龙宗加（西湖大学工学院）

唐贻发 研究员

When the Domain Is Unknown: A Neural-Operator Framework for Free Boundary Problems via Topological Conjugacy

7月13日（周一）15:00-16:00，南楼205

Free boundary problems (FBPs) pervade science and engineering—from phase transitions and tumour growth to coupled multi-physics systems—and their defining difficulty is that the solution domain itself evolves over time and is unknown in advance. Although neural operators have recently delivered order-of-magnitude speed-ups in solving partial differential equations, their theoretical foundation requires the output function to live on a fixed, prescribed domain, leaving FBPs out of reach. This talk introduces the Free Boundary Neural Operator (FBNO). The key idea is to treat an FBP as an infinite-dimensional dynamical system and, through topological conjugacy, relate it to a "conjugate system" defined on a fixed reference domain. FBNO simultaneously learns the flow map of this conjugate system and the homeomorphism linking the two, enabling prediction of both the physical fields and the evolving boundary without any prior geometric knowledge. We further establish an approximation theorem tailored to FBPs that guarantees the theoretical feasibility of this construction. Across the Stefan phase-transition problem, a thermal–structural multi-physics problem, and tumour growth on non-convex geometries, FBNO keeps relative L2 errors below 1% while accelerating inference by up to roughly four orders of magnitude over conventional numerical solvers. In the tumour-growth case, it rapidly predicts tumour size and nutrient distribution, opening a path toward personalized treatment planning. This work offers a new route to free boundary simulation that unites accuracy and efficiency.

报告人简介：龙宗加，西湖大学工学院直博生，研究兴趣集中于科学机器学习与神经算子，关注如何用深度学习方法高效、可靠地求解偏微分方程与复杂物理系统，已在nature 子刊上发表相关研究成果，参与2024年央国企十大国之重器的电力系统基础模型（南网"驭电"项目）的大规模分布式训练，以及面向托卡马克聚变的数据驱动模拟研究。