Inverse Problems, Deep learning, UQ, Operator Learning: Scientific Machine Learning, FNO, ORNN

GMIG studies inverse problems through the lens of deep learning. Following proofs of uniqueness, the Operator Recurrent Neural Network emerged as a powerful architecture for nonlinear recovery. With optimal weights such a network provides a Bayesian estimator. Intrinsic properties of weight matrices guarantee favorable generalization estimates. Current research will make this neural network independent of discretization by introducing neural operators.

GMIG recently obtained and successfully trained a neural operator that maps a wave speed model to the solution of the Helmholtz equation, a key development in learning full-waveform inversion (FWI). Furthermore, GMIG and its collaborators have developed a platform for learning and analyzing unbounded linear operators, enabling uncertainty quantification while also providing convergence rates. In parallel, GMIG has analyzed and obtained conditions for globally injective (one-to-one) ReLU networks, fitting well into the study of inverse problems.

GMIG extended these networks to injective flows, named TRUMPETs, and established a universal approximation property. These flows train much faster than their traditional counterparts, enabling the realistic estimation of posterior distributions in nonlinear (seismic) inverse problems. Via implicit neural representations, GMIG is straddling the boundary of non-uniqueness in the analysis of inverse and partial recovery.

Through a proven track record of impactful research, GMIG has demonstrated that inverse problem solving has implications well beyond theoretical corners of mathematics and computing. In applying principles of deep learning to this type of problem solving, GMIG has developed algorithms that fuel technological advances for its industrial partners.

Further reading: TRUMPETS: Injective Flows for Inference and Inverse problems