Operator-Informed Gaussian Processes for Complex Helmholtz Wavefields: From Synthetic Benchmarks to In Vivo Brain Elastography

2026-07-27 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Data Science & Analytics · Depth: Expert, extended

Summary

This work introduces an extension of operator-informed Gaussian Process (GP) regression for complex-valued Helmholtz wavefields, particularly in dissipative media where the squared wavenumber κ² is complex. The method "realifies" the complex operator into an equivalent coupled real block, enabling inference with standard real-valued GP conditioning and a family of priors including diagonal, intrinsic coregionalization (ICM), and multiscale linear model of coregionalization (LMC) variants. On 1D, 2D, and 3D synthetic benchmarks, the solver achieves accuracy comparable to finite-difference and neural-network baselines using significantly fewer interior constraints. Applied to in vivo brain magnetic resonance elastography (MRE) data, a multiscale LMC prior reconstructs the shear curl field to a Pearson correlation of 0.77, exceeding the 0.75 target. The performance gain stems from the multiscale kernel, not real-imaginary coupling. However, the method faces a low-frequency accuracy ceiling due to physical model mismatch and currently lacks calibrated posterior uncertainty.

Key takeaway

For AI Scientists and Machine Learning Engineers developing physics-informed models for complex wave phenomena, this GP extension offers a robust approach for dissipative media. You should prioritize multiscale kernels over explicit real-imaginary coupling in your priors, as this drives accuracy gains. Be aware that posterior uncertainty may not be fully calibrated against model mismatch, serving more as an indicator of information sparsity. Consider integrating local residual-scaling noise models for improved uncertainty quantification.

Key insights

Operator-informed GP regression can solve complex Helmholtz problems by realifying the operator, enabling uncertainty quantification.

Principles

• Realifying complex operators enables standard GP conditioning.
• Multiscale priors improve accuracy for complex wavefields.
• Operator-induced coupling is sufficient for Re/Im dependence.

Method

The LinPDE-GP framework is extended by transforming complex-coefficient linear operators into coupled 2x2 real blocks. This allows standard real-valued GP conditioning with various priors, incorporating PDE residuals and boundary traces as linear information.

In practice

• Use multiscale kernels for improved complex wavefield reconstruction.
• Prioritize higher drive frequencies for better model fit in MRE.
• Recognize posterior variance as an interpolation-uncertainty map.

Topics

• Gaussian Processes
• Helmholtz Equation
• Physics-Informed Machine Learning
• Complex Wavefields
• Magnetic Resonance Elastography
• Uncertainty Quantification

Code references

• bydeng01/linpde-gp
• marvinpfoertner/linpde-gp

Best for: AI Scientist, Machine Learning Engineer, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.