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）的大规模模型开发高效可靠的计算算法，计算机图形学和机器人技术中基于物理的仿真，偏微分方程的应用，推理问题和优化。重点是时间和空间中的约束微分问题，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中的基于物理的仿真中。我有一个终身的兴趣在探索离散和连续过程之间的关系，我也有兴趣在几何积分，包括保守的离散化的时间依赖微分问题。 虽然我的研究重点是不同应用领域之间的知识和专业知识的转移，但重点将放在计算机图形学，视觉，生物学和金融等领域出现的特定应用程序的框架内。在接下来的五年里，我将致力于以下几个方面的工作：1。优化算法和神经网络的微分方程模型2.捕捉约束微分方程结构和解细节的神经网络方法3.计算机动画中基于物理模拟的数值积分器4.处理物理学中的接触和摩擦--基于动画5.学习可变形物体的属性和其他逆问题几乎我所有的活动都是合作的，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.