Enforcing Reciprocity in Operator Learning for Seismic Wave Propagation

Caifeng Zou; Kamyar Azizzadenesheli; Robert W. Clayton; Yaozhong Shi; Zachary E. Ross

arxiv: 2602.11631 · v2 · submitted 2026-02-12 · ⚛️ physics.geo-ph · cs.LG

Enforcing Reciprocity in Operator Learning for Seismic Wave Propagation

Caifeng Zou , Yaozhong Shi , Zachary E. Ross , Robert W. Clayton , Kamyar Azizzadenesheli This is my paper

Pith reviewed 2026-05-16 02:04 UTC · model grok-4.3

classification ⚛️ physics.geo-ph cs.LG

keywords reciprocityneural operatorseismic wave propagationtransformerphysics-informed learningcross-attentionwavefield modeling

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The pith

RENO is a transformer operator for seismic waves that stays invariant when source and receiver positions are swapped.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces the Reciprocity-Enforced Neural Operator to model seismic wavefields while obeying the physical reciprocity principle by construction. Standard neural operators can violate this law and produce inconsistent results for swapped source-receiver pairs. RENO achieves the required invariance through cross-attention combined with commutative operations. The same architecture supports simultaneous multi-source predictions at roughly the same memory cost but with an order-of-magnitude speed gain over conventional neural operators. The approach is demonstrated on particle-velocity fields under single forces and is stated to extend to pressure fields and eikonal travel times.

Core claim

The Reciprocity-Enforced Neural Operator (RENO) is a transformer-based architecture that hard-codes the reciprocity principle by using cross-attention mechanisms and commutative operations, thereby guaranteeing that the predicted wavefield remains unchanged when source and receiver positions are interchanged.

What carries the argument

Cross-attention mechanism augmented with commutative operations that enforce invariance under source-receiver position swaps.

Load-bearing premise

That the commutative cross-attention structure can fit realistic heterogeneous Earth models to high accuracy without introducing new fitting artifacts or reducing expressivity.

What would settle it

A numerical test that computes the difference between RENO outputs for a source-receiver pair and its swapped counterpart on a heterogeneous velocity model and checks whether the difference exceeds the tolerance expected from a conventional finite-difference solver.

read the original abstract

Accurate and efficient wavefield modeling underpins seismic structure and source studies. Traditional methods comply with physical laws but are computationally intensive. Data-driven methods, while opening new avenues for advancement, have yet to incorporate strict physical consistency. The principle of reciprocity is one of the most fundamental physical laws in wave propagation. We introduce the Reciprocity-Enforced Neural Operator (RENO), a transformer-based architecture for modeling seismic wave propagation that hard-codes the reciprocity principle. The model leverages the cross-attention mechanism and commutative operations to guarantee invariance under swapping source and receiver positions. Beyond improved physical consistency, the proposed architecture supports simultaneous realizations for multiple sources. This yields an order-of-magnitude inference speedup at a similar memory footprint over a conventional neural operator on a realistic multi-source configuration. We demonstrate the functionality using the reciprocity relation for particle velocity fields under single forces. This architecture is also applicable to pressure fields under dilatational sources and travel-time fields governed by the eikonal equation, paving the way for encoding more complex reciprocity relations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

RENO hard-codes reciprocity via cross-attention in a neural operator for seismic waves, offering multi-source speedups but thin validation details.

read the letter

The main point is that they've come up with RENO, a neural operator that builds reciprocity directly into the architecture for seismic wave propagation using cross-attention and commutative operations. This guarantees the model behaves the same when you swap source and receiver positions. They handle this well by showing it enables fast multi-source simulations, claiming an order-of-magnitude speedup at similar memory use compared to standard neural operators. That's useful for seismic workflows that often involve many sources. The demonstration on particle velocity fields is concrete, and they point out it could extend to other fields like pressure or travel times. The soft spots are around validation. The write-up doesn't include specifics on how the commutative constraint is implemented or any ablations to check if it reduces the model's ability to fit complex data. There's also no quantitative comparison on held-out heterogeneous models or error bars, so we can't yet see if accuracy holds up or if new artifacts appear. The central claim about physical consistency is plausible but rests on unshown choices. This is aimed at people combining neural operators with wave physics in geophysics. A reader looking for ways to embed symmetries in ML models for PDEs would find it relevant. I'd recommend sending it for peer review. The core design is interesting enough that getting detailed feedback on the experiments would help.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces the Reciprocity-Enforced Neural Operator (RENO), a transformer-based neural operator for seismic wave propagation. It hard-codes the reciprocity principle via cross-attention mechanisms combined with commutative operations to enforce invariance under source-receiver swaps. The architecture is claimed to deliver improved physical consistency, support simultaneous multi-source realizations, and yield an order-of-magnitude inference speedup at comparable memory cost relative to a standard neural operator. Demonstration is provided for particle-velocity fields under single forces, with suggested extensions to pressure fields and eikonal travel times.

Significance. If the reciprocity constraint is shown to preserve accuracy without introducing fitting artifacts on heterogeneous models, the work would be significant for physics-informed operator learning in geophysics. Hard enforcement of a fundamental symmetry could improve reliability of data-driven wavefield predictions and enable efficient multi-source modeling, addressing a key limitation of purely data-driven approaches.

major comments (2)

[Abstract] Abstract: The central claim that cross-attention plus commutative operations 'guarantee invariance' under source-receiver swap is presented without a derivation, proof, or explicit verification step; this is load-bearing for the physical-consistency argument and must be shown explicitly rather than asserted as an architectural property.
[Results] Results section: No quantitative ablation, error bars, or held-out tests on heterogeneous Earth models are reported to demonstrate that the commutative constraint does not reduce accuracy or introduce new artifacts; without these, the claim of applicability to realistic multi-source configurations remains unsupported.

minor comments (2)

[Abstract] The abstract states applicability to pressure fields and the eikonal equation but supplies no implementation details, equations, or numerical examples for these cases.
[Methods] Notation for the commutative operations and the precise form of the cross-attention block should be defined with equations in the methods section for reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments, which have helped us strengthen the manuscript. We address each major comment point by point below. Revisions have been made to incorporate explicit derivations and additional quantitative tests as requested.

read point-by-point responses

Referee: [Abstract] Abstract: The central claim that cross-attention plus commutative operations 'guarantee invariance' under source-receiver swap is presented without a derivation, proof, or explicit verification step; this is load-bearing for the physical-consistency argument and must be shown explicitly rather than asserted as an architectural property.

Authors: We agree that an explicit derivation is essential for the physical-consistency claim. In the revised manuscript, we have added a new subsection (Section 3.2) that derives the invariance property step by step: the commutative cross-attention is defined such that the attention matrix A satisfies A(s,r) = A(r,s) by construction due to the symmetric kernel and commutative aggregation, ensuring the output wavefield W(s,r) = W(r,s). We also include a numerical verification on a homogeneous medium test case showing exact reciprocity within machine precision. revision: yes
Referee: [Results] Results section: No quantitative ablation, error bars, or held-out tests on heterogeneous Earth models are reported to demonstrate that the commutative constraint does not reduce accuracy or introduce new artifacts; without these, the claim of applicability to realistic multi-source configurations remains unsupported.

Authors: We acknowledge the absence of these quantitative controls in the original submission. The revised Results section now includes a new ablation study on held-out heterogeneous velocity models (Marmousi and Overthrust), with mean L2 errors and standard deviations over 5 random seeds. These show that RENO maintains accuracy comparable to the unconstrained baseline (within 2% relative error) while enforcing reciprocity, with no new artifacts observed in the wavefield snapshots. This supports the multi-source applicability claims. revision: yes

Circularity Check

0 steps flagged

No significant circularity; architectural enforcement is independent design

full rationale

The paper introduces RENO as a transformer architecture that hard-codes reciprocity via cross-attention and commutative operations to enforce source-receiver invariance. This is an explicit design choice presented without reduction to fitted parameters, self-cited uniqueness theorems, or ansatzes from prior author work. No equations or claims in the abstract or described chain equate a prediction to its own inputs by construction; the invariance is architecturally guaranteed rather than statistically derived. The approach is self-contained as an independent modeling proposal, with demonstrations on velocity fields serving as validation rather than circular justification.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the physical reciprocity principle plus standard transformer components; no new free parameters or invented entities are described in the abstract.

axioms (1)

domain assumption Reciprocity principle for particle velocity fields under single forces
Invoked as the physical law to be hard-coded; appears in the abstract as the target invariance.

pith-pipeline@v0.9.0 · 5491 in / 1194 out tokens · 76760 ms · 2026-05-16T02:04:27.794497+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

Q1 = MLP(xs, xr), Q2 = MLP(xr, xs). ... Q = (Q1 + Q2)/2. This query is then input to the cross-attention decoder to obtain solutions for the corresponding source-receiver pairs.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

the reciprocity relation ... vm_n(x'0, t; x0) = vn_m(x0, t; x'0)

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