Recent advancements in large language models (LLMs) based on transformer architectures have sparked significant interest in understanding their inner workings. In this paper, we introduce a novel approach to modeling transformer architectures using highly flexible non-autonomous neural ordinary differential equations (ODEs). Our proposed model parameterizes all weights of attention and feed-forward blocks through neural networks, expressing these weights as functions of a continuous layer index. Through spectral analysis of the model's dynamics, we uncover an increase in eigenvalue magnitude that challenges the weight-sharing assumption prevalent in existing theoretical studies. We also leverage the Lyapunov exponent to examine token-level sensitivity, enhancing model interpretability. Our neural ODE transformer demonstrates performance comparable to or better than vanilla transformers across various configurations and datasets, while offering flexible fine-tuning capabilities that can adapt to different architectural constraints.

Differential equations play important roles in modeling complex physical systems. Recent advances present interesting research directions by combining differential equations with neural networks. By including noise, stochastic differential equations (SDEs) allows us to model data with uncertainty and measure imprecision. There are many variants of noises known to exist in many real-world data. For example, previously white noises are idealized and induced by Brownian motions. Nevertheless, there is a lack of machine learning models that can handle such noises. In this paper, we introduce a generalized fractional white noise to existing models and propose an efficient approximation of noise sample paths based on classical integration methods and sparse Gaussian processes. Our experimental results demonstrate that the proposed model can capture noise characteristics such as continuity from various time series data, therefore improving model fittings over existing models. We examine how we can apply our approach to score-based generative models, showing that there exists a case of our generalized noise resulting in a better image generation measure.

Choosing a proper set of kernel functions is an important problem in learning Gaussian Process (GP) models since each kernel structure has different model complexity and data fitness. Recently, automatic kernel composition methods provide not only accurate prediction but also attractive interpretability through search-based methods. However, existing methods suffer from slow kernel composition learning. To tackle large-scaled data, we propose a new sparse approximate posterior for GPs, MultiSVGP, constructed from groups of inducing points associated with individual additive kernels in compositional kernels. We demonstrate that this approximation provides a better fit to learn compositional kernels given empirical observations. We also provide theoretically justification on error bound when compared to the traditional sparse GP. In contrast to the search-based approach, we present a novel probabilistic algorithm to learn a kernel composition by handling the sparsity in the kernel selection with Horseshoe prior. We demonstrate that our model can capture characteristics of time series with significant reductions in computational time and have competitive regression performance on real-world data sets.