Simulation-based techniques such as variants of stochastic Runge-Kutta are the de facto approach for inference with stochastic differential equations (SDEs) in machine learning. These methods are general-purpose and used with parametric and non-parametric models, and neural SDEs. Stochastic Runge-Kutta relies on the use of sampling schemes that can be inefficient in high dimensions. We address this issue by revisiting the classical SDE literature and derive direct approximations to the (typically intractable) Fokker-Planck-Kolmogorov equation by matching moments. We show how this workflow is fast, scales to high-dimensional latent spaces, and is applicable to scarce-data applications, where a non-parametric SDE with a driving Gaussian process velocity field specifies the model.

Simulation-based techniques such as variants of stochastic Runge–Kutta are the de facto approach for inference with stochastic differential equations (SDEs) in machine learning. These methods are general-purpose and used with parametric and non-parametric models, and neural SDEs. Stochastic Runge–Kutta relies on the use of sampling schemes that can be inefficient in high dimensions. We address this issue by revisiting the classical SDE literature and derive direct approximations to the (typically intractable) Fokker–Planck–Kolmogorov equation by matching moments. We show how this workflow is fast, scales to high-dimensional latent spaces, and is applicable to scarce-data applications, where a non-parametric SDE with a driving Gaussian process velocity field specifies the model.

Graph generation is fundamental in diverse scientific applications, due to its ability to reveal the underlying distribution of complex data, and eventually generate new, realistic data points. Despite the success of diffusion models in this domain, those face limitations in sampling efficiency and flexibility, stemming from the tight coupling between the training and sampling stages. To address this, we propose DeFoG, a novel framework using discrete flow matching for graph generation. DeFoG employs a flow-based approach that features an efficient linear interpolation noising process and a flexible denoising process based on a continuous-time Markov chain formulation. We leverage an expressive graph transformer and ensure desirable node permutation properties to respect graph symmetry. Crucially, our framework enables a disentangled design of the training and sampling stages, enabling more effective and efficient optimization of model performance. We navigate this design space by introducing several algorithmic improvements that boost the model performance, consistently surpassing existing diffusion models. We also theoretically demonstrate that, for general discrete data, discrete flow models can faithfully replicate the ground truth distribution - a result that naturally extends to graph data and reinforces DeFoG's foundations. Extensive experiments show that DeFoG achieves state-of-the-art results on synthetic and molecular datasets, improving both training and sampling efficiency over diffusion models, and excels in conditional generation on a digital pathology dataset.

Non-linear partial differential Kolmogorov equations are successfully used to describe a wide range of time dependent phenomena, in natural sciences, engineering or even finance. For example, in physical systems, the Allen-Cahn equation describes pattern formation associated to phase transitions. In finance, instead, the Black-Scholes equation describes the evolution of the price of derivative investment instruments. Such modern applications often require to solve these equations in high-dimensional regimes in which classical approaches are ineffective. Recently, an interesting new approach based on deep learning has been introduced by E, Han, and Jentzen [1], [2]. The main idea is to construct a deep network which is trained from the samples of discrete stochastic differential equations underlying Kolmogorov's equation. The network is able to approximate, numerically at least, the solutions of the Kolmogorov equation with polynomial complexity in whole spatial domains. In this contribution we study variants of the deep networks by using different discretizations schemes of the stochastic differential equation. We compare the performance of the associated networks, on benchmarked examples, and show that, for some discretization schemes, improvements in the accuracy are possible without affecting the observed computational complexity. Algorithms based on the theory of deep learning have become essential in a wide variety scientific disciplines. In this paper we are concerned with applications to the solution of high-dimensional semi-linear parabolic partial differential equations (PDEs). The importance of such PDEs in finance, mathematics, natural science and engineering, cannot be understated and vast amounts of efforts have been deployed to develop numerical solution methods.

The diffusion approximation of stochastic gradient descent (SGD) in current literature is only valid on a finite time interval. In this paper, we establish the uniform-in-time diffusion approximation of SGD, by only assuming that the expected loss is strongly convex and some other mild conditions, without assuming the convexity of each random loss function. The main technique is to establish the exponential decay rates of the derivatives of the solution to the backward Kolmogorov equation. The uniform-in-time approximation allows us to study asymptotic behaviors of SGD via the continuous stochastic differential equation (SDE) even when the random objective function $f(\cdot;\xi)$ is not strongly convex.