Inducing Mixture Consistency in Nonlinear Aerosol Set Encoders by Training

Linear aerosol operations such as mixing and spatial coarse-graining require latent representations that commute with population addition. Scale--shape encoders guarantee this property by design; we test whether it can instead be induced in a Deep Sets encoder whose nonlinear outer map breaks mixture consistency. We train with separate representation-level consistency and mixture-output losses on particle-resolved aerosol populations evaluated in 64-way mixtures. For every tested nontrivial training arity, the consistency loss reduces the median encoder gap from $10^6$--$10^7$ to approximately $10^{-7}$, a relative gap at double-precision roundoff, and the learned outer map is affine to within roundoff at sampled convex mixtures. Mixture-output training reduces diagnostic error for both nonlinear and linear-by-construction encoders, which track closely. These results show that training can retrofit empirical mixture consistency into a nonlinear encoder, although architectural linearity remains preferable when an exact guarantee is available.