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In recent years we have witnessed a growth in mathematics for deep learning, which has been used to solve inverse problems of partial differential equations (PDEs). However, most deep learning-based inversion methods either require paired…

We propose and study a strongly coupled PDE-ODE system with tissue-dependent degenerate diffusion and haptotaxis that can serve as a model prototype for cancer cell invasion through the extracellular matrix. We prove the global existence of…

In this paper, we propose a robust end-to-end multi-modal pipeline for place recognition where the sensor systems can differ from the map building to the query. Our approach operates directly on images and LiDAR scans without requiring any…

Monocular Depth Estimation (MDE) serves as a core perception module in autonomous driving systems, but it remains highly susceptible to adversarial attacks. Errors in depth estimation may propagate through downstream decision making and…

We propose a data-driven framework to learn interaction kernels in stochastic multi-agent systems. Our approach aims at identifying the functional form of nonlocal interaction and diffusion terms directly from trajectory data, without any a…

Binary regression models are commonly used in disciplines such as epidemiology and ecology to determine how spatial covariates influence individuals. In many studies, binary data are shared in a spatially aggregated form to protect privacy.…

We propose a framework to study tipping points in reaction-diffusion equations (RDEs) in one spatial dimension, where the reaction term decays in space (asymptotically homogeneous) and varies linearly with time (nonautonomous) due to an…

The neural ordinary differential equation (neural ODE) model has attracted increasing attention in time series analysis for its capability to process irregular time steps, i.e., data are not observed over equally-spaced time intervals. In…

Aggregation-diffusion equations are foundational tools for modelling biological aggregations. Their principal use is to link the collective movement mechanisms of organisms to their emergent space use patterns in a concrete mathematical…

Rotary Positional Embeddings (RoPE) have become the standard for Large Language Models (LLMs) due to their ability to encode relative positions through geometric rotation. However, we identify a significant limitation we term ''Spectral…

Vegetation patterns are a characteristic feature of semi-arid regions. On hillsides these patterns occur as stripes running parallel to the contours. The Klausmeier model, a coupled reaction-advection-diffusion system, is a deliberately…

Diffusion Transformer (DiT) models have achieved unprecedented quality in image and video generation, yet their iterative sampling process remains computationally prohibitive. To accelerate inference, feature caching methods have emerged by…

Spatial L{\'{e}}vy-like flights are introduced as a way in the absorbing phase transitions to produce non-local interactions. We utilize the autoencoder, an unsupervised learning method, to predict the critical points for $(1+1)$-d directed…

Monocular Depth Estimation (MDE) aims to predict pixel-wise depth given a single RGB image. For both, the convolutional as well as the recent attention-based models, encoder-decoder-based architectures have been found to be useful due to…

The quasi-steady state approximation and time-scale separation are commonly applied methods to simplify models of biochemical reaction networks based on ordinary differential equations (ODEs). The concentrations of the "fast" species are…

Motivation: Modeling biological signaling networks using ordinary differential equations (ODEs) has proven to be a powerful technique for generating insight into cellular dynamics, but it typically requires estimating rate parameters based…

Current Deep Learning methods for environment segmentation and velocity estimation rely on Convolutional Recurrent Neural Networks to exploit spatio-temporal relationships within obtained sensor data. These approaches derive scene dynamics…

We investigate predator-prey school interactions in aquatic environments using a stochastic differential equation (SDE)-based, particle-level model that incorporates attraction, repulsion, alignment, and environmental noise. Two predation…

This paper presents a semi-supervised hyperspectral unmixing solution that integrate the spatial information in the abundance estimation procedure. The proposed method is applied on a nonlinear model based on polynomial postnonlinear mixing…

The study of a particle with position-dependent effective mass (pdem), within a double heterojunction is extended into the complex domain --- when the region within the heterojunctions is described by a non Hermitian ${\cal{PT}}$ symmetric…