Neural networks & partial differential equations
Keywords: ConvNet, ResNet, PDE, heat equation, optimal control, flow of vector field, rotation & scale-invariant,
Even bigger is the claim to explain (deep) neural networks by way of partial differential equations (PDE).
Instead of presenting a polished theory, if there is one?
I'd like to stay close to the path that led me to this.
Naturally many have noticed it before, including (but not limitited)
Weinan E, A proposal on machine learning via dynamical systems (2017)
Lars Ruthotto, Eldad Haber, Deep Neural Networks Motivated by Partial Differential Equations (2018)
Ricky T. Q. Chen, et al., Neural Ordinary Differential Equations (2018)
A workshop on PDE and Inverse Problem Methods in Machine Learning (2020)
This course examines electric and magnetic quasistatic forms of Maxwell's equations applied to dielectric, conduction, and magnetization boundary value problems. Topics covered include: electromagnetic forces, force densities, and stress tensors, including magnetization and polarization; thermodynamics of electromagnetic fields, equations of motion, and energy conservation; applications to synchronous, induction, and commutator machines; sensors and transducers; microelectromechanical systems; propagation and stability of electromechanical waves; and charge transport phenomena. Acknowledgments The instructor would like to thank Thomas Larsen and Matthew Pegler for transcribing into LaTeX the homework problems, homework solutions, and exam solutions.
This course will give a detailed introduction to learning theory with a focus on the classification problem. It will be shown how to obtain (pobabilistic) bounds on the generalization error for certain types of algorithms. The main themes will be: * probabilistic inequalities and concentration inequalities * union bounds, chaining * measuring the size of a function class, Vapnik Chervonenkis dimension, shattering dimension and Rademacher averages * classification with real-valued functions Some knowledge of probability theory would be helpful but not required since the main tools will be introduced.
R. Neal. (1992)cite arxiv:hep-lat/9208011Comment: 15 pages, 4 figures (only one of which is present), New version with corrected LaTex, Submitted to J. of Comp. Physics.