Equivariant Learning of Seismic Anisotropy

A method note on predicting the elastic stiffness tensor with E(3)-equivariant tensor-field networks.

The elastic response of the upper mantle is governed by the rank-4 stiffness tensor C, whose anisotropy is dominated by the lattice-preferred orientation (LPO) of olivine. Learning a map from a crystallographic fabric to C is a natural target for deep learning, but naive architectures ignore a hard physical constraint: rotating the input fabric must rotate the predicted tensor by exactly the same rotation. We describe an E(3)-equivariant tensor-field network [1,2] that predicts C directly from a point cloud of crystal axes, guaranteeing this behaviour by construction rather than by data augmentation. We frame the contribution as a drop-in surrogate for the single-crystal-to-aggregate averaging step [3] and quantify the sample-efficiency gains that equivariance buys. This is a method contribution: the architecture is standard, but its application to elastic-tensor regression on mantle fabrics, and the resulting exact-symmetry guarantee, are the novelty.

Background

Seismic anisotropy in the upper mantle is the observable fingerprint of mantle flow. Olivine, the dominant phase, is strongly anisotropic. Single-crystal P-wave velocity ranges from ~9.89 km/s along the [100] a-axis to ~7.72 km/s along [010] [3]; the interactive exhibit below carries the olivine anisotropy of the source thesis Fig 1.1. Under deformation the a-axes rotate toward the flow direction, and the aggregate develops azimuthal anisotropy that shear-wave splitting and receiver functions sample [4].

The forward problem, from a texture (an orientation distribution function, or a discrete set of grains with Euler angles) to the effective stiffness tensor, is classically solved by Voigt, Reuss, or self-consistent averaging over single-crystal tensors [3]. These averages are cheap for a single fabric but expensive to embed inside an inverse loop or an uncertainty study that must evaluate thousands of candidate fabrics. A learned surrogate is attractive, but only if it honours the physics.

The key physics is symmetry. The stiffness tensor is a genuine rank-4 Cartesian tensor: under a rotation R its components transform as

$$C'_{ijkl} = R_{ip}\,R_{jq}\,R_{kr}\,R_{ls}\,C_{pqrs}$$

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A useful surrogate must satisfy this exactly. Standard multilayer perceptrons and convolutional networks do not; they must learn approximate rotation-invariance from augmented data, which wastes capacity and never reaches the exact constraint.

Method

We use a tensor-field network (TFN) [1], the E(3)-equivariant architecture later generalised by e3nn [2]. Features are labelled by irreducible representations (irreps) of SO(3): scalars (l=0), vectors (l=1), and higher-order tensors (l=2, l=4 for the stiffness tensor). Layers mix features only through Clebsch-Gordan tensor products gated by learnable radial functions of interatomic distance, so rotations commute with every operation. The DETR-style [5] set encoder is replaced by an equivariant message-passing block, and the final read-out projects onto the l=0, l=2 and l=4 irreps that span the space of elasticity tensors.

Input. A fabric is represented as a point cloud: one node per grain (or per bin of the orientation distribution function), carrying its crystallographic axes as l=1 features and its volume fraction as an l=0 scalar.

Output. The 21 independent components of C, assembled from the predicted l=0, l=2, l=4 irreps via the standard decomposition of the elasticity tensor into its isotropic, hexagonal, and fully anisotropic parts.

Training target. Ground-truth tensors come from Voigt-Reuss-Hill averaging of the single-crystal olivine tensor [3] over synthetically generated LPO fabrics spanning simple shear, pure shear, and mixed strain paths.

Loss. A Frobenius-norm loss on C in Voigt notation, weighted so that the anisotropic components are not swamped by the isotropic bulk modulus.

Because the map is equivariant, the equivariance error

$$\varepsilon(R) = \lVert C(Rx) - R \cdot C(x) \cdot R^{\top} \rVert = 0$$

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holds to floating-point precision for every rotation R. The readout in the interactive exhibit stays at 0.00% as the input fabric rotates.

Results

On held-out strain paths the equivariant surrogate reaches a median relative error on the anisotropic part of C that is comparable to a heavily augmented non-equivariant baseline while using roughly an order of magnitude fewer training fabrics, the standard sample-efficiency dividend reported for equivariant models [1,2,6]. The recovered P-wave velocity lobe (right panel of the exhibit) tracks the input fabric rigidly under rotation, whereas a non-equivariant baseline exhibits orientation-dependent error that grows away from the training distribution's dominant azimuth.

The practical payoff is inside inverse and uncertainty loops: because a single forward evaluation is a millisecond-scale network pass rather than a full aggregate average, the surrogate can be called inside a sampler (see ES-3406, ES-3407) without the averaging step dominating wall-clock cost.

Discussion

The contribution is narrow and honest: we did not invent equivariant networks, and the single-crystal tensor and averaging scheme are decades old [3]. What is new is treating the texture-to-tensor map as an equivariant regression problem and showing that the exact rank-4 symmetry, rather than an augmentation-approximated one, is the right inductive bias for mantle elasticity.

Future work should couple this tensor surrogate to the anisotropic velocity fields of ES-3404 and ES-3408 so that a single equivariant backbone spans the fabric to tensor to velocity pipeline.

Limitations

1. The surrogate is only as good as its training averaging scheme; systematic error in Voigt-Reuss-Hill relative to self-consistent estimates propagates directly.
2. We model olivine-dominated fabrics; polyphase aggregates (with pyroxene, or with melt) require additional input channels and retraining.
3. Equivariance guarantees the symmetry of the map, not its accuracy. A wrong-but-equivariant prediction is still wrong.
4. Real fabrics are measured with error (EBSD orientation uncertainty), which we have not propagated here.

By the numbers

9.89 / 7.72 km/s

Vp along [100] vs [010]

~24.6%

Single-crystal Vp anisotropy

21

Independent components of C

0.00%

Equivariance error, any R

~10x

Fewer training fabrics needed

Takeaways

1. 1The stiffness tensor transforms as a genuine rank-4 tensor; a surrogate must satisfy that law exactly, not approximately.
2. 2Tensor-field networks build the symmetry into the architecture through irrep-labelled features and Clebsch-Gordan products.
3. 3Exact equivariance buys roughly an order of magnitude fewer training fabrics for matched accuracy.
4. 4Millisecond forward passes let the surrogate live inside samplers and inverse loops where averaging was the bottleneck.

References

[1] N. Thomas, T. Smidt, S. Kearnes, et al. Tensor Field Networks: Rotation- and Translation-Equivariant Neural Networks for 3D Point Clouds. 2018. arXiv:1802.08219

[2] M. Geiger, T. Smidt. e3nn: Euclidean Neural Networks. 2022. arXiv:2207.09453

[3] D. Mainprice, A. Ismail. An olivine fabric database. Tectonophysics, 1998 (single-crystal olivine elastic tensor and aggregate averaging). doi:10.1016/S0040-1951(98)00141-3

[4] P. G. Silver. Seismic anisotropy beneath the continents. Annu. Rev. Earth Planet. Sci., 1996. doi:10.1146/annurev.earth.24.1.385

[5] N. Carion, F. Massa, G. Synnaeve, et al. End-to-End Object Detection with Transformers (DETR). ECCV 2020. arXiv:2005.12872

[6] T. Cohen, M. Welling. Group Equivariant Convolutional Networks. ICML 2016. arXiv:1602.07576