Monotone transport-map latent space

Integrate this at the input and output interface of a one-dimensional conservative neural surrogate, such as an SSM, temporal Transformer, or latent MLP predicting u_{t+1} from u_t. First normalize every nonnegative state to equal mass: u\leftarrow Mu/(\sum_i u_i\Delta x), where M is the desired mass, usually 1. Choose a fixed reference density r on the same grid, compute its CDF F_r, and choose K reference probabilities p_k=(k-0.5)/K. For every snapshot u, compute the discrete CDF C_i=\sum_{m\le i}u_m\Delta x and obtain the quantile samples q_k=\inf\{x_i:C_i\ge p_k\}; use linear interpolation between grid points to reduce quantization error. Feed q=(q_1,\ldots,q_K) or a low-rank projection of q into the dynamics network. A practical model predicts residual transport-map changes, q_{t+1}=q_t+\Delta t\,f_ heta(q_t,a_t), rather than predicting q_{t+1} directly. Decode by monotone interpolation of q(p) and push forward the reference measure: estimate u on cells from the inverse map, or use the differentiable histogram formula u(x_i)\approx r(x_j)\Delta x/\max(\Delta q_j,\epsilon) for intervals where q maps reference cell j to target cells. During training use both transformed-space loss L_q=K^{-1}\sum_k|q_{t+1,k}-\widehat q_{t+1,k}|^2 and physical mass/nonnegativity checks. The CDT and quantile definitions are fixed mathematically; the reference density, K, interpolation, and model architecture are engineering choices. First test on a translating Gaussian or square pulse and then 1D Burgers before shock formation, using a small GRU or SSM. Compare against the same model on Eulerian grid values at equal parameter count and latent dimension. Success is lower rollout error and faster loss descent at the same latent width; for pure translation, a two-mode linear latent baseline should nearly match the CDT model while Eulerian POD should require many modes.