Simon Arridge: Nonlinear Inverse Problems with Learning

Abstract

Several problems in imaging are based on recovering coefficients of a PDE, resulting in a non-linear inverse problem that is typically solved by an iterative algorithm with the gradient obtained by an adjoint state method. When the forward problem is time-varying this corresponds to the method of time-reversal which convolves a forward and time-reversed field with the derivative of the spatial operator (sometimes called the “imaging condition”). Applications include full-waveform imaging (FWI) in Ultrasound Computed Tomography, PhotoAcoustic Tomography (PAT) and time-resolved Diffuse Optical Tomography (tDOT). Within Learned Physics approaches time reversal corresponds to the Neural ODE method for learning the time-derivative of an ODE parameterised by a neural network. By combining the trained network with symbolic regression an interpretable model can be discovered. In this talk I will discuss application of these methods for solving some forward and inverse problems in imaging.