方程组的求解问题在几何造型系统等有着广泛的应用。本项目以CAD中求交、干涉检测、最近距离计算、PDE造型等为例，研究非线性方程组的几何求解理论与方法。具体内容包括：(1)非线性函数新型包围盒的理论和方法研究，可望为任意非线性方程组提供稳定高效的几何裁剪方法及其较坚实的理论基础；(2)操作直观、局部控制性好、高收敛阶的PDE方程包围盒求解技术，使得带参数PDE方程结合深度学习的求解成为可能；(3)直接逼近曲面/曲面间交线高逼近阶的近似算法，可望显著减少待计算交点的数目并提升计算稳定性与效率；(4)非线性方程组高效稳定的几何迭代求解方法；(5)基于拓扑推理修正的求交算法，以提高拓扑成环正确性与系统稳定性。本项目若成功实施，可望为任意非线性方程组提供稳定高效的几何裁剪方法，显著提升CAD中求交、最近距离计算、偏微分方程求解、拓扑成环等问题的效率、精度和稳定性；同时可望更好地满足虚拟现实等应用。

Solving a system of non-linear equations has wide applications in computer graphics and geometric design. This project investigates geometric methods and relevant theories for solving a system of non-linear equations using typical examples in Computer Aided Design (CAD), such as curve/surface and surface/surface intersection, collision detection, minimum distance computation and the solution of Partial Differential Equations (PDE). Main issues to be addressed in the project include: (1) Bounding box theory and methods of non-linear functions, curves and surfaces, which is promising for developing efficient and robust geometric clipping methods for solving a system of non-linear equations; (2) Intuitive and locally controllable bounding box with high approximation order and with further integration of novel deep learning techniques for solving partial differential equations; (3) Efficient and stable algorithms for directly computing surface/surface intersections; (4) Fast, robust and iterative geometric methods for solving a system of nonlinear equations; (5) Intersection algorithms based on topology evolution for improving the reliability and robustness of the resulting solutions. Upon successful completion, it is expected to provide a set of robust and efficient geometric clipping methods for solving an arbitrary system of nonlinear equations, and to improve computational stability, efficiency and accuracy of algorithms for computating intersections, minimum distance and collision detection, and for solving partial differential equations. In addition, it is also expected to meet real-time computational requirements for virtual reality and other applications.