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Mathematical approach on deep neural networks
Deep neural networks underpin applications from robotics to vehicles. Ensuring their reliability in real-world scenarios is vital. This means clear specification, resilience against disruptions like adversarial attacks, and consistent performance. Beyond adversarial challenges, our research delves into the mathematical frameworks of graph neural networks and deep generative models, aiming to bridge machine intelligence with rigorous mathematical understanding.
Scientific machine learning for numerical PDEs
Scientific machine learning (SciML) emerges at the intersection of classical scientific computing and modern machine learning techniques. It offers innovative ways to tackle complex problems, including the solution of partial differential equations (PDEs) which are foundational in many scientific disciplines. Traditional numerical methods for PDEs, while robust, can be computationally intensive. SciML leverages deep learning architectures, like neural networks, to approximate solutions to these PDEs with potentially greater efficiency. This fusion promises accelerated scientific discoveries by bridging the gap between data-driven insights and domain-specific knowledge.
Numerical computations for the humid atmosphere
The inviscid hydrostatic primitive equations of the atmosphere with humidity and saturation are considered in the presence of topography. Because of its high accuracy, the hydrostatic equation is well accepted as a fundamental equation of the atmosphere, and is considered as a starting point for studying the extremely complicated atmospheric phenomena, and for predicting the weather and possible climate changes. The main focus has been to develop effective simulations such as numerical weather predictions using primitive equations of the atmosphere with saturation, which introduces a change of phase from the physical point of view. One purpose of numerical simulations is to study the rain shadow effect; a region in the leeward of mountains that receives less rainfall than the region windward of the mountains (see video below).
Deep generative models for optimal device design
Deep generative models (DGMs) are neural networks trained to emulate complex probability distributions using numerous samples. They have applications like optimal device design by predicting efficient configurations. Recently, DGMs have become a significant focus in artificial intelligence research, leading to innovations like realistic image and voice generation. Yet, there are challenges in designing and understanding their efficiency. Our work provides a mathematical framework for DGMs, covering approaches like normalizing flows, variational autoencoders, and diffusion models, complemented by experimental insights.
Numerical method for singularly perturbed problems
Attention is given to the singularly perturbed boundary value problems with small parameter, which are suitable linearizations of the Navier-Stokes or Boussinesq equations. The numerical computations of the convection-dominated equations raise substantial difficulties since, without care, the computed solutions are highly oscillatory when the diffusive term is small because the singularity at the boundary propagates inside the domain along the characteristics. The computational difficulties can be overcome with the use of a correction function usually called "corrector", which resolves the discrepancy between the zero boundary conditions and those of the outer solutions.
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