AI advances mathematics: concretely in practice, unconsciously in theory There are three long-standing pursuits in mathematics for any complex domains: A unified theoretical framework that covers both forward and inverse problems A solver for function composition A solution to

"AI advances mathematics: concretely in practice, unconsciously in theory" There are three long-standing pursuits in mathematics for any complex domains: A unified theoretical framework that covers both forward and inverse problems A solver for function composition A solution

Groups, Rings, and Fields (群, 环, 域) This really surprised us. What this means is that a neural network thinks more like a human, more specifically, like a mathematician. In classical mechanics or fluid dynamics, we compute intrinsic physical quantities like stress, strain,

**Primitive Neural Network Mathematics** Here are 5 core questions in our mind: 1. Why does a Transformer require high-dimensional token representations? 2. Why can neural networks fit or mimic even random data so well? 3. How do networks handle the stochastic nature of the world

*** Neural Network Learnability *** Our study concludes with seemingly contradictory properties of neural networks: abundance, redundancy and abstraction. Traditional abstraction (e.g., in human reasoning or symbolic logic) often moves upward: from specifics to general

Main point of this paper: High-order interactions in curved manifolds. This is closer to Deep Manifold theory than any of the 1,000+ papers I’ve reviewed. I’ve collected a list of related “high order” concepts: High-order Regularization High-order Feature Interaction High-order

I spent 11 years (1988–1999) working on numerical computation for discontinuous media such as sand, gravel, and rock. Since 2016, I’ve been focused on AI. The emergence of Physics-Informed Neural Networks (PINNs) and their associated neural operators naturally caught my

agree ? AI is fooling our brain, ego, and vanity effortlessly with her inner beauty and a touch of evil foreplay

** Historical Lesson: FEM, UC Berkeley, and the Supercomputing Era ** Professor Ray W. Clough, in the Department of Civil Engineering at UC Berkeley, is considered one of the founding figures of the Finite Element Method (FEM). In 1960, he published a seminal paper that formally

** Neural Network Mirage ** There are many research efforts suggesting, or at least assuming, that neural networks can be explained through a variety of established mathematical and statistical frameworks. Some view them as the Monte Carlo method or Gaussian fields, others as

I have tremendous respect for Prof. Zienkiewicz for his contributions to FEM and its mathematics. Yet, it was also an old wound, one that not many people knew about. Over the past 12 months, especially as we have gained a clearer sense of neural network mathematics. I see his

What I’m seeing is that most enterprises are adopting a copy-paste approach to generative AI, simply replicating generic models without ML depth. Many so-called GenAI leadership are "GenAI in name only," (AINO) Like any promising new technology, it’s best to work backwards—start

real intelligence is in real world... Most AI researchers are NOT living in real world

Stationary, Symmetry and Neural Plasticity

Elon, Demis's criticism of Yann LeCun's interview is unnecessarily harsh, especially when there is no commonly accepted definition of intelligence, general intelligence, or universal intelligence. If you listen to LeCun full interview, He makes a fair number of good points.

Deep Manifold Mathematical Lineage