Operator-Informed Gaussian Processes for Complex Helmholtz Wavefields: From Synthetic Benchmarks to In Vivo Brain Elastography
For researchers in wavefield inference and medical elastography, this provides a probabilistic solver for complex Helmholtz problems, though the uncertainty is not yet calibrated.
The authors extend operator-informed Gaussian process regression to complex-valued Helmholtz problems, enabling inference of complex wavefields from sparse data. On benchmarks, the method is competitive with baselines at lower cost, and on in vivo brain elastography it reconstructs the shear curl field with a correlation of 0.77, exceeding a 0.75 target.
The Helmholtz equation governs time-harmonic wave propagation, and in dissipative media a complex modulus renders its squared wavenumber $κ^2$ complex. Inferring such fields from sparse, noisy data calls for solvers that also quantify their own uncertainty. Physics-informed Gaussian-process (GP) regression supplies this by returning a posterior over the solution, yet operator-conditioned formulations have been developed almost exclusively for real-valued fields. We extend operator-informed GP regression to complex-valued Helmholtz problems by realifying the complex operator into an equivalent coupled real block, which enables inference with standard real-valued GP conditioning. The construction admits a family of priors, from a proper diagonal prior to coregionalized and multiscale variants, and conditions on PDE residuals and boundary traces. On benchmark problems in one to three dimensions, the solver is competitive with finite-difference and neural-network baselines at a far smaller interior-constraint budget. Unlike those deterministic baselines, it returns a posterior over the complex wavefield rather than a point estimate. Applied to \textit{in vivo} brain magnetic resonance elastography, a proper multiscale prior reconstructs the shear curl field to a correlation of $0.77$ with measurement, above a $0.75$ target. The gain arises from the multiscale kernel rather than from real--imaginary coupling. We further identify a low-frequency accuracy ceiling set by model mismatch and a posterior uncertainty that is not yet calibrated. Calibrated uncertainty therefore emerges as the central next step for probabilistic wavefield inference in dissipative media.