Deep Learning for Scientific Computing - Two Stories on the Gap Between Theory & Practice - Ben Adcock

Explore deep learning applications in scientific computing through two case studies highlighting the gap between theory and practice. Delve into high-dimensional function approximation and inverse problems for imaging, examining limitations of current approaches in stability, generalization, and practical performance. Discover recent theoretical advancements demonstrating the potential of deep neural networks to match best-in-class schemes in both settings. Gain insights into achieving robust, reliable, and improved practical performance in scientific computing using deep learning techniques. Learn about challenges in parametric modeling, limited performance for smooth function approximation, and unpredictable generalization in inverse problems. Understand the universal instability theorem and its implications for deep learning in scientific applications.

Intro
Main collaborators
Deep Learning (DL) for scientific computing
This talk two stories on the theory-practice gap
Parametric modelling
Challenges
MLFA: examining the practical performance of DNNS
Limited performance for smooth, univariate approximation
Balancing architecture size
Smooth, multivariate functions
Piecewise smooth function approximation
Theoretical insights
DNN existence theory for holomorphic functions
Practical DNN existence theorem: Hilbert-valued case
Discussion
Deep learning for inverse problems
Further examples
These are not rare events
Unpredictable generalization
The universal instability theorem
Hallucinations in practice
Construction: unravelling and restarts
FIRENETS example
Conclusions