CIF: Small: Towards a Control Framework for Neural Generative Modeling

产生式机器学习是一种神经网络，它对输入数据进行训练，然后生成具有相似特征的新数据。特别是，生成性机器学习已经被用于创建图像，最近的工作集中在基于扩散的神经网络上，该神经网络由经过训练以逐渐消除噪声的图像到图像转换网络来驱动。本项目采用控制论方法，为理解基于扩散的产生式机器学习提供了一个理论框架。通过将扩散模型的操作框定为最优控制问题，研究人员试图建立与偏微分方程域和随机微分方程域的基本联系，目的是了解生成模型的可控性、可表现性、计算复杂性和稳健性。与目前严重依赖经验设计、理论基础有限的扩散模型相比，该项目寻求在降低计算成本的情况下极大地改善对生成性网络的训练。鉴于目前在许多应用中对生成模型的广泛兴趣，该项目具有连接多个技术社区的潜力，特别是考虑到其理论重点。研究人员还计划将本科生纳入研究努力，并将生成性机器学习的伦理和社会分支纳入他们的教育活动。这个关于神经生成性建模的控制框架的项目将在生成性建模的新背景下将几个不同的智力链编织在一起-即随机轨迹和系综的控制，偏微分方程组(PDE)的控制，以及用于多尺度图像分析的PDE的经典理论。该研究计划围绕三个主要方向展开：(1)扩散过程的控制；(2)密度空间的控制；(3)图像偏微分方程的控制。第一个方向将通过利用扩散过程的最优控制技术来开发用于生成性建模的第一原理框架。第二个方向将建立在这个框架的基础上，将生成性建模表述为概率密度空间中的控制问题，潜在地绕过了显式使用随机微分方程。这将把产生式建模的设置与描述在流畅作用下概率密度的演化的Liouville类型偏微分方程组的最优控制问题联系起来。最后，第三个方向将把这次调查的范围扩大到从图像处理的多尺度分析公理中产生的更一般的PDE类别。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Generative machine learning are neural networks that are trained on input data so as to then generate new data with similar characteristics. In particular, generative machine learning has been used for the creation of images, and recent work has focused on diffusion-based neural networks driven by an image-to-image translation network trained to gradually remove noise. This project adopts control-theory methodologies to provide a theoretical framework for understanding diffusion-based generative machine learning. By framing the operation of diffusion models as an optimal control problem, the investigators seek to establish a foundational link to the domains of partial and stochastic differential equations with the aim to understand generative models in terms of their controllability, expressiveness, computational complexity, and robustness. In contrast to current diffusion models which rely heavily on empirical design with limited theoretical foundation, the project seeks to greatly improve training for generative networks at a reduced computational cost. Given the current widespread interest in generative models in numerous applications, the project has the potential to bridge multiple technical communities, particularly given its theoretical focus. The investigators also plan to include undergraduates in the research endeavors as well as to incorporate ethical and societal ramifications of generative machine learning in their educational activities.This project on a control framework for neural generative modeling will weave together several distinct intellectual strands in the novel context of generative modeling – namely, control of stochastic trajectories and ensembles, control of partial differential equations (PDEs), and classical theories of PDEs for multiscale image analysis. The research program is articulated around three major directions: (1) control of diffusion processes; (2) control in the space of densities; (3) control of image PDEs. The first direction will develop a first-principles framework for generative modeling by drawing on the techniques of optimal control of diffusion processes. The second direction will build on this framework to phrase generative modeling as a control problem in the space of probability densities, potentially bypassing explicit use of stochastic differential equations. This will connect the setting of generative modeling to the problem of optimal control of PDEs of Liouville type that describe the evolution of probability densities under the action of smooth flows. Finally, the third direction will expand the scope of this inquiry to the more general class of PDEs arising from the axioms of multiscale analysis in the context of image processing.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.