A Transformer-based deep direct sampling method is proposed for solving a class of boundary value inverse problem. A real-time reconstruction is achieved by evaluating the learned inverse operator between carefully designed data and the reconstructed images. An effort is made to give a specific example to a fundamental but critical question: whether and how one can benefit from the theoretical structure of a mathematical problem to develop task-oriented and structure-conforming deep neural network? Specifically, inspired by direct sampling methods for inverse problems, the 1D boundary data are preprocessed by a partial differential equation-based feature map to yield 2D harmonic extensions in different frequencies as different input channels. Then, by introducing learnable non-local kernel, the approximation of direct sampling is recast to a modified attention mechanism. The proposed method is then applied to electrical impedance tomography, a well-known severely ill-posed nonlinear inverse problem. The new method achieves superior accuracy over its predecessors and contemporary operator learners, as well as shows robustness with respect to noise.

This research shall strengthen the insights that the attention mechanism, despite being invented for natural language processing tasks, offers great flexibility to be modified in conformity with the a priori mathematical knowledge, which ultimately leads to the design of more physics-compatible neural architectures.

This is a joint work with Ruchi Guo (UCI) and Shuhao Cao (University of Missouri-Kansas City).

We introduce mathematical methods for reducing complexity of deep neural networks 
in the context of computer vision for mobile and IoT applications such as sparsification and differentiable architecture search. We also describe applications in infectious disease prediction, and a deep learning and optimal 
transport (the deep particle) method in predicting invariant measures of 
stochastic dynamical systems arising in partial differential 
equation (PDE) modeling of transport in chaotic flows (e.g. rapid stirring of coffee and milk, raging forest fires in the wind). 

Living cells exhibit many forms of spatial-temporal dynamics, including recently-discovered traveling waves. Cells use these traveling waves to organizer their insides, to improve cell-cell communication, and tune their ability to move around the body. Some of these traveling waves arise from excitability (positive feedback) and non-local coupling (dynamics that spread spatially on timescales much faster than the timescale of wave motion). In collaboration with graduate students at UC Irvine, our research has studied two traveling waves involving the mechanics of the cytoskeletal protein actin: one that is approximately equivalent to a reaction-diffusion system [Barnhart et al, 2017, Current Biology], and one that is not [Manakova et al, 2016, Biophys J]. For the non-reaction-diffusion wave, we demonstrate conditions for wave travel analogous to ones previously derived for reaction-diffusion waves. We also demonstrate the existence of a "pinned" regime of parameter space absent in the equivalent reaction-diffusion system.