Physical Sensitivity Kernels Can Emerge in Data-Driven Forward Models: Evidence From Surface-Wave Dispersion

Ziye Yu; Yuqi Cai; Xin Liu

cs.LGApr 5, 2026

Physical Sensitivity Kernels Can Emerge in Data-Driven Forward Models: Evidence From Surface-Wave Dispersion

Authors: Ziye Yu, Yuqi Cai, Xin Liu

Organizations: Institute of Geophysics, China Earthquake Administration, Beijing 100081, China · Laboratory of Seismology and Physics of Earth’s Interior, School of Earth and Space Sciences, University of Science and Technology of China, Hefei 230026, China · Institute of Advanced Technology, University of Science and Technology of China, Hefei 230088, China

Abstract

Data-driven neural networks are increasingly used as surrogate forward models in geophysics, but it remains unclear whether they recover only the data mapping or also the underlying physical sensitivity structure. Here we test this question using surface-wave dispersion. By comparing automatically differentiated gradients from a neural-network surrogate with theoretical sensitivity kernels, we show that the learned gradients can recover the main depth-dependent structure of physical kernels across a broad range of periods. This indicates that neural surrogate models can learn physically meaningful differential information, rather than acting as purely black-box predictors. At the same time, strong structural priors in the training distribution can introduce systematic artifacts into the inferred sensitivities. Our results show that neural forward surrogates can recover useful physical information for inversion and uncertainty analysis, while clarifying the conditions under which this differential structure remains physically consistent.

Figures & tables

Wave	Period (s)	MAE	MAPE (%)	Cosine	Correlation	$N$
		Dispersion Accuracy		Kernel Similarity
Rayleigh	2–5	0.00963	0.605	0.9776	0.9771	3,000
	5–10	0.01212	0.727	0.9820	0.9811	5,000
	10–20	0.01301	0.778	0.9675	0.9611	10,000
	20–30	0.01344	0.651	0.9329	0.9041	10,000
	30–45	0.01401	0.527	0.8883	0.7889	15,000

Table 1: Statistical summary of dispersion prediction accuracy and sensitivity kernel similarity.

Figure 1: Influence of training priors on surrogate-derived sensitivity kernels for a representative velocity model. (a) Weak-prior ensemble, in which velocity models are broadly distributed without imposing specific structural features. The representative model used for all kernel calculations is shown by the black curve; shading indicates the P10–P90 and P25–P75 ranges. (b)–(c) Rayleigh- and Love-wave sensitivity kernels computed from the surrogate trained on the weak-prior ensemble, compared with theoretical kernels. The surrogate gradients closely follow the theoretical kernels, with peak sensitivity shifting systematically with period, consistent with physical expectations. (d) Strong-LVZ-prior ensemble, in which a low-velocity zone (LVZ) is imposed in the upper mantle. Horizontal bands and lines indicate the mean and ±1 $\sigma$ ranges of the Moho depth and LVZ center inferred from the prior. (e)–(f) Rayleigh- and Love-wave sensitivity kernels from the surrogate trained on the strong-prior ensemble, using the same representative model as in (a). In contrast to (b)–(c), the surrogate gradients exhibit persistent depth-localized anomalies near the imposed LVZ depth that remain approximately fixed across periods, deviating from theoretical kernels whose sensitivity shifts with period. This comparison demonstrates that the physical consistency of surrogate-derived gradients depends on the training prior, with strong structural priors introducing non-physical features in the inferred sensitivity kernels.

Figure 2: Surrogate-based gradient inversion and local posterior uncertainty estimated from the Fisher information matrix. Left panels show recovered shear-wave velocity models for four representative samples, including the true model (black), initial model (gray dashed), and inverted model (red). Shaded regions denote $\pm 2\sigma$ uncertainty derived from the inverse Fisher matrix. Right panels show corresponding dispersion fits for Rayleigh and Love waves before and after inversion. The inversion successfully recovers the main structural features of the true model and achieves close agreement with observed dispersion curves. The posterior uncertainty increases with depth, reflecting reduced sensitivity and increased non-uniqueness in deeper regions. These results demonstrate that gradients derived from the neural surrogate provide a physically meaningful approximation of the forward operator Jacobian and can be used for both inversion and uncertainty quantification.

Neural Surrogate Modeling

Neural surrogate modeling uses learned models to approximate costly simulations, enabling faster prediction, design, control, and inference.

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PDE Surrogate Modeling Jacobian Regularization PDE Inverse Problems Neural Operators

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Neural operators provide fast surrogates for partial differential equation (PDE) solvers, but their reliability can degrade for high-dimensional spatial inputs and inverse or repeated inference. State-only training constrains solution values but not the learned input--output response. We study sensitivity-constrained neural operators (SC-NOs), which augment standard training with sampled solver-derived Jacobian supervision. Selected sensitivities from differentiable solvers or discrete adjoints are matched during training, allowing response information to be amortized across minibatches without imposing the full Jacobian at every update. We evaluate SC-NO on advection--diffusion and RANS--Spalart--Allmaras benchmarks, input-dimensionality scaling tests, long-horizon autoregressive rollout, and a shallow-water Tohoku tsunami source-inversion case. Sensitivity supervision improves forward prediction and yields larger gains in gradient-based inverse reconstruction of distributed fields. Scaling experiments show an improved accuracy--cost tradeoff for high-dimensional gridded inputs, while ablations indicate that state values and Jacobian information provide complementary supervision. In the tsunami case, SC-FNO reconstructs gridded seafloor deformation from sparse early gauge observations and forecasts subsequent wave propagation in a near-real-time proof-of-concept workflow. These results support sampled sensitivity supervision as a practical way to improve neural PDE surrogates when forward accuracy, inverse stability, robustness, and computational cost must be considered together.

Abstract

Neural Surrogate Modeling

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