A central challenge in sensory neuroscience is to understand neural computations and circuit mechanisms that underlie the encoding of ethologically relevant, natural stimuli. In multilayered neural circuits, nonlinear processes such as synaptic transmission and spiking dynamics present a significant obstacle to the creation of accurate computational models of responses to natural stimuli. Here we demonstrate that deep convolutional neural networks (CNNs) capture retinal responses to natural scenes nearly to within the variability of a cell's response, and are markedly more accurate than linear-nonlinear (LN) models and Generalized Linear Models (GLMs). Moreover, we find two additional surprising properties of CNNs: they are less susceptible to overfitting than their LN counterparts when trained on small amounts of data, and generalize better when tested on stimuli drawn from a different distribution (e.g. between natural scenes and white noise). An examination of the learned CNNs reveals several properties.

By combining robust regression and prior information, we develop an effective robust regression method that can resist adaptive adversarial attacks. Due to the widespread existence of noise and data corruption, it is necessary to recover the true regression parameters when a certain proportion of the response variables have been corrupted. Methods to overcome this problem often involve robust least-squaresregression.

Differential equations play important roles in modeling complex physical systems.

Laplace learning is a semi-supervised method, a solution for finding missing labels from a partially labeled dataset utilizing the geometry given by the unlabeled data points. The method minimizes a Dirichlet energy defined on a (discrete) graph constructed from the full dataset. In finite dimensions the asymptotics in the large (unlabeled) data limit are well understood with convergence from the graph setting to a continuum Sobolev semi-norm weighted by the Lebesgue density of the data-generating measure. The lack of the Lebesgue measure on infinite-dimensional spaces requires rethinking the analysis if the data aren't finite-dimensional. In this paper we make a first step in this direction by analyzing the setting when the data are generated by a Gaussian measure on a Hilbert space and proving pointwise convergence of the graph Dirichlet energy.

A new paradigm in machine learning for scientific computing is focused on designing learning algorithms and methods for continuum problems. This paradigm is referred to as operator learning and has received considerable interest in the last few years [5,7,18,20,23-25,27,30,34,36]. The basic task may be posed as learning a map between infinite-dimensional function spaces, i.e., learning an operator F: X Y, where, for example, X and Y are real, separable Hilbert spaces. Operator learning naturally arises in many scientific problems where one wants to learn how a continuum model, often described by partial differential equations (PDEs), maps inputs, such as parameters or boundary conditions, to outputs, such as states or observables. A prototypical example to keep in mind is learning parameter-to-solution maps of parametric PDEs [1,2,11]. In contrast to more classical surrogate modeling, which typically focuses on learning finite-dimensional parameter-to-solution maps for some fixed discretization, operator learning directly aims to learn/approximate the continuum map F: X Y itself. Thus, the inputs and outputs are functions (not vectors) and the goal is to directly design discretization-invariant methods [7,23]. From a statistical perspective, this naturally leads to a nonparametric regression problem in which both the object of interest (the operator) and the observations (finite number of noisy samples) are infinite-dimensional.

A central challenge in sensory neuroscience is to understand neural computations and circuit mechanisms that underlie the encoding of ethologically relevant, natural stimuli. In multilayered neural circuits, nonlinear processes such as synaptic transmission and spiking dynamics present a significant obstacle to the creation of accurate computational models of responses to natural stimuli. Here we demonstrate that deep convolutional neural networks (CNNs) capture retinal responses to natural scenes nearly to within the variability of a cell's response, and are markedly more accurate than linear-nonlinear (LN) models and Generalized Linear Models (GLMs). Moreover, we find two additional surprising properties of CNNs: they are less susceptible to overfitting than their LN counterparts when trained on small amounts of data, and generalize better when tested on stimuli drawn from a different distribution (e.g. between natural scenes and white noise). An examination of the learned CNNs reveals several properties.

Moreover, we find two additional surprising properties of CNNs: they are less susceptible to overfitting than their LN counterparts when trained on small amounts of data, and generalize better when tested on stimuli drawn from a different distribution (e.g. between natural scenes and white noise). An examination of the learned CNNs reveals several properties. First, a richer set of feature maps is necessary for predicting the responses to natural scenes compared to white noise.

Electroencephalogram (EEG) artifact detection in real-world settings faces significant challenges such as computational inefficiency in multi-channel methods, poor robustness to simultaneous noise, and trade-offs between accuracy and complexity in deep learning models. We propose a hybrid spectral-temporal framework for real-time detection and classification of ocular (EOG), muscular (EMG), and white noise artifacts in single-channel EEG. This method, in contrast to other approaches, combines time-domain low-pass filtering (targeting low-frequency EOG) and frequency-domain power spectral density (PSD) analysis (capturing broad-spectrum EMG), followed by PCA-optimized feature fusion to minimize redundancy while preserving discriminative information. This feature engineering strategy allows a lightweight multi-layer perceptron (MLP) architecture to outperform advanced CNNs and RNNs by achieving 99% accuracy at low SNRs (SNR -7) dB and >90% accuracy in moderate noise (SNR 4 dB). Additionally, this framework addresses the unexplored problem of simultaneous multi-source contamination(EMG+EOG+white noise), where it maintains 96% classification accuracy despite overlapping artifacts. With 30-second training times (97% faster than CNNs) and robust performance across SNR levels, this framework bridges the gap between clinical applicability and computational efficiency, which enables real-time use in wearable brain-computer interfaces. This work also challenges the ubiquitous dependence on model depth for EEG artifact detection by demonstrating that domain-informed feature fusion surpasses complex architecture in noisy scenarios.