6 papers
A Limit Theory of Foundation Models: A Mathematical Approach to Understanding Emergent Intelligence and Scaling Laws
Jun Shu, Junxiong Jia, Deyu Meng +1
Emergent intelligence have played a major role in the modern AI development. While existing studies primarily rely on empirical observations to characterize this phenomenon, a rigo…
Consistency of Variational Inference for Nonlinear Inverse Problems of Partial Differential Equations
Shaokang Zu, Junxiong Jia, Deyu Meng
We investigate the convergence rates of variational posterior distributions for statistical inverse problems involving nonlinear partial differential equations (PDEs). Departing fr…
Stochastic gradient descent based variational inference for infinite-dimensional inverse problems
Jiaming Sui, Junxiong Jia, Jinglai Li
This paper introduces two variational inference approaches for infinite-dimensional inverse problems, developed through gradient descent with a constant learning rate. The proposed…
Sequential Monte Carlo with Gaussian Mixture Approximation for Infinite-Dimensional Statistical Inverse Problems
Haoyu Lu, Junxiong Jia, Deyu Meng
By formulating the inverse problem of partial differential equations (PDEs) as a statistical inference problem, the Bayesian approach provides a general framework for quantifying u…
Nonparametric Prior Learning in Differential Equation Modeling
Junxiong Jia, Deyu Meng, Zongben Xu +1
This paper addresses Bayesian inference related to partial differential equations (PDEs), particularly nonparametric regression constrained by PDEs. To effectively encode prior inf…
Functional normalizing flow for statistical inverse problems of partial differential equations
Yang Zhao, Haoyu Lu, Junxiong Jia +1
Inverse problems of partial differential equations are ubiquitous across various scientific disciplines and can be formulated as statistical inference problems using Bayes' theorem…