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

Modeling wave propagation in dissipative media—where attenuation is represented by a complex-valued modulus—requires solving the Helmholtz equation with complex wavenumbers. Traditional numerical solvers often struggle with the highly oscillatory nature of these fields and the pollution effect, while existing physics-informed machine learning methods (like PINNs) typically provide only point estimates without reliable uncertainty quantification. This paper introduces a complex-valued extension of the LinPDE-GP framework, which treats the solution of linear partial differential equations (PDEs) as a Bayesian inference problem. The authors propose a "real2" representation, which maps the complex-valued Helmholtz operator into an equivalent coupled 2x2 real-valued block system. This allows the use of standard real-valued GP conditioning machinery, avoiding the complexities of implementing complex Hermitian adjoints while still providing a full posterior distribution over the complex wavefield.

The proposed solver is shown to be competitive with finite-difference and neural-network baselines across one- to three-dimensional benchmarks, while requiring significantly fewer interior constraints. By incorporating a family of priors—ranging from simple diagonal priors to multiscale linear models of coregionalization (LMC)—the framework effectively captures the underlying wave physics. When applied to in vivo brain magnetic resonance elastography (MRE) data, the multiscale LMC prior successfully reconstructed the shear curl field with a correlation of 0.77, exceeding the target threshold of 0.75. The authors identify that the performance gain in this application stems primarily from the multiscale kernel structure rather than the coupling of real and imaginary channels.

This work bridges a critical gap between signal processing, which has a mature framework for complex-valued regression, and computational physics, which requires principled uncertainty quantification for wavefield inference. By providing a probabilistic solver that returns a posterior distribution rather than a single deterministic estimate, this approach offers a more conservative and robust tool for scientific workflows where material parameters are uncertain and measurements are sparse or noisy. The identification of uncalibrated uncertainty as a primary limitation provides a clear roadmap for future developments in probabilistic wavefield modeling.