Polynomial Chaos Expansion Based Nonlinear Filtering of Stochastic Processes

What the math gives to ML

The paper's transferable asset is a coefficient-space uncertainty propagation method: instead of sampling many trajectories or linearizing a nonlinear stochastic transition, it represents the state as an orthogonal polynomial chaos expansion and projects the nonlinear SDE update onto each basis function. The resulting coefficient recursion preserves non-Gaussian uncertainty induced by nonlinear dynamics while requiring only a fixed number of coefficient tensors. Its LMMSE measurement update can be inserted into a neural world model or recurrent state estimator as a differentiable uncertainty module, replacing an EKF or particle ensemble. The most promising first test is a low-order PCE latent filter for nonlinear sequence prediction, measuring whether it improves calibration and stability at lower rollout cost than particle filtering.