Periodic Reporting for period 2 - SWING (Signals, Waves, and Learning: A Data-Driven Paradigm for Wave-Based Inverse Problems)
Reporting period: 2021-07-01 to 2022-12-31
Scattering of waves governs fundamental questions in science, from imaging molecules to fine-tuning concert hall acoustics. Efficient scattering computations rely on sparse representations of wavefields. Spurred by the empirical successes of deep learning, the emphasis has recently shifted to data-driven modeling. However, unlike signal-theoretic implementations that come with sharp approximation guarantees, it remains unclear whether the popular deep learning structures can represent important scattering operators. In SWING, we address this question by leveraging advances in signal processing and machine learning. We propose theory and algorithms for the upcoming, learning-based wave of breakthroughs in forward and inverse scattering. SWING is built on three research thrusts: 1. To design efficient computational structures with approximation guarantees for learning scattering operators. We will focus on minimal structures for Fourier integral operators which model key problems. 2. To treat learning for inverse scattering as a sampling problem and derive practical sample complexity results. We will explore connections between learning theory and stability of inverse problems, and examine the regularization roles of data, physics and nonlinearity. 3. To apply our techniques to two classes of inverse problems: (i) emerging modalities in molecular imaging, giving rise to problems in geometry and unlabeled sampling; and (ii) seismic tomography of Earth and Mars, with data-driven discretizations of scattering operators playing a central role. With the growth of wave-based sensing, there is an urgency to quantify the limits of the data-driven paradigm in scattering problems. The power of data in fitting models is indisputable: it is certainly the next frontier. We believe, however, that the best designs combine data-based models with an understanding of the underlying physics.
We worked on deep learning methods and theory for inverse problems in wave-based imagiing. We designed a new deep neural network, the FIONet, which solves forward and inverse problems in wave imaging with unprecedented generalization while being trained on very small datasets. It does so by emulating the natural ray geometry of wave-based imaging. On the theoretical side, we proved that probabilistic transformers, a randomized version of the the popular transformers, achieve universal approximation while implementing arbitrary _exact_ constraints. This is the first such result in the literature for any network. Constrained approximation is of clear significance. For example, in inverse problems the constraints may signify physical behavior; in mission critical applications a neural network's output should not even slightly deviate from a prescribed range; in statistical applications the output should be a structured object like a low-rank covariance matrix. Finally, we contributed new neural architectures that are provably injective, a property of paramount importance for well-posedness of many problems in scientific inference and imaging which guarantees that to each set of measurements there corresponds a unique object. We proved that these architectures can approximate arbitrary data distributions with low-dimensional manifold structure. We demonstrated applications to problems in medical and seismic imaging and electromagnetic inverse scattering.
All three results described above—FIONet, injective networks, and universal approximation under constraints—progress significantly beyond the state of the art. In the following months we will extend the geometric ideas that underlie the FIONet to other physical phenomena. We will then move the focus to unlabeled inverse problems and cryo-electron tomography. We expect to soon publish exciting results on ab initio travel time tomography, and further deepen that direction with theory and applications to spectral data. Finally, we will address multiple scattering problems such as scattering and boundary control.