Current Research Directions
AI for Quantum Physics and Chemistry
We employ the tools of deep learning to solve problems in quantum physics and chemistry. The idea is to use neural networks less in the traditional learning sense, and more as a function class with desirable properties from the Approximation Theory point of view. We are working on several problems:​
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Computing quantum ground states by combining Variational Monte Carlo with an exotic normalizing flow based on determinantal point processes.
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Generation of desired quantum correlations in SPDC between photon-pairs via design of nonlinear photonic crystals and pump beams.
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Enhancing Path Integral Monte Carlo via generative models.
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Solving inverse problems in quantum dynamics using diffusion.
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AI approaches to finding ground states in the Bose-Hubbard and Fermi-Hubbard models.
AI for Partial Differential Equations
PDEs are used throughout science, in particular in physics. There are many standard techniques for solving PDEs numerically, but these are often quite time consuming. AI approaches promise acceleration of the solution:
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Low-rank neural operators. A neural operator maps from the problem's specification (e.g. initial conditions or boundary conditions) to the PDE's solution. We have designed operators which are both expressive and efficient, based on a singular value decomposition of the operator.
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Physics Informed Neural Networks allow one to solve an (unlearned) PDE directly, by performing gradient descent on the expectation of the residual. We have generalized these methods to work in high dimensions.
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We have applied such techniques to the solution of certain bottleneck problems in quantum optics, based on the Nonlinear Schrodinger Equation.
Atomistic and Molecular AI
We investigate both molecular generation and structure determination. Some topics are within the purview of my work at Genesis Molecular AI.
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Design of new molecules. We are interested in generating small molecules which can bind to a given protein, mainly for the purpose of therapeutics. We design conditional diffusion and flow models to achieve this end.
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Incorporation of physics into atomistic generative models. Unlike many areas of AI, we actually understand the underlying rules of the molecular world, which are dictated by the laws of physics. It is therefore of interest to blend diffusion / flow models with physics.
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Molecular structure determination. A longstanding goal is to extract molecular structure efficiently from Cryo-EM data. We are designing deep learning pipelines to enable this.
Previous Research Directions
Novel Scientific Imaging Modalities
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Nanoscale imaging using optical microscopy beyond the diffraction limit. We developed deep learning based algorithms which allow for the 3D imaging of the inner workings of a cell - at the level of individual proteins - in high resolution.
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Full sound-speed inversion ultrasound. While ordinary (b-mode) ultrasound imaging essentially captures the boundaries of objects, our new technique allows for the dense computation of material properties at each point in space. This allows ultrasound to mimic much more expensive CT imaging.
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Cheap spatial transcriptomics. ST provides a means to extract the expression of many genes or proteins in a spatially varying manner. Using AI, we achieved this directly from whole slide microscopy images, at a fraction of the cost and time required by standard ST.
Computational Algebraic Topology
Algebraic topology is considered to be one of the purer areas of mathematics, but recently researchers have considered both computational aspects of the field, as well as outright applications. In our work, we were interested in both:
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On the computational side, we examined the problem of finding the optimal basis of a homology group, under various definitions of optimality; under some definitions, the problem admits a solution with low-degree polynomial complexity, while under other definitions it is NP-complete.
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On the applications side, we incorporated persistent homology into various types of segmentation problems, based both on differential equations and Markov Random Fields.
Mathematical Methods for Vision & Learning
Our investigations focused on an array of disparate techniques:
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Techniques based on geometric partial differential equations, which were used in the context of curve, surface, and manifold evolution.
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Techniques based on combinatorial optimization, which were employed to minimize functions - e.g. to solve for optical flow or segmentation - which could naturally be framed in a discrete formalism.
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​Inverse problems: deep learning approaches to inverse problems which rely on classical approaches with provable properties.
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​Information theoretic treatment of generative AI: theoretical tradeoffs between uncertainty and (perceptual) quality.
AI for Biology and Medicine
We focused on applications in 3 different areas:
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Virtual staining of pathology slides. Using conditional diffusion models, we developed techniques that map from hyperspectral microscopy images - based on autofluorescence imaging - to tens of stains simultaneously.
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Automated detection of disease in endoscopy and surgery. We developed a state of the art detector for polyps in colonoscopy, with very low false alarm rate. We also performed monocular 3D reconstruction of the colon, to determine which areas of the colon wall had been missed.
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Analysis of EKG and EEG signals. We combined the Koopman Theory of nonlinear dynamical systems with deep learning to achieve state of the art performance in various tasks.