发表机构
College of Computer Science and Technology（计算机科学与技术学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对偏微分方程联邦学习缺乏可迁移非IID数据定义的问题，提出PDE-Dirichlet协议，揭示解空间异质性对联邦学习动态的影响，为评估非IID联邦PDE学习提供通用基础。
AI 中文摘要
联邦科学机器学习允许各机构在不集中本地物理数据的情况下训练神经代理模型，但针对偏微分方程（PDE）的研究缺乏可迁移的非独立同分布（non-IID）数据定义。现有协议会根据方程特定规则对坐标、系数、边界条件或几何结构进行划分。本文提出solution-space PDE-Dirichlet协议，该协议将连续监督响应转换为可复用的解空间分箱，并通过分箱几何上的最优传输量化客户端间的实际分离程度。我们推导了总体分配异质性与Dirichlet浓度之间的精确反比关系，明确了响应异质性引发梯度分歧、局部更新离散化及参数发散的条件。在7项受控及公开的PDE任务、3类神经算子族、5个随机种子的实验中，更低的浓度会持续增大实际解空间距离与优化异质性。最终误差的退化程度具有任务依赖性：低粘度Burgers方程受影响最大，在异质性最高的设置下误差达4.157个百分点；而额外通信或更平滑的动态特性可在参数持续分离的情况下缩小最终误差差距。这些结果将可复现的几何机制与任务依赖的泛化结果区分开来，为评估非IID联邦PDE学习提供了通用基础。
英文摘要
Federated scientific machine learning enables institutions to train neural surrogates without centralizing local physical data, yet studies of partial differential equations (PDEs) lack a transferable definition of non-independent and identically distributed data. Existing protocols partition coordinates, coefficients, boundary conditions, or geometries according to equation-specific rules. Here, we introduce solution-space PDE-Dirichlet, a protocol that converts continuous supervised responses into reusable solution bins and quantifies the realized separation between clients through optimal transport over the geometry of these bins. We derive an exact inverse relation between population allocation heterogeneity and the Dirichlet concentration, and we establish conditions under which response heterogeneity induces gradient disagreement, local-update dispersion, and parameter divergence. Across seven controlled and public PDE tasks, three neural-operator families, and five random seeds, a lower concentration consistently increases the realized solution distance and optimization heterogeneity. The degradation in final error is task dependent: the largest effect occurs for low-viscosity Burgers, reaching 4.157 percentage points under the most heterogeneous setting, whereas additional communication or smoother dynamics can reduce the final gap despite persistent parameter separation. These results distinguish a reproducible geometric mechanism from task-dependent generalization outcomes and provide a common basis for evaluating non-IID federated PDE learning.
