Monotone CDT autoencoder bottleneck

Use this as the bottleneck of a convolutional or MLP autoencoder for nonnegative 1D fields, especially when the downstream model must decode many transported states. At preprocessing time, normalize each training field u to mass M and compute K quantile samples \widehat q_k=F_u^\dagger(p_k) at p_k=(k-0.5)/K; these are the CDT samples in probability coordinates. The encoder E_\phi maps \widehat q to z\in\mathbb R^d. The decoder produces h=D_ heta(z)\in\mathbb R^K, then applies the exact monotonic parameterization \delta_k=softplus(h_k)+epsilon and q_k=q_min+L cumulative_sum(delta)_k/sum(delta), which is the adapted formula above. Decode q into a physical density by treating q as a piecewise-linear monotone map from reference quantile p to spatial location x: for each reference interval [p_k,p_{k+1}], distribute its mass (p_{k+1}-p_k) across the physical grid cells intersected by [q_k,q_{k+1}], or use a differentiable triangular-kernel push-forward. Train with CDT-space MSE plus physical reconstruction L1 and the explicit mass penalty; monitor the fraction of negative reconstructed cells and total-mass error, which should be zero or near machine precision by construction. For a stronger reduced model, compute an SVD of training q vectors and let E initially be the first d POD coefficients, then fine-tune the encoder and decoder. The first cheap experiment should use advected Gaussian mixtures and square pulses on a 256-point grid, comparing this decoder with a standard convolutional autoencoder at d=2,4,8. Evaluate reconstruction Wasserstein error, physical L1 error, negative-mass violations, and storage. A successful result is substantially lower error at d=2 or d=4 and elimination of decoder-induced negativity; for a translating single pulse, d=2 should be sufficient because the transformed trajectory is affine in the translation.