Nicole Tianjiao Yang

Paul Hagemann, Sophie Mildenberger, Lars Ruthotto et al.

Score-based diffusion models (SBDM) have recently emerged as state-of-the-art approaches for image generation. Existing SBDMs are typically formulated in a finite-dimensional setting, where images are considered as tensors of finite size. This paper develops SBDMs in the infinite-dimensional setting, that is, we model the training data as functions supported on a rectangular domain. In addition to the quest for generating images at ever-higher resolutions, our primary motivation is to create a well-posed infinite-dimensional learning problem that we can discretize consistently on multiple resolution levels. We thereby intend to obtain diffusion models that generalize across different resolution levels and improve the efficiency of the training process. We demonstrate how to overcome two shortcomings of current SBDM approaches in the infinite-dimensional setting. First, we modify the forward process using trace class operators to ensure that the latent distribution is well-defined in the infinite-dimensional setting and derive the reverse processes for finite-dimensional approximations. Second, we illustrate that approximating the score function with an operator network is beneficial for multilevel training. After deriving the convergence of the discretization and the approximation of multilevel training, we demonstrate some practical benefits of our infinite-dimensional SBDM approach on a synthetic Gaussian mixture example, the MNIST dataset, and a dataset generated from a nonlinear 2D reaction-diffusion equation.

Eric Gelphman, Deepanshu Verma, Nicole Tianjiao Yang et al.

Optimal feedback control with implicit Hamiltonians poses a fundamental challenge for learning-based value function methods due to the absence of closed-form optimal control laws. Recent work~\cite{gelphman2025end} introduced an implicit deep learning approach using Jacobian-Free Backpropagation (JFB) to address this setting, but only established sample-wise descent guarantees. In this paper, we establish convergence guarantees for JFB in the stochastic minibatch setting, showing that the resulting updates converge to stationary points of the expected optimal control objective. We further demonstrate scalability on substantially higher-dimensional problems, including multi-agent optimal consumption and swarm-based quadrotor and bicycle control. Together, our results provide both theoretical justification and empirical evidence for using JFB in high-dimensional optimal control with implicit Hamiltonians.

Wei Deng, Yu Chen, Nicole Tianjiao Yang et al.

Diffusion models have become the go-to method for large-scale generative models in real-world applications. These applications often involve data distributions confined within bounded domains, typically requiring ad-hoc thresholding techniques for boundary enforcement. Reflected diffusion models (Lou23) aim to enhance generalizability by generating the data distribution through a backward process governed by reflected Brownian motion. However, reflected diffusion models may not easily adapt to diverse domains without the derivation of proper diffeomorphic mappings and do not guarantee optimal transport properties. To overcome these limitations, we introduce the Reflected Schrodinger Bridge algorithm: an entropy-regularized optimal transport approach tailored for generating data within diverse bounded domains. We derive elegant reflected forward-backward stochastic differential equations with Neumann and Robin boundary conditions, extend divergence-based likelihood training to bounded domains, and explore natural connections to entropic optimal transport for the study of approximate linear convergence - a valuable insight for practical training. Our algorithm yields robust generative modeling in diverse domains, and its scalability is demonstrated in real-world constrained generative modeling through standard image benchmarks.

Eric Gelphman, Deepanshu Verma, Nicole Tianjiao Yang et al.

Neural network approaches that parameterize value functions have succeeded in approximating high-dimensional optimal feedback controllers when the Hamiltonian admits explicit formulas. However, many practical problems, such as the space shuttle reentry problem and bicycle dynamics, among others, may involve implicit Hamiltonians that do not admit explicit formulas, limiting the applicability of existing methods. Rather than directly parameterizing controls, which does not leverage the Hamiltonian's underlying structure, we propose an end-to-end implicit deep learning approach that directly parameterizes the value function to learn optimal control laws. Our method enforces physical principles by ensuring trained networks adhere to the control laws by exploiting the fundamental relationship between the optimal control and the value function's gradient; this is a direct consequence of the connection between Pontryagin's Maximum Principle and dynamic programming. Using Jacobian-Free Backpropagation (JFB), we achieve efficient training despite temporal coupling in trajectory optimization. We show that JFB produces descent directions for the optimal control objective and experimentally demonstrate that our approach effectively learns high-dimensional feedback controllers across multiple scenarios involving implicit Hamiltonians, which existing methods cannot address.

Yu Chen, Wei Deng, Shikai Fang et al.

The Schrödinger bridge problem (SBP) is gaining increasing attention in generative modeling and showing promising potential even in comparison with the score-based generative models (SGMs). SBP can be interpreted as an entropy-regularized optimal transport problem, which conducts projections onto every other marginal alternatingly. However, in practice, only approximated projections are accessible and their convergence is not well understood. To fill this gap, we present a first convergence analysis of the Schrödinger bridge algorithm based on approximated projections. As for its practical applications, we apply SBP to probabilistic time series imputation by generating missing values conditioned on observed data. We show that optimizing the transport cost improves the performance and the proposed algorithm achieves the state-of-the-art result in healthcare and environmental data while exhibiting the advantage of exploring both temporal and feature patterns in probabilistic time series imputation.