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Life sciences · Preprint
arXiv · September 4, 2026
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This preprint introduces solution-space PDE-Dirichlet, a protocol for quantifying heterogeneity in federated learning of PDEs via optimal transport over solution bins. Controlled experiments across seven synthetic tasks demonstrate that lower Dirichlet concentration increases solution distance and optimization heterogeneity, with task-dependent performance degradation (largest effect 4.157 percentage points on low-viscosity Burgers). The work provides a reproducible geometric framework for evaluating non-IID federated PDE learning but remains computational and has not been peer reviewed.
Controlled computational study; multi-task federated learning simulation. Synthetic PDE tasks (Burgers equation, and six others specified as 'public PDE tasks'); no human or patient population.. Intervention: Solution-space PDE-Dirichlet protocol with varying Dirichlet concentration (lower concentration = higher heterogeneity); federated neural-operator training across heterogeneous solution bins.. Compared with: Gradient disagreement, local-update dispersion, and parameter divergence under different heterogeneity levels; effect of additional communication and smoother dynamics on final error gap..
Lower Dirichlet concentration consistently increases realized solution distance and optimization heterogeneity across seven PDE tasks. Largest final error degradation occurs for low-viscosity Burgers equation, reaching 4.157 percentage points under the most heterogeneous setting. Inverse relation established between population allocation heterogeneity and Dirichlet concentration.
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A methodological and computational study introducing a new protocol for federated PDE learning with controlled experiments across seven tasks, but no clinical or clinical-adjacent outcomes, no peer review, and no comparison to established clinical practice.
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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.
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