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Enforcing Reciprocity in Operator Learning for Seismic Wave Propagation

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

TL;DR

The paper addresses the challenge of physically consistent and efficient seismic wavefield modeling by proposing RENO, a transformer-based neural operator that hard-codes reciprocity. By using a reciprocity block that averages queries from two source–receiver permutations, RENO enforces invariance under swapping source and receiver, enabling simultaneous multi-source inference in a frequency-domain Helmholtz setting and achieving significant speedups with similar memory usage. Empirical results show exact reciprocal generalization, faster convergence, and up to ~30x inference speedup over a reciprocity-unenforced baseline, highlighting practical benefits for multi-source seismic tasks and inversions. The approach advances physics-informed ML in seismology by combining discretization-agnostic input processing (GNO) with a symmetry-preserving architecture, with potential extensions to broader reciprocity relations and more complex wave phenomena.

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 without crosstalk issues. This yields an order-of-magnitude inference speedup at a similar memory footprint over an reciprocity-unenforced neural operator on a realistic 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.

Enforcing Reciprocity in Operator Learning for Seismic Wave Propagation

TL;DR

The paper addresses the challenge of physically consistent and efficient seismic wavefield modeling by proposing RENO, a transformer-based neural operator that hard-codes reciprocity. By using a reciprocity block that averages queries from two source–receiver permutations, RENO enforces invariance under swapping source and receiver, enabling simultaneous multi-source inference in a frequency-domain Helmholtz setting and achieving significant speedups with similar memory usage. Empirical results show exact reciprocal generalization, faster convergence, and up to ~30x inference speedup over a reciprocity-unenforced baseline, highlighting practical benefits for multi-source seismic tasks and inversions. The approach advances physics-informed ML in seismology by combining discretization-agnostic input processing (GNO) with a symmetry-preserving architecture, with potential extensions to broader reciprocity relations and more complex wave phenomena.

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 without crosstalk issues. This yields an order-of-magnitude inference speedup at a similar memory footprint over an reciprocity-unenforced neural operator on a realistic 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.
Paper Structure (10 sections, 3 equations, 4 figures, 2 tables)

This paper contains 10 sections, 3 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reciprocity relations
  4. Neural operators
  5. Methods
  6. Experiments
  7. Reciprocal generalization
  8. Learning behavior
  9. Computational efficiency
  10. Discussion and Conclusions

Figures (4)

  • Figure 1: Schematic of the reciprocity relation for single vertical forces at sources and vertical particle velocity responses at receivers.
  • Figure 2: RENO architecture. Blue circles denote concatenation. $A$ in the blue circle denotes an averaging operation. $Q$ represents queries and $KV$ represents key-value pairs. All positions are encoded using sinusoidal embeddings.
  • Figure 3: Reciprocal experiments where sources and receivers are swapped. The star marks the source, which becomes the receiver marked with a triangle in the reciprocal experiment. The first row shows the waveform solutions from the finite difference solver, taken as ground truth. The second and third rows show the predictions from the reciprocity-enforced and unenforced neural operators, respectively. Both models are trained on a single simulation shown on the left.
  • Figure 4: (a) Loss and (b) reciprocal error during training for the reciprocity-enforced and unenforced models. Validation examples of reciprocal waveforms at (c) epoch 10 and (d) epoch 100.