Abstract:  Mathematical theorem proving has long been a goal in artificial intelligence (AI), but progress has been slow due to the scarcity of human-generated proofs that can be translated into machine-readable formats. Geometry, in particular, poses unique challenges due to the difficulty of representing geometric figures and relations in a symbolic format that machines can easily process. Researchers at Google DeepMind have recently proposed a novel architecture, AlphaGeometry, which utilizes a neuro-symbolic approach. This system integrates a neural language model trained on a large corpus of synthetic theorems and proofs with a symbolic reasoning engine that ensures logical deductive correctness. On the IMO-AG-30 benchmark, AlphaGeometry achieved remarkable accuracy, solving 25 out of 30 problems, nearly matching the gold medalist benchmark of 25.9/30. In this talk, I will provide an overview of the construction and integration of the symbolic reasoning engine and the neural language model in AlphaGeometry. Specifically, I will detail how the symbolic reasoning engine, comprising the Deductive Database (DD) and Algebraic Reasoning (AR), operates. The DD manages formal geometric deductions by applying predefined theorems and axioms, while AR complements this by resolving complex algebraic expressions inherent in geometric proofs. Furthermore, I will explain the Neural Language Model utilized in AlphaGeometry, highlighting how it differs from known large language models such as GPT-4. This comparison will shed light on the potential of general-purpose language models in solving mathematical problems, offering insights into the broader capabilities of large language models in Math.

Abstract:  In this talk,  I will present some new developments of transformers in the time series.  In addition, I will introduce the state space model, in particular Mamba and its application in time series if time allows.

Multiscale time dependent partial differential equations (PDE) are challenging to  compute by traditional mesh based methods especially when their solutions develop  large gradients or concentrations at unknown locations.  Particle methods, based on microscopic aspects of the PDEs, are mesh free and self-adaptive,  yet still expensive when a long time or a resolved computation is necessary. 

We present DeepParticle, an integrated deep learning, optimal transport (OT),  and interacting particle (IP) approach, to speed up  generation and prediction of PDE dynamics through two case studies on transport  in fluid flows with chaotic streamlines:  

1) large time front speeds of Fisher-Kolmogorov-Petrovsky-Piskunov equation (FKPP); 

2) Keller-Segel (KS) chemotaxis system modeling bacteria evolution in the presence of a chemical attractant. 

Analysis of FKPP reduces the problem to a computation of principal eigenvalue of an advection-diffusion operator. A normalized Feynman-Kac representation  makes possible a genetic IP algorithm to  evolve the initial uniform particle distribution to a large time invariant measure  from which to extract front speeds. The invariant measure is parameterized  by a physical parameter (the Peclet number). We train a light weight deep neural network with local and global skip connections  to learn this family of invariant measures. The training data come  from IP computation in three dimensions at a few sample Peclet numbers. 

The training objective being minimized is a discrete Wasserstein distance in OT theory. The trained network predicts a more concentrated invariant measure at a larger Peclet number  and also serves as a warm start to accelerate IP computation. The KS is formulated as a McKean-Vlasov equation (macroscopic limit) of a stochastic IP system. The DeepParticle framework extends and  learns to generate various finite time bacterial aggregation patterns.