# Transformer Meets Boundary Value Inverse Problems

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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).

# Optimal transport and the Monge-Ampere equation

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The optimal transport problem asks: What is the cheapest way to transport goods (e.g. bread in bakeries) to desired locations (e.g. grocery stores)? Although simple to state, this problem is tricky to solve. Optimal transport is closely related to a nonlinear PDE called the Monge-Ampere equation, and important questions about optimal transport can be approached using this connection. In this talk we will discuss optimal transport, its connection to the Monge-Ampere equation, and some recent applications of optimal transport theory in geometric and functional inequalities and meteorology.

# Efficient Deep Neural Networks and a Deep Particle Method for PDEs

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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).

# What is cohomology?

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# Kaehler geometry of molecular surfaces

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I will provide an introduction to the theory of complex manifolds, via the simplest example of the 2-sphere. The talk will center around how molecules needed for drug design can be efficiently described using the complex numbers. No background knowledge will be assumed. This is joint work with D. Cole, S. Hall, and R. Pirie.

# Traveling waves in cells from reaction-diffusion and non-reaction-diffusion systems

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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.

# Applications of Descriptive Set Theory in Dynamical systems

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Many classical problems in dynamical systems have been studied for more than a century. Despite enormous progress, several have resisted solution. Descriptive Set Theory provides the tools to prove impossibility results, both in the quantitative and the qualitative behavior of dynamical systems.

# What is Learning Theory?

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I will talk about some challenging problems and the role of Fourier concentration in an emerging field called learning theory. Suppose we want to learn a certain data. The basic question we ask is what is the minimal amount of information needed to learn (predict) the data with high precision and probability, and what is the role of Fourier analysis in these questions.