. 2
• Shared object-local Transolver blocks encode responses and incident traces, while relative-position-aware slice attention predicts inter-object coupling.Sharing parameters across scatterers supports configurations with different object counts and arrangements.Source: §1 · p. 3
• Predicted interface pressure and operator-derived normal flux define equivalent sources for near-field and far-field reconstruction across shapes and topologies.The canonical sampling layout and coupling network remain unchanged as local response operators incorporate geometric and topological variation.Source: §1 · p. 3
• Experiments assess input representations, interaction architectures, held-out familiar geometries, unseen Peanut and Torus families, and representative field reconstructions at a fixed frequency.The reconstructions examine how interface prediction errors relate to recovered spatial distributions and interference patterns.Source: §1 · p. 3

2.1. The exterior scattering problem

• The exterior problem concerns sound-hard objects in 3D, with each object enclosed alone by a disjoint canonical ball of the same physical radius.The objects are centered at their respective centers, and the canonical interfaces provide the common setting for the local responses.Source: §2.1 · p. 3

2.2. A local response on the canonical interface

• Canonical pressure and flux traces define equivalent sources whose summed contributions recover the scattered near field and far-field amplitude.The exterior representation retains the outgoing field generated by each enclosed object; regular incident and other-object fields contribute zero to that representation.Source: Figure 1 · p. 4
• The local DtN map is defined by a mixed Laplacian problem and is uniquely solvable when k^2 is outside the operator spectrum.The map depends on object geometry and physical wavenumber, but not scene position or incident direction.Source: §2.2 · p. 4
• Spherical-harmonic probes establish a common input basis and canonical directions a common output grid for sampled responses.The weighted projection uses sampled harmonics and sphere-area quadrature; FEM constructs the response from projected weak reactions.Source: §2.2 · p. 4

2.3. Constructing the discrete response by internal elimination

• Eliminating interior FEM degrees of freedom yields the discrete DtN response as a Schur action, with one factorization reused for probe solves.The boundary degrees of freedom are separated from the remaining degrees of freedom, including those on the Neumann obstacle boundary.Source: §2.3 · pp. 4–5
• Mass-matrix-weighted least squares projects FEM reactions onto output harmonics before evaluating flux coefficients on the canonical grid.The construction maps weak reactions through modal projection and then canonical sampling.Source: §2.3 · p. 5

2.4. Converting the DtN response to scattered pressure

• The projected DtN response is converted to a canonical scattered-pressure operator whose columns represent responses to individual regular spherical-wave excitations.The response is defined by the FEM discretization, modal spaces, and quadrature, and is supplied to ESPC-Transolver with incident trace and scene features.Source: §2.4 · p. 5
• The retained Foldy–Lax system couples each object’s outgoing waves into regular excitations at the others, while the network predicts coupled canonical pressure traces.The off-diagonal translation blocks represent inter-object wave transfer; the retained system requires I − TU to be nonsingular.Source: §2.4 · p. 5

3. ESPC-Transolver: a compositional Transolver architecture

• Shared local blocks encode each object's canonical response and incident-field input, while relative-position-aware interactions exchange information between objects.Local processing and inter-object updates are interleaved; a shared decoder predicts one complex scattered-pressure value per canonical node.Source: §3 · p. 6, §3.1 · p. 6, Figure 2 · p. 6

3.1. Object-compositional organization

• Local Physics-Attention blocks aggregate canonical nodes into learned slices, update slice tokens, and return updates to the nodes.Local-block parameters are shared across objects, while successive blocks have distinct parameters.Source: §3.1 · pp. 6–7

3.2. Source-wise slice interactions

• The interaction module pairs object-specific slices with position descriptors and separately normalizes source-object slice messages before summing them at each target.The interaction module is inserted after local blocks 1, 2, and 3; its shared edge MLP processes relative-position descriptors.Source: §3.1 · p. 7, §3.2 · pp. 7–8

3.1. Object-compositional organization

• Shared local parameters and summed source-object messages make predicted traces equivariant to object permutations.Consistently reordering input objects reorders their predicted traces in the same way.Source: §3.1 · p. 7

3.3. Prediction and operator-weighted supervision

• Training combines scene-normalized losses, including an auxiliary flux error that weights trace errors through the projected local flux operator.Scattered-field targets supervise scene coupling, and training-set normalization scales remain fixed at evaluation.Source: §3.3 · p. 8

4.1. Pressure and flux as equivalent surface sources

• Predicted total pressure traces and local-operator fluxes define equivalent sources whose summed outgoing fields reconstruct the exterior near field and far-field amplitude.The translation phase preserves interference between radiation from different object centres.Source: §4.1 · p. 8, §4.3 · p. 9

4.2. Recovering the outgoing coefficients

• The outgoing coefficients are recovered by eliminating the regular incoming contribution from the retained modal pressure and flux relations.The implementation projects predicted total traces onto the retained Q-mode space.Source: §4.2 · p. 9

4.3. Near-field and far-field evaluation

• The trace-to-field error bound separates propagated canonical-trace prediction error from retained-mode representation discrepancy relative to the numerical reference.Near- and far-field metrics use planar-grid and spherical-quadrature norms, respectively.Source: §4.3 · p. 9

5.1. Experimental setup

• The evaluation uses fixed-frequency, three-dimensional sound-hard scattering; training includes 10,000 scenes from basic shapes and 10,000 from 11 CAD geometries.Tests cover held-out instances of familiar sphere, ellipsoid, and box families, plus unseen Peanut and Torus families.Source: §5.1 · p. 10

5.2. Experiment 1: Comparison of response representations under a fixed compositional backbone

• ESPC-Transolver has lower errors than all other models on held-out familiar geometry families; Transolver-Geo-strong's 3.10% mean error beats Transolver-Geo-matched's 7.77% there, but their ordering reverses on Peanut and Torus.Canonical-response conditioning also yields lower mean and median errors than both geometry-conditioned alternatives on the complete Peanut and Torus test sets.Source: Table 1 · p. 11, Figure 4 · p. 11, §5.2 · p. 11
• Near-field complex-valued errors are lower than both geometry-conditioned models in all three illustrated scenes, and far-field reconstruction errors are also lower for ESPC-Transolver.The reconstructions include exterior amplitude and phase patterns, including interference from multiple wave interactions.Source: Figure 6 · p. 13, Table 2 · p. 14, §5.2 · p. 14

5.3. Experiment 2: Comparison of neural network architectures

• 2.21% versus 4.97% on Same-geometry structure, 5.27% versus 7.46% on Peanut, and 15.44% versus 16.05% on Torus: ESPC-Transolver has the lowest mean errors among evaluated architectures.The advantage over Transolver is concentrated in familiar families and Peanut, with a smaller difference on Torus.Source: §5.3 · p. 14, Table 3 · p. 15
• ESPC-Transolver requires 1.088 s for 100 predictions, versus 0.540 s for Transolver, illustrating an accuracy–inference-time trade-off.The measurements include neural-network forward inference time.Source: Table 4 · p. 15, §5.3 · p. 15

• ESPC-Transolver separates object-specific scattering responses from scene-dependent interactions to predict acoustic multiple scattering.The method uses shared object-local blocks and relative-position-aware attention.Source: §6 · p. 15
• Across evaluated sound-hard scattering problems, ESPC-Transolver achieves lower interface and canonical scattered-field prediction errors than the geometry-conditioned baselines and alternative architectures, including on unseen Peanut and Torus geometries.Source: §6 · p. 15
• Near- and far-field reconstructions show that predicted interface pressure and normal flux recover exterior pressure distributions and multiple-scattering interference patterns.Source: §6 · p. 15

• Table A.5 lists network configurations and total trainable parameters, counting shared-module parameters only once.Source: Table A.5 · p. 17, Appendix A · p. 17

• Radial-occupancy features are evaluated at radial sampling locations, and feature normalization uses training-set means and variances with a 10^-4 variance floor.Source: Appendix A.1 · p. 17
• Surface encoders process 2,048 area-sampled points and normals, using local-neighbor and whole-object features projected to a 128-dimensional conditioning space.Local features aggregate the 16 nearest neighbors; global features combine max and mean pooling.Source: Appendix A.1 · p. 17
• PTv3 and Erwin use specified regression-head widths, while PIBNet constructs a multiscale graph with triangulation, tree, and reciprocal connections.The passage specifies PTv3 and Erwin heads as 64 →64 →2 and 32 →64 →2, respectively.Source: Appendix A.1 · p. 18

Appendix A.2. Training configuration

• All models use AdamW for 200 epochs with batch size 16, cosine annealing from 10−4 to 10−7, weight decay 10−5, and gradient clipping at norm 1.Source: Appendix A.2 · p. 18
• The auxiliary loss weight is 0.1 except for PTv3 and Erwin, where it is zero.Source: Appendix A.2 · p. 18
• The flux-loss stabilization parameter εq is 10−6 times median flux energy over training scenes.Source: Appendix A.2 · p. 18
• Response degrees of freedom and global scattered-field output are normalized with uncentered training-data RMS scales, then restored to physical scales for evaluation.Source: Appendix A.2 · p. 18

Appendix A.3. Numerical discretization and implementation details

• The numerical discretization uses graded, curved quadratic finite-element meshes, with spherical-harmonic modes ordered by degree and order.The excerpt refers to retained modal dimensions in Eq. (12) but does not provide their values.Source: Appendix A.3 · p. 18

Appendix A.4. Error metrics

• Exterior-field errors are normalized by reference-field norms; near-field and far-field errors use uniform planar sampling and spherical quadrature weights, respectively.Source: Appendix A.4 · p. 18

Appendix B.1. Finite-dimensional projection and coupling consistency

• The retained solution is expressed as (I−TU)bQ = Tainc, and the recovery bound assumes I−TU is nonsingular.The conversion also accounts for finite-quadrature projection of higher output modes.Source: Appendix B.1 · pp. 18–19

Appendix B.2. Local-response and coupled-system sensitivity

• Local-response sensitivity is separated from coupled amplification, whose spectral-norm factor is 1/σmin(I−TU).Source: Appendix B.2 · p. 19

Appendix B.3. Trace-prediction and exterior-representation errors

• The exact decomposition of exterior-representation error separates neural trace error from finite-mode recovery and numerical-representation discrepancy.Source: Appendix B.3 · p. 19