Computational methods involving differential equations in computer graphics, machine learning and inference problems

我研究的总体目标是为涉及学习（神经网络）、计算机图形学和机器人技术中基于物理的模拟、应用中的偏微分方程、推理问题和优化的大规模模型开发高效可靠的计算算法。重点是在时间和空间的约束微分问题，数据驱动的学习和推理（逆）问题，神经网络，非线性优化算法和数据同化方法的方法。这些问题出现在计算机图形学、图像处理、金融、模型校准和分布参数估计等基于物理的仿真中。我一生都对探索离散过程和连续过程之间的关系感兴趣，我也对几何积分感兴趣，包括对时间相关微分问题的保守离散化。虽然我的研究重点是在不同应用领域之间的知识和专业知识的转移，但在这个框架内，重点将放在计算机图形学、视觉、生物学和金融等领域的特定应用上。在接下来的五年里，我希望在以下几个方面进行工作：优化算法与神经网络的微分方程模型2。捕获约束微分方程结构和解细节的神经网络方法计算机动画中基于物理模拟的数值积分器在基于物理的动画中处理接触和摩擦学习可变形物体属性和其他逆问题我几乎所有的活动都是协作的，并涉及HQP。

The general objective of my research is to develop efficient and reliable computational algorithms for large-scale models involving learning (neural networks), physics-based simulation in computer graphics and robotics, partial differential equations in applications, inference problems, and optimization. The focus is on methods for constrained differential problems in time and space, data driven learning and inference (inverse) problems, neural nets, nonlinear optimization algorithms, and data assimilation approaches. Such problems arise in physics-based simulation in computer graphics, image processing, finance, model calibration, and distributed parameter estimation. I have a lifelong interest in exploring the relationship between discrete and continuous processes, and I'm also interested in geometric integration, including conservative discretizations for time--dependent differential problems. While my research focuses on the transfer of knowledge and expertise amongst different areas of application, emphasis will be placed within this framework on particular applications that arise in areas such as computer graphics, vision, biology, and finance. Over the next five years I expect to work on the following topics: 1. Differential equation models for optimization algorithms and neural nets 2. Neural net methods for capturing structure and solution detail of constrained differential equations 3. Numerical integrators for physics-based simulations in computer animation 4. Handling contact and friction in physics--based animation 5. Learning deformable object properties and other inverse problems Almost all of my activities are collaborative and involve HQP.