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In this article, we review the mathematical foundations of convolutional neural nets (CNNs) with the goals of: (i) highlighting connections with techniques from statistics, signal processing, linear algebra, differential equations, and optimization, (ii) demystifying underlying computations, and (iii) identifying new types of applications. CNNs are powerful machine learning models that highlight features from grid data to make predictions (regression and classification). The grid data object can be represented as vectors (in 1D), matrices (in 2D), or tensors (in 3D or higher dimensions) and can incorporate multiple channels (thus providing high flexibility in the input data representation). CNNs highlight features from the grid data by performing convolution operations with different types of operators. The operators highlight different types of features (e.g., patterns, gradients, geometrical features) and are learned by using optimization techniques. In other words, CNNs seek to identify optimal operators that best map the input data to the output data. A common misconception is that CNNs are only capable of processing image or video data but their application scope is much wider; specifically, datasets encountered in diverse applications can be expressed as grid data. Here, we show how to apply CNNs to new types of applications such as optimal control, flow cytometry, multivariate process monitoring, and molecular simulations.

 

Liquid crystals (LCs), when supported on reactive surfaces, undergo changes in ordering that can propagate over distances of micrometers, thus providing a general and facile mechanism to amplify atomic-scale transformations on surfaces into the optical scale. While reactions on organic and metal substrates have been coupled to LC ordering transitions, metal oxide substrates, which offer unique catalytic activities for reactions involving atmospherically important chemical species such as oxidized sulfur species, have not been explored. Here we investigate this opportunity by designing LCs that contain 4′-cyanobiphenyl-4-carboxylic acid (CBCA) and respond to surface reactions triggered by parts-per-billion concentrations of SO2 gas on anatase (101) substrates. We used electronic structure calculations to predict that the carboxylic acid group of CBCA binds strongly to anatase (101) in a perpendicular orientation, a prediction that we validated in experiments in which CBCA (0.005 mol%) was doped into a LC (4’-n-pentyl-4-biphenylcarbonitrile). Both experiment and computational modeling further demonstrated that SO3-like species, produced by a surface-catalyzed reaction of SO2 with H2O on anatase (101), displace CBCA from the anatase surface, resulting in an orientational transition of the LC. Experiments also reveal the LC response to be highly selective to SO2 over other atmospheric chemical species (including H2O, NH3, H2S, and NO2), in agreement with our computational predictions for anatase (101) surfaces. Overall, we establish that the catalytic activities of metal oxide surfaces offer the basis of a new class of substrates that trigger LCs to undergo ordering transitions in response to chemical species of relevance to atmospheric chemistry. 

We report how analysis of the spatial and temporal optical responses of liquid crystal (LC) films to targeted gases, when per-formed using a machine learning methodology, can advance the sensing of gas mixtures and provide important insights into the physical processes that underlie the sensor response. We develop the methodology using O3 and Cl2 mixtures (representative of an important class of analytes) and LCs supported on metal perchlorate-decorated surfaces as a model system. Whereas O3 and Cl2¬ both diffuse through LC films and undergo redox reactions with the supporting metal perchlorate surfaces to generate similar ini-tial and final optical states of the LCs, we show that a 3-dimensional convolutional neural network (3D CNN) can extract feature information that is encoded in the spatiotemporal color patterns of the LCs to detect the presence of both O3 and Cl2 species in mixtures as well as to quantify their concentrations. Our analysis reveals that O3 detection is driven by the transition time over which the brightness of the LC changes, while Cl2 detection is driven by color fluctuations that develop late in the optical response of the LC. We also show that we can detect the presence of Cl2 even when the concentration of O3 is orders of magnitude greater than the Cl2 concentration. The proposed methodology is generalizable to a wide range of analytes, reactive surfaces and LCs, and has the potential to advance the design of portable LC monitoring devices (e.g., wearable devices) for analyzing gas mixtures us-ing spatiotemporal color fluctuations.