arXiv Machine Learning
Sep 1

Sensitivity-Constrained Neural Operators for Data-Efficient Forward and Inverse Modeling of Partial Differential Equation Systems

The paper introduces Sensitivity‑Constrained Neural Operators (SC‑NOs), which augment standard neural operator training with sampled Jacobian supervision from differentiable solvers or discrete adjoints. By matching selected sensitivities during training, SC‑NOs improve forward prediction accuracy and significantly enhance gradient‑based inverse reconstruction for distributed fields. Experiments on advection–diffusion, RANS–Spalart–Allmaras, high‑dimensional gridded inputs, and a shallow‑water tsunami source‑inversion case demonstrate that SC‑NOs achieve a better accuracy–cost trade‑off and enable near‑real‑time wave‑propagation forecasting from sparse observations.

By Abdolmehdi Behroozi, Chaopeng Shen, Daniel Kifer, Kathryn Lawson
arXiv Statistics ML
Sep 17

A Continuous-Time Ensemble Kalman-Bucy Smoother for Causal Inference and Model Discovery

The paper presents an ensemble Kalman–Bucy smoother (EnKBS) for continuous‑time data assimilation of nonlinear dynamical systems, reconstructing conditional distributions from ensemble moments without needing tangent‑linear or adjoint models. It demonstrates that EnKBS achieves exact smoothing mean and covariance in the infinite‑ensemble limit for linear‑Gaussian systems and incorporates regularization techniques like covariance localization and inflation for high‑dimensional problems. The method is applied to Bayesian inference of causal relationships in a dyadic trigger‑feedback model and to an iterative learning algorithm that uncovers the structure and hidden parameters of a reduced‑order model of midlatitude atmospheric circulation, all with small ensembles under partial observations.

By Zhang Jiang (University of Wisconsin-Madison), Marios Andreou (University of Wisconsin-Madison), Sebastian Reich (University of Potsdam), Nan Chen (University of Wisconsin-Madison)