Model-space learning of stochastic seawater intrusion using conditional normalizing flows

Quantifying uncertainty in seawater intrusion in heterogeneous coastal aquifers is computationally demanding, as it involves the repeated solution of the coupled nonlinear variable-density flow and transport equations over many realizations of the high-dimensional parameter space. We propose a grid-free surrogate modeling framework that combines model-space learning with conditional normalizing flows. The hydraulic head and solute concentration fields are projected onto a finite-dimensional model space spanned by Chebyshev or Fourier bases, or represented by Kolmogorov–Arnold Networks whose activation functions are parameterized by the same bases, so that each random solution is represented by a low-dimensional coefficient vector. A conditional normalizing flow with parameter-dependent Gaussian base distribution then learns the conditional probability distribution of the coefficients given the statistics of the heterogeneous conductivity field and the boundary conditions. We derive an error estimate that decomposes the surrogate error into a model-space truncation term and a statistical term controlled by the Wasserstein distance between the true and learned coefficient distributions. The framework is demonstrated on steady-state seawater intrusion in confined and unconfined aquifers. The surrogate accurately captures the non-Gaussian features of the coefficient distributions–skewness in the confined case and pronounced multimodality in the unconfined case. Although its accuracy decreases for parameter configurations at the boundary of the sampled domain, a one-time offline training phase enables rapid sampling of solution statistics.