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

Area of Science:

• Computational physics
• Applied mathematics
• Wave propagation modeling

Background:

• The Helmholtz equation describes wave propagation in dissipative media, where complex moduli lead to complex wavenumbers.
• Inferring wavefields from sparse, noisy data requires solvers that quantify uncertainty.
• Existing physics-informed Gaussian-process (GP) regression is primarily for real-valued fields.

Purpose of the Study:

• To extend operator-informed GP regression for complex-valued Helmholtz problems.
• To enable uncertainty quantification in wavefield inference for dissipative media.
• To apply the method to complex real-world problems like magnetic resonance elastography.

Main Methods:

• Realifying the complex Helmholtz operator into an equivalent coupled real block for standard GP conditioning.
• Developing a family of priors, including diagonal, coregionalized, and multiscale variants.
• Conditioning on partial differential equation residuals and boundary traces.

Main Results:

• The solver is competitive with finite-difference and neural-network methods on benchmark problems, requiring fewer interior constraints.
• It provides a posterior over the complex wavefield, unlike deterministic baselines.
• Applied to brain magnetic resonance elastography, it achieved a 0.77 correlation for the shear curl field, exceeding the 0.75 target.

Conclusions:

• The developed method effectively extends GP regression to complex wavefield inference.
• Multiscale kernels contribute to performance gains, not real-imaginary coupling.
• Future work should focus on calibrated uncertainty quantification and addressing low-frequency accuracy limitations.