Solution-space heterogeneity shapes federated learning dynamics across partial differential equations

To address the lack of a universal definition for non-independent and identically distributed data in federated learning for partial differential equations (PDEs), researchers proposed a new protocol called solution-space PDE-Dirichlet. This protocol converts continuous supervised responses into reusable solution regions and uses optimal transmission to quantify the degree of separation between clients. Experiments were conducted on seven controlled and public PDE tasks, three families of neural operators, and five random seeds. The results showed that lower Dirichlet concentrations continuously increased the distance between solutions and the heterogeneity of optimization; the final error degradation depended on the specific task. For example, in the low-viscosity Burgers equation, the maximum error reached 4.157 percentage points under the most heterogeneous settings. Although additional communication or smoother dynamics could reduce the final gap, parameter separation still persisted. These results distinguished between reproducible geometric mechanisms and task-related generalization results, providing a common basis for evaluating non-independent and identically distributed federated PDE learning.