We are at the cross-roads in Computational Mathematics! The machine learning revolution is real this time around and is changing our field in a fundamental way! We may experience the sudden death of FEM and other classical numerical methods, and the rise of new and simpler methods using Deep Learning. No more we have to spend days in building elaborate grids or agonizing over solution smoothness and precise boundary conditions. Instead, we will be able to produce realistic solutions for non-sterilized computational problems in diverse physical and biological sciences. Most importantly, we will be able to discover new equations from all this data!
At CRUNCH we lead the way in this new revolution on machine learning for scientific computing on diverse applications. The postdocs, the students, and the visitors of CRUNCH lead this revolution with bold spirit and no fear for new directions and new challenging applications!
The thrust of the research of the CRUNCH group is the development of data-driven stochastic multiscale methods for physical and biological applications, specifically numerical algorithms, visualization methods and parallel software for continuum and atomistic simulations in biophysics, soft matter and functional materials, fluid and solid mechanics, biomedicine and related applications. Machine Learning for Scientific Computing is a new (disruptive) area that we emphasize, i.e., encoding conervation laws into kernels to build Physics-informed Learning Machines. Previously, numerical methods developed at CRUNCH are spectral/hp element methods, multi-element polynomial chaos, stochastic molecular dynamics (DPD), and spectral and high-order methods for fractional partial differential equations. The CRUNCH group has pioneered such methods, e.g. the spectral element method on unstructured meshes (1995), generalized polynomial chaos (gPC) for uncertainty quantification, rigorous coarse grained molecular methods (2010), and poly-fractonomials for fractional operators (2015). More recently we have focused on numerical Gaussian Processes that allow us to solve PDEs from noisy measurements only without the tyranny of building elaborate grids! We hace also employed deep neural networks to solve complex PDEs in continuos space-time domains for first time! Wee are also interested to employ fractional operators and Gaussian processes to discover Hidden Physics Models -- our group has pionerred this!
Funding is currently provided by DOE, AFOSR, DARPA, ARO, ARL, NIH and NSF.